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seminars - Critical norm blow-up for the energy supercritical nonlinear heat equation
We consider the critical norm blow-up problem for the nonlinear heat equation with power type nonlinearity |u|^{p-1}u in R^n.
In the Sobolev supercritical range p>(n+2)/(n−2), we show that if the maximal existence time T is finite, the scaling critical L^q norm of the solution becomes infinite at t=T. The range of p is
optimal in view of known examples of blow-up solutions with the bounded critical norm for the Sobolev critical case. This is a joint work with Jin Takahashi (Tokyo institute of technology).
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Hello Quantum
So the previous blogs gave an introduction to what quantum computing is and further went on to give a basic idea on two single qubit gates, Hadamard and Pauli-X aka the bit flip, There are many
companies currently researching and investing in quantum computing, big names like IBM, Microsoft, Google, and startups like Rigetti and Xanadu. Quantum Information Science Kit, Qiskit is IBM’s
quantum computing framework to create and manipulate quantum programs.
Over the past few blog posts I posted the physics and math behind operations on qubits, this blog will focus on writing some code, to be specific we’ll create a Bell State, so what is a bell state?
A bell state is an entangled state of two qubits
A bell state is achieved using a CNOT gate and a Hadamard gate, a Hadamard gate puts a qubit into a state of superposition, the CNOT gate, that is the controlled NOT is a gate which acts on two
qubits, one qubit acts as the control bit while the other is the target qubit. Now on applying the CNOT gate to a 2 qubit register, if the control bit is |1> the target bit flips that is if the
target bit is |0> it becomes a |1> and if it is |1> it becomes a |0> and if the control bit is |0> the target bit is unchanged.
A bell state is achieved using a CNOT gate and a Hadamard gate, a Hadamard gate puts a qubit into a state of superposition, the CNOT gate, that is the controlled NOT is a gate which acts on two
qubits, one qubit acts as the control bit while the other is the target qubit. Now on applying the CNOT gate to a 2 qubit register, if the control bit is |1> the target bit flips that is if the
target bit is |0> it becomes a |1> and if it is |1> it becomes a |0> and if the control bit is |0> the target bit is unchanged.
Now with that settled let’s write some code. I’ll be using Qiskit for coding, you can find more details on Qiskit and how to install it at qiskit.org
Now let’s code
The code below should be executed in a Jupyter Notebook
from qiskit import QuantumCircuit,QuantumRegister
from qiskit import ClassicalRegister,Aer,execute
%matplotlib inline
from qiskit.tools.visualization import plot_histogram
q = QuantumRegister(2)
c = ClassicalRegister(2)
circ = QuantumCircuit(q,c)
backend = Aer.get_backend('qasm_simulator')
job = execute(circ,backend=backend,shots=1000)
result = job.result()
Now what exactly happened here?
First we import QuantumCircuit, QuantumRegister, ClassicalRegister,Aer and execute from qisit
Aer is a high performance simulator for quantum circuits , next we create a 2 qubit quantum register and a 2 bit classical register, then we compose a circuit
Each qubit is accesed via indexing q[0] denotes the 0thqubit and so on. In the next step we put the qubit at 0 to a state of superposition by applying the Hadamard gate, so that the qubit has a 50%
chance of collapsing into 1 and a 50% to collapse into a 0 while measuring. We also apply a CNOT with qubit0 as control and qubit1 as target ( in the circuit diagam q30 and q31 respectively). Now we
measure the qubits. Here an interesting thing happens, if qubit0 on measurement yields 0 qubit1 will also be 0 and if qubit0 yields 1 qubit1 will also yield 1. This is what we get in the histogram
below, we do this a 1000 times and we get a 46.6% chance of the classical register being in 00 and a 53.4% chance that it is in 11. Weird isn’t it, somehow our two qubits have managed to communicate
with each other so that when we measure qubit0 to be 0 qubit1 automatically becomes a 0 and if qubit0 is 1 qubit1 is a 1. This is an entangled state.
Due to the superposition, measurement of the qubit0 will collapse it into one of its basis states with a given probability.Because of the entanglement, measurement of one qubit will assign one of two
possible values to the other qubit instantly.Bell states can be generalized to represent specific quantum states of multi-qubit systems, such as the GHZ state for 3 subsystems. Understanding of the
Bell states is essential in analysis of quantum communication
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Eric Weinstein: A Conversation (YouTube Content)
Eric Weinstein: A Conversation was a livestream of Into the Impossible hosted by Brian Keating, dicussing Geometric Unity with guest Eric Weinstein.
#GeometricUnityâ #EricWeinsteinâ #ThePortalâ Eric Weinstein, host of the Portal Podcast, reveals Geometric Unity, his provocative new Theory of Everything. First discussed in 2013, later
explored on the Joe Rogan Experience and Lex Fridmanâ s podcast, I am delighted Eric revealed the published version FIRST on The INTO THE IMPOSSIBLE Podcast.
See this Collection of Videos in Support of Geometric Unity https://pullthatupjamie.com
Watch Weinsteinâ s â April Foolâ sâ 2020 episode of The Portal, where he explains his theory of Geometric Unity: https://youtu.be/Z7rd04KzLcgâ
Watch Weinsteinâ s latest interview on The Joe Rogan Experience.
The Portal Group's Transcript Completion Project generates and edits transcripts for content related to Eric Weinstein and The Portal Podcast. Completed transcripts are available to read on The
Portal Blog and The Portal Wiki. If you would like to contribute, you can make direct edits to the wiki, or you can contact Aardvark or Brooke on The Portal Group Discord Server.
This transcript was generated with Otter.ai by Aardvark#5610 from this content's YouTube version. It was edited by Aardvark#5610. Further corrections and contributions were provided by pyrope#5830.
Brian Keating: [Inaudible] And we are pleased to bring you a longtime friend of the campus, a friend of this cosmologist, a friend of physics, and that is none other than Dr. Eric Weinstein, who
today is joining us from an undisclosed location, but maybe we'll get into that. We have already 114 people watching with many thumbs up. One thumbs down from my mother, Mom, how could you do that?
That's wrong, Mom. Don't do that. It's just to me she said, not to you. Eric, how are you doing today?
Eric Weinstein: I'm well Brian, good to be with you.
Brian Keating: It's great to be with you. It's been four months exactly, or three months exactly, since we last conversed via this medium when we had on our mutual friends Max Tegmark and Garrett
Lisi. And that was of course very enjoyable for me, to go over this, go over some of the long standing questions I've been having in this exploration of the multiverse of brilliant minds that grace
me with their presence on the Into the Impossible podcast.
I have been not a stranger to the work that you've been working on. Some say it is the work of a lifetime. Some say it is revolutionary, and could have tremendous implications. Some have questions
about it, because of its far reaching implications. And we're talking about universal theories of everything, perhaps a new one created by today's guest, Eric Weinstein. And that goes by the moniker
Geometric Unity (GU), and I've been fascinated with this ever since I heard about it probably ten years ago, almost ten years ago now. And today, I thought it would be fun to get Eric on the show, as
he has promised to at least be interested in coming on to discuss recent developments that the listenership of this fine podcast would be interested to know. And as you know Eric, we go deep.
Should a Theory of Physics Say Anything about God?[edit]
So first of all, I want to say thank you, and I want to ask you what is new in the theories of everything space? In particular, we're hearing a lot of talk nowadays from people like Michio Kaku, who
will be a guest on my podcast next week, about "the God equation". And my first question to you, which is always of interest to me personally, is why does a theory of physics have anything to say
about God, or any relevance to God whatsoever? Before we get into the nitty gritty details.
Eric Weinstein: Well, there are two things that I think that are up, which areâ one is man's rigorous attempt to understand his circumstance necessarily intersects you with God, which is a
traditional explanation for why is everything here, and the other is that God sells. Part of the problem is that if you name something "the really important particle that we just discovered", that's
not going to sell as many books as "the God particle". And so we want to know God's thoughts, we want God particles, and then we back away from them. We claim, "No no no, I didn't mean God particle,
I meant god damn particle." This is a game that we play with the public, where we try to amp the public up and get them hot and bothered, and once they're sufficiently in a lather, we try to educate
them about the real nature of the universe. So think about it from a computer perspective as syntactic sugar. We're pouring God all over something hoping that people will swallow it.
Brian Keating: And of course, I'm holding up on the screen right nowâ on my screen I'm sharing a highlighted section from that great work of literature known as A Brief History of Time. This was one
of the books that got me interested in cosmology and astronomy by the late great Stephen Hawking, who passed away exactly three years ago on Einstein's birthday, on Pi Day, at least here in the
United States the 14th of March. By the way, do you know what other famous figure he shares his demise date with Eric?
Eric Weinstein: I don't.
Brian Keating: A Jewish intellectual by the name of Karl Marx. So Karl had an impact on universal capitalism, and Einstein, of course, was born that day, and Hawking died that day. And in the final
paragraph of the book, he says if we can discover, if we can all have part in a final theory, "Then we shall all, philosophers, scientists, and just ordinary people, be able to take part in the
discussion of the question of why is it that we and the universe exist. If we find the answer to that, it would be the ultimate triumph of human reasonâ for then we would know the mind of God." And,
as you said, these things sell. It was rumored that he said every equation cuts your audience/readership in half, every mention of God doubles it. So at some level, there's conservation here, but I
was always taughtâ
Eric Weinstein: [Inaudible]
Brian Keating: Sorry?
Eric Weinstein: Go ahead.
Brian Keating: I was always taught physics is not for "Why" questions, and yet there it is. He's bringing up "Why" questions. What do you make of that? Can physics provide the "Why"?
Eric Weinstein: I feel like, when you are talking in these terms, you are reasonably confident that the person is not trying to read the mind of God, because one wouldn't trifle with God in such a
way. I believe that in some sense, if you're really worried that the telephone is connected, you'd probably speak about this differently. You might be humorous, you mightâ I don't know. There's
something about the fact that we talk in this way, and it feels to me like when Moses is seeing a bush that is not consumed by flame but appears to be in flame, he knows pretty well that you should
be a little careful. And I just, I don't understand the impulse to constantly God-ify everything.
Experiment and the Scientific Method[edit]
Brian Keating: And certainly, I'll have on Michio Kaku next week on the Into the Impossible podcast, and I hope maybe we'll get a cameo from you. But in his book, he writes something very
provocative. And he says at the end of his book, he quotes those lines from Stephen Hawking, which is kind of like this infinite regress, which kind of strains credulity, so to speak. But he says, at
one point he says, "It's not fair to test string theory, to ask to test string theory experimentally, because we don't know its final principles." But the same, I claim, could have been said about
quantum mechanics. Do we know the final principles of quantum mechanics? Does that immunize it from experimental test?
Eric Weinstein: Again, these are the same questions over and over again. There's something very wrong about the simplistic nature of the scientific method, and the relationship between experiment and
theory, and instance and idea. We're effectively playing through the exact same set of problems: where we hold up one theory to some sort of experimental threshold, we give a pass to another theory.
We're all the time pretending that we're not actually doing what we're really doing, which is observing who believes in what theory.
One of the reasons string theory got such a boost is that the brilliance of the initial volunteers for the first string revolution around 1984 were so good that we were inclined to give them a huge
pass, at least at first. And then, we have this differential application where the string theorists become paradoxically the most persnickety about what is a prediction, because they don't want to
give up the fact that they aren't really making predictions. So if you, for example, predict internal quantum numbers of the next particles to be found, but you don't come up with an energy
threshold, and you don't say what will invalidate your theory, they get angry. Because, in fact, what we've done is we've given them an asymmetric relationship with the scientific method through
special pleading.
So we have a really unfortunate situation, which is that we have highly simplistic Popperians, highly simplistic devotees of the scientific methodâ and I really think that people need to go back to
Dirac's 1963 Scientific American article to understand that the real issue is very weird, and we haven't really talked about it. There were three big names in the 20th century in my mind, who
contributed something like physical law. And leaving Dr. Mills out of it for the moment, I would say that Einstein, Dirac, and Yang tower, not necessarily that they're the best physicists, although I
think I could make a pretty good claim in all three cases, but that physical law is different than the consequences of physical law. And the people who seem to do well with physical law employ
mechanisms that would drive Sabine Hossenfelder to distraction. They talk a lot about beauty, and elegance, and simplicity. And what Dirac said was don't force people who come up with new physical
laws to play the game of agreement with experiment, because the instance of an idea can easily be off and not agree with experiment.
And then you have a problem whereby you're pushing people initially. The instant you open your mouth, "Say what it is that would invalidate your theory so we'll know that you're wrong if you're
wrong." And I don't know who this is intended to fool. It's completely irresponsible. And what it is, is an attempt to constantly take anyone who would come forward with an idea and put them
instantaneously on the defensive. I think that the right thing to do is to sit people down and say you're supposed to be adults. If we look at our history, everybody who has proposed new physical law
and gotten it right had errors. Einstein didn't get the divergence free part. He was vague before that with Grossman. Famously, Dirac's theory of quantum electrodynamics took almost 20 years before
the renormalization revolution supplied the ability to compute with it. We had a confusion between the bare and the dressed mass. And famously, the degeneracy between the electron and the proton: we
had two particles that Dirac claimed to be anti-particles, because he was too timid to suggest a positron and an antiproton, which Heisenberg [inaudible] has the mass asymmetry. Yang's theory, if
left massless, wouldn't come up with the right rates for beta decay if you didn't impart mass to the W and Z, to the intermediate vector bosons.
So I think that you have a situation by which new ideas are always not properly instantiated, and the community that is constantly trying to make sure that... I think that the idea is that people are
foolish enough to play this game with the most aggressive members of the community, because the implication is if you won't come up with a testable prediction that invalidates your theory, you're
anti scientific and we have no time for this. And so people, well like, you know, with the [math]\displaystyle{ \text{SU}(5) }[/math] theory, they immediately said okay, well it predicts proton
decay. Well, grand unification is a larger idea, and some versions and instantiations do predict proton decay, and some do not. So what are you going to say about that? I think that the problem is
that we're not in an adult phase where we've faced up to the fact that we have almost 50 years of stagnation, and what you're seeing with this proliferation of new claimants to have fundamental
theories is, in part, that string theory has finally weakened itself, and the aging of the particular cohortâ which is Baby Boomers, who are the string theory proponentsâ they've gotten weak enough
that effectively other people feel emboldened. And I think Stephen Wolfram said this recently, that in a previous era, he would have expected to have been attacked. But we've been waiting around for
so long that perhaps the political economy of unification and wild ideas has changed somewhat.
Approaches to a Theory of Everything[edit]
Brian Keating: And before we get off the subject of the "Why" questions, I do like a framework that I've heard you, and almost no one besides you, portray laws of fundamental physics, and that's
using the good old fashioned mechanism we were all taught in high school journalism: the "Five W" approach. And I wonder if we could start there, with why that is a good deconstructivist approach to
ascertaining the realm of validity of a physical law, of a purported new theory, a theory of everythingâ which I dislike that moniker as you know. But nevertheless, can you talk about that framework
and how, for our up and coming but bright listeners, of which there are many currently watching right now, how you approach that using the "Five W's", why that's so important, and then maybe that
will segue into a description of the actual physical instantiation of that framework.
Eric Weinstein: I will point out that how has the "W" on the end. Yeah, I think that... I usually do it as "Where" and "When", "Who" and "What", "How" and "Why". And let's just say, first of all,
what we generally speaking mean by a theory. What we're usually talking about is a way in which waves can propagate and interact in various media. All right? The theories of the world are theories of
waves and interaction. Waves imply a medium. So the "Where and When" is sort of a particular kind of a substrate, usually, which Einstein imbued with the name spacetime, "Where" being space, "When"
being time. The "Who" and the "What" I take to be fractional spin and integral spin particles; every particle that we know of that's fundamental is one or the other. So, let's say that the "What" is
the fermionic, fractional spin particles, and the "Who" is the integral spin, generally speaking force particles, non-gravitationalâ but then we also have to throw in the Higgs and the metric for
spin-0 and spin-2. And then there's the "How" and the "Why", the "How" would be the equations of motion, and the "Why" would be the Lagrangian that generates the equations of motion.
And so, in some sense, it's not surprising then that the theory has to conform to the basic idea of when you're trying to tell somebody something, these are the questions that we want to ask, and
it's a surprisingly tight mapping. And I just find that people can better remember that, because very often what we've done is we've taught people to focus on the wrong things when we talk about
fundamental physics. They're overly focused on entanglement. They're very focused on quantum measurement. They have no idea about bundles, they don't have ideas about symmetry groups or why symmetry
groups are important. And so, for some reason, when people learn about theories of everything, they're very animated, but they're very animated as to on the grounds of what has sold books recently.
Limits and Methods for Constructing the Universe[edit]
Brian Keating: That's right. And we have no shortage of multiverses, double-slit experiments, spooky action at a distance, and other invocations of this gentleman [Albert Einstein]. I point out that
Einstein is Weinstein with[out] a W. Okay, you have a fascination with W's, obviously. So, I want to go starting with Einstein, to something that I know is very influential to you. And it's sort of a
provocative question that has inspired you, apparently. And that was a question, a stylized question, posed to Ernst Strauss by Albert Einstein, regarding the amount of freedom present in our field
theoretic universe. What is that question?
Eric Weinstein: Well, the question is how much freedom is there in what we take to be the Standard Modelâ and I'm sorry, I'm using a term of art accidentally. How much freedom is there to construct
the universe? And is this one of many that could have been constructed, or is it effectively unique? And are we talking about the God concept, if you will, as a design constraint, where things are
the way they are because they could not be otherwise? And I think that it's a very interesting question, because in some sense, I don't know that he meant it this way, but I took it to be a research
Brian Keating: And in terms of it providing this direction for you, is the question itself the research direction? Or is the overarching theme, of sort of freedom, flexibility within physical laws,
the programmatic kind of marching orders that you took unto yourself?
Eric Weinstein: It's an interesting question. I mean, I think that what I don't understand is that people talk about theories of everything casually, as if a theory of everything is sort ofâ it may
not be a very artful term. It's sort of theories of all the rules, not what can be played with once one knows all the rules. I guess what I take it to mean is that we have a problem of even
conceiving of what a non-effective theory would be. Well, what is an ultimate theory? I mean, I think that in large measure I see two kind of canonical versions. One of them I would sort of associate
with Garrett Lisi's E8 idea, although I don't believe that that works. You start with something incredibly rich that exists by necessity, like a large exceptional Lie group, or maybe a large finite
group, or something that is somehow distinguished. And then you attempt to milk it for peculiarities that can be identified with our world, and that's how you get the richness of our world. Whether
or not you believe in Garrett's theory, I do think it's emblematic of an approach.
Another approach is closer to embryology, where you start with something that is deceptively simple, like a single fertilized egg. Then you ask, does that attempt, in some sense, to bootstrap itself
into the totality of existence? And that's much closer to what I ended up doing. I mean, I considered Garrett's E8 thing before I ever met Garrett, because E8 is spinorial, it's chiral, it has lots
of stylized things that seem to fit our world, but I couldn't figure out how to really make it into a theory, and then I went the other direction. I think it's pointless to ask why is there something
rather than nothing, unless I'm mistaken. I think that the point of a fundamental theory is to get the scientists to accept the initial input is so uninteresting to go beyond that they put down their
pens, and the theologians and philosophers take over.
If you imagine that the initial input to the universe is just 4 dimensions, for example, I don't think that many scientists would be motivated to say "Why are there 4 dimensions?" at a scientific
level, because that sort of begs theâ it's not enough of a clue for anything to proceed scientifically. I mean, maybe all versions have multiple dimensions, maybe there's 17 dimensions too. So I
think that in large measure, the gambit that I'm trying to follow, as misguided as it sounds, is, "Is four dimensions on its own, in the form of a manifold with a few extra mild conditions," like a
single unique spin structure, something like that... Orientable. "Is a nice 4-dimensional manifold sufficient to start the universe from effectively no other major assumptions?" And that's how crazy
this is.
Brian Keating: So when you say "this", we're talking about Geometric Unity. A reminder, we're talking to Eric Weinstein, Dr. Eric Weinstein, proprietor of The Portal Podcast. And you can find his
YouTube channel at nobani88, which is a cryptic reference to the year I had my first kiss. I don't know why it's called that, but it should be The Portal, we'll get that fixed. Eric, if in the
meantime, could you tilt your webcam down just a tiny bit, so your head is not at the bottom of the of the frame? That would make itâ Yes, very good. Very nice.
What is Fundamental?[edit]
So, what is fundamental? I've had these conversations just recently on my podcast with Dr. Stephen Meyer, who you know is a proponent of the intelligent design hypothesis, I'm not going to get into
that. I am a critic of that, and then we are yet good friends. But he makes the case that in things like the Guth Vilenkin conjecture, or in the Lawrence Krauss universe from nothing, we always start
with the laws of nature and instantiation thereof. So too with debates I've had with Sean Carroll, a friend of mine, and a greatly respected mentor in the field. That God could have chosen to start
the universe with an empty Hilbert space is his conjecture, and therefore, there's a simpler universe than the one we inhabit. We're not going to talk about Sean necessarily, we're not going to talk
about Stephen Meyer. But I want to talk about what is the fundamental element, the yealm, the thing from which emerges spacetime? Or is the spacetime, or observerse if we can go there now, is that
truly fundamental, or is it emergent? What comes first, the observerse or the observer?
Eric Weinstein: Well first of all, I mean, let me just say a few words. What we're talking about is crazy. And I think it's really important to just own up to the fact that for people who want sober
physics, this is probably not the channel for you today. Now... No, I mean, I take this stuff very seriously because I don't like the bullshitty aspect. And we're using April 1st as a contrivance,
because I think that many people are induced to self-inhibit, because particular members of the community are incredibly aggressive in making it extremely expensive to explore ideas. And I'd like to
think that living outside the community, I could start a tradition to make it at least inexpensive one day a year to throw the middle finger to those people who like to play Simon Says games, or
reputational destruction games. Nowâ
Brian Keating: A purge. A purge for physics.
Eric Weinstein: Well, there should be many more such days, and I'd love to get there. But let's at least start with one a year. So, this is my second year round trying to hit this. Look, I believe,
that at some level, that the initial ingredient may just be a 4-dimensional manifold. And then things emerge from that. A 4-dimensional manifold with a little bit of extra structureâ but that's why
this is crazy.
Brian Keating: So it starts from very modest inputs, and from such modest inputs comes a rather extravagant universe. Let's talk about the inputs. I don't know how closely you want to follow, if you
want to share screens or anything like that, we're free to do that. What are the inputs? There are the players, the matter players, there are the gauge bosons, there are new predictions, there are
new concepts that Geometric Unity has provided. And so the question, I guess first of all, is how close do we want to follow this prescription of what has been portrayed in the past? And/or do we
want to talk about what is new in the preceding year since the last episode of April Fool's purge podcasting began with The Portal special episode?
Eric Weinstein: Well, it's very interesting to consider that we've had a year where there's been a fair amount of interest in it. And, let's be honest, very little of the interest has been
particularly detailed. I would have thought that maybe what I said was ununderstandable. And then, oddly, a paper purporting to critique the theory managed to demonstrate that they had understood
fairly well what I had said, and that it was understandable. Unfortunately, there was one named author and a [sic] imaginary friend, and I don't respond well to people posing behind pseudonyms. But
part of that was constructive. And, you knowâ I'm attempting to share a bit of screen now.
Brian Keating: Okay. Here, I'll add that. Okay. So now it's full, you've got the full screen on the paper.
Forgotten Problems in Physics[edit]
Eric Weinstein: So effectively, what I'm asking is, can a manifold [math]\displaystyle{ X^4 }[/math] produce the baroque structure of the Standard Model? Nowâ and gravity. And if you think back to
the famous mug popular in the CERN gift shop, there really isn't that much going on in the Standard Model if you group terms in particular ways. But there's a lot of weirdness. Why the Lorentz group,
why [math]\displaystyle{ \text{SU}(3) \times \text{SU}(2) \times \text{U}(1) }[/math] for the internal symmetries generating the forces, why three families? I thought that something that many younger
viewers may not be aware of is that things really changed around 1983, '84. If you think about the original anomaly cancellation of Green and Schwarz in 1984, I believe, you could ask what was
physics like right before that moment? And I think it's absolutely shocking, because we don't realize the extent to which the string theorists really redefined what the major problems in physics
were. I think most people in the post-string era somehow believe that the major issue is quantum gravity. And I don't really, I just find it astounding, because that's really what the string
theorists were selling.
So this is from Murray Gell-Mann's address to the second Shelter Island conference, where they're trying to recapture the magic from Ram's Head Inn after World War II, when the young physicists were
invited toâ feeling that they had done well on the engineering project that was the Manhattan Project, they were buoyed in their confidence. And years later in 1983, Murray Gell-Mann says well, what
are the big problems? "As usual, solving the problems of one era has shown up the critical questions for the next. The first ones that come to mind looking at the standard theory of today are," and
then, I think this is absolutely shocking and indicates the extent to which the current generation has really given up on doing what we would typically have called physics, relegating the things that
are relevant to the physical universe that we see usually to the realm of particle phenomenology.
Okay, so what are his big questions? Why this particular structure for the families? In particular, why flavor chiral with left- and right-handed particles being treated differently by the weak
force, rather than say vectorlike ones left and right transformable into being treated the same? Next, why three families? That generalizes Robbie's famous question "Who ordered that?" as if the
universe was a Jewish deli, commenting on the muon. How many sets of Higgs bosons are there? We talk about the Higgs boson, but maybe there are multiple sets and there are multiple different scales
at which symmetry is broken and mass is imparted through soft mass mechanisms. Lastly, why [math]\displaystyle{ \text{SU}(3) \times \text{SU}(2) \times \text{U}(1) }[/math]? Remember, [math]\
displaystyle{ \text{SU}(3) }[/math] is the color force for the strong force, but [math]\displaystyle{ \text{SU}(2) }[/math] here is weak isospin, which has not yet become the W and Z's. And this
[math]\displaystyle{ \text{U}(1) }[/math] is weak hypercharge, which has not yet become electromagnetism through symmetry breaking. And in some sense, I just feel sort of sad that we don't think of
these as questions because we know not to ask them.
And somehow we got convinced that we were being called to quantize gravity, not necessarilyâ if gravity is geometric, you could just as easily have said should we be geometrizing the quantum? And if
we geometrize the quantum, you would notice that this era would have been triumphant, because that's really what happened. We didn't do a lot of physics, but we really did put the framework of
physicsâ that is quantum field theory, quantum measurement, classical field theoryâ all in very geometric frameworks. In fact, I would say that there were three really big revolutions, although we
don't talk in these terms. One was the discovery by Simons and Yang of the Wuâ Yang dictionary, I'm blanking on Wu's name, which Is Singer was also instrumental in taking to Oxford. Then there's the
geometric quantization revolution, where the quantum was understood to be intrinsically geometric because the Heisenberg uncertainty relations should emerge from the curvature tensor of a prequantum
line bundle, but the sections being the states of a vector space once polarization is taken into account. And then lastly, the geometric quantum field theory revolution, in which we came to
understand the quantum field theory really isn't about the physical world, that it gets applied in one particular set of inputs to the physical world, but it's actually a mature mathematical
enhancement of bordism theory from topology, strangely. So, those three major revolutions all went exactly counter to quantized gravity. They said, "Let's geometrize the quantum instead," and so they
Brian Keating: And didâ how successful should we regard this? The resulting byproduct or lack thereof progress, lack thereof in the intervening...?
Eric Weinstein: This is very unpleasant to have to say this, but I think that we are talking about a great era with heroes. The top hero among them is undoubtedly Ed Witten. But I do believe that
Yang and Simons, I think Yang and Simon's discovery of Ehresmannian bundle theory, which has a precursorâ and I'm blanking on the gentleman's name (Robert Hermann), all the self published books from
from the '60s. It'll come to me, but there was a man in Boston who probably got there a little bit earlier. And then I would say that you have accidental physicists. Dan Quillen, for example, did a
huge amount to talk about connections on determinant bundles and the like, which come out of various quantization procedures, particularly with Berezin integration of fermion sectors. So I think that
a lot of things got done to shore up what we do to mature input into a quantum theory. It just, it wasn't physics, per se. It was sort of the mathematics of physics. And I think that that was very
frustrating, which is, you know, it's sort of, to physicists it's yeoman's work. They wanted to go to Stockholm, and they ended up winning the first Fields Medal won by a physicist, and I thinkâ
it's weird. It's like, what is your time? Your time is whatever it is that can be done. And they thought their time was to quantize gravity. "Well guess again," nature said, "we have something
incredibly important." So I feel like I'm trying to rescue their legacy. They want to go down as string theorists for the most part. And they want to say that string theory was the most successful of
any claimant, even though it wasn't very successful. And, my feeling isâ
Brian Keating: Now, can you say it's notâ Go ahead.
Eric Weinstein: Well, yes, I feel like we can say that it's not very successful, because they gave us the terms in which we should evaluate it. You know, I remember being told "Give us 10 years,
we'll have the whole thing cleaned up. Don't worry your pretty little head, we'll be fine," or, "We have a finite number of theories to check." And then lo and behold, there's a continuum, or why is
it called string theory when there are branes involved? And it was because if you asked once upon a time, they'd say, "Well, it's not like mathematicians think about higher-dimensional objects beyond
strings." There was an explanation for why there were no branes. And, you know, thatâ yes, string theory has failed in its own terms. Now is it salvageable, are there pivots beyond? Yeah, sure. I'm
not saying that they didn't stumble on a tremendous amount of structure, maybe that structure ultimately carries the day. But I do think that the idea that they're entitled to this many pivots
without having to become self-reflective is preposterous. And I think many people feel that way, and they know that they might pay for such a statement with their career. And since I've prepaid, it
falls to people like me and to you, perhaps, to say look, the string theorists weren't able to confront their failure.
The Grand Nature of Physics[edit]
Brian Keating: When we talk about these things in rather, some say, grandiose terms, I think sometimes we do lose sight.
Eric Weinstein: [Inaudible] I really don't want to use the word grandiose. Like, are we going to talk about grandiose unified theory? Let's be honest about it. Physics is the most honest way to ask
the most grand questions in the universe.
Brian Keating: Absolutely.
Eric Weinstein: If physics is grandiose, then we've got real problems. Then grand doesn't exist. And if grand doesn't exist, then grandiose doesn't exist. So, my feeling is no. This is the actual
grand quest, and we're not going to back off it and be pussies about it. This is not grandiose, this is the real deal.
Brian Keating: I was thinking, speaking of myself, being a self-aggrandizement of seeking these ultimate questions, but we do, and I was gonna give physics a good deal of credit, because we do ask
these ultimate questions. And yet, of course, day to day basis, I remember wanting to help you out as little [a] role I could play in the exposition of this magnificent opus that you're working on,
and saying, "Eric, this is great, but I got a bunch of kids that I gotta go pick up." And you said "Well, maybe that's why we'll never get off the planet, because guys and girls"â
Eric Weinstein: Everybody has to pick up their dry cleaning.
Brian Keating: Every time you gotta pick up your dry cleaningâ but when we lose sight of it, I find with my colleagues, and I'll speak, because I doubt many of them are listening. I really don't
feel like they're that curious, intellectually. I think it is a job. I think their their job is the dry cleaning. And I can sort of prove that in some ways, because I often hear them say things like
well, Eric is a showman, he's a podcaster. He's a host, and he's had training, and he's very smooth, and he can speak well. And I say "Well, do you think he do you think he emerged from the womb like
that? And by the way, Mister or Missus Professor, Doctor Professor, you have got a lot of training in quantum field theory and string theory yourself. That was presumably a challenge for you. You
didn't emerge womb-like, you know, from the caverns of the womb, knowing quantum field theory, so you had to work at that." So it's all about prioritization. Why do you think physicists aren't more
troubled by the lack of progress, that our mutual friend Sabine has pointed out, in the last 50 years, at least in fundamental physics? My colleagues will rightfully point out tremendous advances in
cosmological theory, in condensed matter theory, etc. But why isn't that more troubling? I think the answer is we're not that curious. You have a vision of us that's maybe more more refined than I
think we deserve, and that's because you're not a professional physicist.
Eric Weinstein: Look, I feel very similar about my feelings about physics as an outsider to the way I view the UK. When I go to the UK, very often they they seem to be defeated, because they lost
their empire which they should never have had in the first place. But my feeling is if you really look at the UK, it's an amazing place, and any outsider should be able to see that. I guess what I
think about here is that any outsider who really takes physics seriously should be able to see that this is our premier community, intellectually. It is the most accomplished of intellectual
communities. And it's also very badly behaved, and it's fallen on hard times.
It's like seeing a grand family that's forgotten itself, because it has to constantly submit to the arXiv. We now have the snarXiv, as you know. The snarXiv is filled with papers that are
indistinguishable as a Turing test from arXiv papers. I think I looked for like, I don't know, the Gell-Mannâ Nishijima formula on hep-th, and I realized that people really weren't doing physics.
You know, there's certain things that you would have to do if you were going to do physics. I don't mean to say that no physics is going on, but my God, it's really people that just don't believe
anymore. I think that when you're talking about almost 50 years of a particular kind of failure in fundamental physics, where theories and predictions effectively become accepted as being the likely
explanations for the universe. We're getting to the point where everybody who's contributed to the Standard Model after this year will be over 70.
Understanding Geometric Unity[edit]
Brian Keating: What do you say to the younger people who say they can't understand it, they can't comprehend Geometric Unity. Our friend Sabine, she can't understand it. Is it too complex for her?
Eric Weinstein: No, there's a bunch of different games. One game is the "I can't understand all this fancy pants stuff." Another game is "Be hyper specific so we can invalidate you." There's another
game, which is, "Well, we know that you don't know quantum field theory really well, so what energy level do these things kick in at?" And I find all of this incredibly dispiriting and exhausting,
because it's also transparent. We can say what Geometric Unity actually is. We can draw a picture and people can get it. And in fact, I was talking to my good buddy Joe Rogan earlier today, and a
particular group of people that listen to my podcast put up a site for Joe called pullthatupjamie.com. If you want to navigate to pullthatupjamie.comâ in part, this is below Sabine's level. But I'm
happy to, you know, if you got her on the horn, she could understand what's being said.
Brian Keating: Yeah, I have no doubt about that. The question is, when we talk in the language of bundles, of fibers, etc, at what level do people kind of lose the physics for the geometry, for the
pure mathematical? And I thinkâ
Eric Weinstein: Let's walk the first step, and then let's watch people who are technically capable claim that they can't follow what's going on, because I don't think it's true.
Brian Keating: So...
Eric Weinstein: You have [math]\displaystyle{ X^n }[/math] for a manifold of n-dimensions. Make it orientable with a particular orientation, make it have a unique spin structure, whatever you need to
do to set it up as a decent manifold. Replace that manifold, momentarily, by the bundle of all metric tensors pointwise on the same space. And that way, spacetime would be a particular section of
that bundle. Let me see if I can find a...
So the first thing is that the observerse replaces spacetime. And, again, you're not trying to kill off Einstein, you're trying to recover Einstein from a different structure. So I'm looking...
Okay. So right here, I've got a 4-dimensional manifold. Imagine that I'm interested in looking at the bundle of all pointwise metrics, which is going to beâ if the base space is 4-dimensional, make
[math]\displaystyle{ 4 = n }[/math]â it will be of dimension [math]\displaystyle{ n^{\frac{n^2 + 3n}{2}} }[/math]. So [math]\displaystyle{ 4^2 }[/math] is 16, plus [math]\displaystyle{ 3n }[/math],
[math]\displaystyle{ 3 \times 4 = 12 }[/math]. So [math]\displaystyle{ 16 + 12 = 28 }[/math], divided by 2 is 14. If you have a [math]\displaystyle{ (1,3) }[/math] metric downstairs, I believe that
you are naturally courting a [math]\displaystyle{ (7, 7) }[/math] or [math]\displaystyle{ (9, 5) }[/math] metric upstairs. And that is the first step in GU, which is that you replace a single space
with one particular metric by a pair of spaces, a total space and a base space of a fiber bundleâ this is in the strong form of GUâ and physics mostly happens upstairs on the bundle of all metrics,
not downstairs on the particular space that got you started.
Here, [math]\displaystyle{ U^4 }[/math] is an open set in [math]\displaystyle{ X^4 }[/math]. Okay, so effectively, what are we saying? We're saying that physics is going to dance on not only the
space of four coordinates, typically [math]\displaystyle{ x }[/math], [math]\displaystyle{ y }[/math], [math]\displaystyle{ z }[/math], and [math]\displaystyle{ t }[/math], or thinking in a
coordinate-independent fashion, simply four parameters, it's also going to dance on the space of rulers and protractors at every given point. And so that structure is the beginning of GU, and then
you can recover Einstein, spacetime, by simply saying that if I have a section of that bundle, that's a spacetime metric.
Brian Keating: So when you say in the simplest form, or in the reduced form of GU, what do you mean?
Eric Weinstein: Well, I gave three forms of GU. One form is the trivial form, in which you have the second space [math]\displaystyle{ Y }[/math] the same as the first space [math]\displaystyle{ X }[/
math]. That means that you can easily recover everything Einstein did as a form of Geometric Unity by trivially making the observerse irrelevant. You're just repeating the same space twice, and
you've got one map between them called the identity, and now you're back in your old world. So without loss of generality, you cover that. Another one is a completely general world, which I thinkâ
What did we call it here... Well, I called the middle one the Einsteinian one, where you actually make the second space [math]\displaystyle{ Y }[/math] the space of metrics. And that's the one that I
think is the most interesting, but I don't want to box myself in, because I don't want to play these games of Simon's "You said this," or "You said that." You know, I can play the lawyerly game as
well as anyone if that's what we are really trying to do. I thought we were trying to do physics.
The thing that I'm trying to get at here is that I believe you and I are somehow having a pullback of a 14-dimensional conversation right now. My guess is that there is a space, with a [math]\
displaystyle{ (7, 7) }[/math] metric, probably more likely than a [math]\displaystyle{ (9, 5) }[/math] metric, on 14 dimensions, where not only are the waves that are relevant going over the original
coordinates [math]\displaystyle{ x_1 }[/math] through [math]\displaystyle{ x_4 }[/math], they're also going through four ruler coordinates on the tangent bundle of the original [math]\displaystyle{ x
}[/math] coordinates. So there are 4 rulers to measure the 4 directions, and then there are also going to be 6 protractors. Because if you name the directions John, Paul, George, and Ringo, you'd
have John with Paul, John with George, John with Ringo, Paul with George, Paul with Ringo, George with Ringo. Right? And so, those 6 protractors are actually degrees of freedom for the fields, and
the fields live on that space.
Then the question is why do we perceive 4 dimensions and complicated fields? And the answer is pullbacks. When you have a metric, you have a map from the base space into the total space, so Einsteinâ
we don't think of it this wayâ is embedding a lifeless space which is without form, [math]\displaystyle{ X^4 }[/math], into a 14-dimensional space before Geometric Unity ever even got on the
scene, and giving him the ability to pull back information, which he may say is only happening on that tiny little slice, that little filament that is the 4-dimensional manifold swimming in a
14-dimensional world with a 10-dimensional normal bundle. But why not imagine that actually the fields are actually spread out over all 14 dimensions, and then all you're seeing is pullback
information downstairs. Now the metric is doing something new that it wasn't doing before. It's pulling back data that is natural to [math]\displaystyle{ Y^{14} }[/math] as if it was natural on X,
but I call this invasive fields versus native fields, just because some species are invasive, and some species are endemic, or native. The interesting thing about the bundle of all spinors, sorry,
the bundle of all metrics, is that it almost has a metric on it. I don't know if I've ever heard anyone mention this.
Brian Keating: The spaceâ repeat that. The space of all metrics almost has a metric on it?
Eric Weinstein: Yeah, nearly.
Brian Keating: Explain?
Eric Weinstein: So in other words, assume that you haven't chosen a metric on [math]\displaystyle{ X^4 }[/math]. What you have then is you have a 10-dimensional subspace along the fibers, which we
can call the vertical space. And that 10-dimensional space at every point upstairs, every point is, in fact, a metric downstairs, being by construction. So that means it imparts a metric on
10-dimensional vectors along the fibers. Now those are symmetric 2-tensors, effectively, because it's a space of metrics. You have this really interesting space here, call that [math]\displaystyle{ V
}[/math]. Well that [math]\displaystyle{ V }[/math] has a Frobenius metric based on the particular metric at which you are looking at the tangent space, which has got a 10-dimensional subspace picked
out. If you map that 10-dimensional subspace into the 14-dimensional tangent space of the manifold [math]\displaystyle{ Y^{14} }[/math], you can take a quotient and call that [math]\displaystyle{ H }
[/math]. And that [math]\displaystyle{ H }[/math] will also have a metric because it's isomorphic to the dual of the pullback of the cotangent bundle downstairs. And the cotangent bundle has a metric
because at that point that you picked in [math]\displaystyle{ Y^{14} }[/math] is itself a metric downstairs.
So now you've got a metric on [math]\displaystyle{ V }[/math], you've got a metric on [math]\displaystyle{ H^* }[/math], and you just don't know how [math]\displaystyle{ H^* }[/math] becomes the
complement to [math]\displaystyle{ V }[/math] and [math]\displaystyle{ T }[/math]. That's the only piece of data you're missing for a metric. So you've got a 4-metric, you've got a 10-metric, the
10-metric is sitting inside of the tangent. The 4-metric is naturally sitting inside of the cotangent bundle. They're weirdly complementary, you've got a metric on the nose but for one piece of data,
which we call a connection. So up to a connection, the manifold [math]\displaystyle{ Y^{14} }[/math] has a metric on it without ever having chosen a metric because it's made out of metric data.
Now spinors have a really interesting property, which I would call an exponential property. That is, the spinor of a direct sum is the tensor product of the spinors on the summands.
Brian Keating: That's not true for any spin, or is that true for any spin, or just half integer, or...?
Eric Weinstein: Well, that's true for anyâ no, it's true for the spin representation. It's not true generically, for any representation. But it allows you to build the spinors on what should be the
total space, because now you've got a 4-dimensional... So, I think it's here at 3.12. If the spinors of a sum are the tensor products of the spinors on the summands, and I create a new bundle, which
is the 10-dimensional vertical bundle inside the tangent bundle direct sum the 4-dimensional bundle inside the cotangent bundle, then the spinors on that thingâ which is isomorphic and in fact
semi-canonically isomorphic to both the tangent bundle and the cotangent bundle, being chimeric, it's isomorphic, but it's not fully canonically. It's only semi-canonically. So spinors on that will
be identifiable with the spinors on [math]\displaystyle{ Y }[/math] as soon as you have a connection that completes this and makes it fully canonically isomorphic.
So the take home message, there is a spin bundle up on the bundle of all metrics, which is nearly the spinors on the tangent bundle, that exists without making a metric choice. And if you're really
serious about quantum gravity, you should be very freaked out about the idea that once you quantize the metric, you've got a whole lot of pain, because the electron and the hadron bundles, and all
the spin-1/2 matter, the medium in which these particles are disturbances, are excitations, doesn't really exist in the absence of a metric choice. If you allow the metric to become quantum and allow
it to blink out, the spin-1, spin-0, and spin-2 particles may be indeterminate between observations. But the bundle itself, the medium, is indeterminate between observations of the metric for
fermions. So now you're in a really different conceptual world. Everybody should want to free fermion bundles from dependence on the metric if they're serious about letting the metric blink out in
some supposed quantum gravity regime.
Implications, Expectations, and Communication[edit]
Brian Keating: Let me ask you about that for a second. So it seems like this is a huge, you know, "Huge if true," I always like to say.
Eric Weinstein: Well we say that, but I don't know whether I just missed one hell of a meeting. I just don't understand why everybody isn't worried aboutâ
Brian Keating: So this is huge, right? This is, what you're saying is that you can get spinorsâ
Eric Weinstein: If I haven't made a boneheaded mistake.
Brian Keating: Well, this is where I'm going to. I don't think you have, but I'm just a simple experimental cosmologist, okay?
Eric Weinstein: I'm just a podcast host.
Brian Keating: I traffic in [the] nuts and bolts of cosmological experiments, telescopes, as you know, detectors and fields. I am out of my depth in many cases, but this struck me like a freaking
thunderbolt that you were derivingâ essentially, spinors can be defined without choosing a metric. That is new. I don't think that any critic, any anonymous, pseudonymous, or anonanononymous person
can really criticize that. I mean, that's just a fact. So why wouldn't physiciâ if it's not true, it would be, you know, almost surprising, but if it is true, why haven't physicists noticed this
before and why aren't they making a bigger deal out of it? Partially, it might be your fault, because you haven't published this.
Eric Weinstein: Blame the victim.
Brian Keating: Who else?
Eric Weinstein: So what I usually hear about this is, people say "Oh, you don't understand, Jean-Pierre Bourguignon told us how to move spinors under variation of the metric." But he's varying the
metric continuously, there's always a metric present. What if there's no metric for a little while?
Brian Keating: Which could be the universe before before God intervened.
Eric Weinstein: Are you going to do a Feynman integral over all variations of the metric? I mean, I don't know what kind of pain you're signing up for, but I'd certainly rather freeâ look, here's
the basic statement. If we're serious about quantum gravity, we should be very serious about trying to get fermions that don't require their bundles to be dependent on the existence of a metric at
all times. And I'm sure that either there's a brilliant explanation that I don't understand, and I'm eager to hear it, or it's a key sign that the community really dropped the ball. Remember, for
example, that the Bohmâ Aharonov effectâ I'm sure that when, when Aharonovâ Bohm said "Hey, shouldn't there be an effect of this zero field strength?" they probably thought 'Have I lost my mind?'
I'm sure that Yang and Lee, when they proposed that maybe the weak force was left-right asymmetric, probably thought, 'Are we were going to be laughed at, did we just not understand what everybody
else understood?' Physics gets things really spectacularly wrong occasionally, and I'm curious to know if this is one of those moments.
Brian Keating: Yeah. I mean, you might also say oh, there's 26 dimensions in heterotic string theory. That can't be right. No, it's only 10, or 11, or 5-brane, m-brane theory. I want to ask another
question, which is frequently used in criticisms, both anonymous and nonymous, which is that this doesn'tâ
Eric Weinstein: I, can I actually, can I just say something? I really don't want to talk about anonymous trolls with PhDs criticizing the theory. And I also don't want to talk about non-constructive
hit jobs on new theories. Last time I checked, physics was in a crisis that some people were admitting to and other people were sweeping under the rug.
Brian Keating: Okay, wellâ
Eric Weinstein: If you have a crisisâ wait wait waitâ if you have a crisis, for God's sakes [sic], open it up. We don't need one more talk from the same crowd of people who have been keynoting
every conference of note for the last 30 years who haven't got the new ideas. Let's at least hear crazier, weirder, wilder people. And if you guys don't have the guts and courage to do it from inside
the community, hear it from a podcast host.
Brian Keating: Okay, well, this is my podcast, and I do want to respond to these criticisms, because for me, I don't find them legitimate. And you can choose to be silent as is your want. No, it's
rare toâ
Eric Weinstein: No, I wish to punish dysfunctional cowards who attempt to snipe, pretending to be helpful. You can do better at it.
Brian Keating: I can do better as well. But I do want to say that this is maybe a general comment, not for pseudonymous and anonymous people, bananymous. But this is a general complaint that I've
heard: it has to reproduce quantum theory. And I think, forget about that with regard to GU, it could be said about other theories, loop quantum gravity, etc. First of all, I think GU does produce
what we would say is a relativistic quantum field theory in the Dirac equation, which is manifestly resplendent and produced and predicted. So I don't want to hear from you just yet, Eric, I do want
to get your response. But this notion that a theory of everything has to subsume anythingâ I said this to our mutual friend, Stephon Alexander, professor at Brown University and esteemed
cosmologist, and close friend to both Eric and myself, I said, "Look, I don't think it's valid to say that any theory of everything, string theory or whatever, has to predict every manifestation of
physics," and this is where I take issue, and I make truck with Professor Kaku, who says things like, "The one-inch-long God equation will predict everything." I don't think that's possible, (A) I
don't think it's useful to think about the goal of physics is to predict every phenomenon in physics.
Eric Weinstein: Because it's an incautious statement. Really what you're trying to say is that there's stuff that you should be able to read off in the basic setup of theory directly, and there's
stuff that you should work your ass off in order to get from the theory. Now, you know, we don't see quarks running around free the way you might imagine, naively, you would if you were looking at
the hadronic part of the Standard Model Lagrangian, and so you have to work pretty hard, I would imagine, in order to find these bound states that we call protons and neutrons, and try to understand
infrared slavery, etc, etc. Now, that's part of the hazard of saying I can predict everything. No, even computationally, you don't think so. Really, it's just a question of, we should be able to
recover everything that we've already done. And actually, I think that that's pretty fair.
Brian Keating: So evenâ
Eric Weinstein: I think there's a dumb way of doing it, where you try to say, "Show me this, or then you don't have anything." And I have to say, I encounter a tremendous amount of that from people
who are old enough to drink, and it gives me pause as to who's raising the young. That's not the issue. The issue is, they're right, they should be saying "Look, here's what we know how to do, and
you should be looking to recover what we already know how to do from what you're saying," and I think that's actually fair. There's a question of should you be able to do everything on day one?
Should you be able to do it when you've been cut off for 27 years working completely on your own under totally weird circumstances, where every month you feel you get farther and farther away from
the literature, and your brain hasn't spoken this language in a million years? Those are questions that I feel likeâ that's really sad, because people don't understand what the cost of isolation is.
I do think, however, that working in a context with competent people who aren't constantly trying to rename everything after themselves, there's no question that that's a reasonable and fair thing,
if we had a collegial world based on a desire to advance our understanding. And I'm happy if I fail at that with a collection of constructive colleagues to say that that's a black mark against the
theory, that's fine.
Brian Keating: Now, when I look at the corresponding, shall we say, implications against string theory, I would say things like the swamp land, the multiverse problem, these may be issues that cause
stillbirth in many people's minds. I've talked to you about Paul Steinhardt, the Einstein Professor of Natural Science at Princetonâ he regards the string theory as essentially bad for society, not
just for physics, not just for science, but bad for society because of the extravagance in a truest sense of the word, in a bad sense of the word, of the multiverse and string landscape. Now I know
you're shaking your headâ go ahead.
Eric Weinstein: No no no. Let me be very clear about it. We're wimping out from what needs to be said, and it's really important the community gets it right. I don't think string theory is a problem.
string theory can't harm anyone, string theory doesn'tâ it's the string theorists when they're in their triumphalist mode, that it's an insufferable state of being. But even then, you know, I'm sure
Feynman was insufferable, and I think Murray Gell-Mann was insufferable, and Pauli was pretty insufferable. We've had insufferable members of our community for a very long time, and we should not be
getting rid of insufferable people. The problem is, what happens when people become insufferable and they don't constantly check in with the unforgiving nature of the universe. I mean, Pauli
predicted the neutrino in an insufferable fashion.
Brian Keating: And apologized. He apologized profusely, "I've done something which should never be done." Now, I asked you though, should string theoryâ let's just be neutral to GU for a second.
Should string theory, from string theory, emerge the Aharonovâ Bohm effect? I mean, a true theory of everything, it would, right?
Eric Weinstein: Look, and if it took a while to recover certain features of the world that you had in an effective theoryâ I mean, look, let's put it this way. If you look at Marshallian demand in
economic theory, should you be able to predict that from the Lagrangian of the universe? No, it's in a different strata of the world. You should be able to predict things that are within the adjacent
strata of the theory, and then you might have to appeal to some higher effective theory.
Look, I want to defend both the string theorists and string theory. These are incredibly smart people who found some real structure, and who never knew when to quit when it came to trumpeting just
how much better string theory is than everything else. Even there, they had a point. They were smarter and deeper, in general, than everyone else. They just weren't as good as they claimed to be, and
they weren't as successful as they claimed to be, and what they did succeed that they didn't want to take credit for, because it was really mathematics done in physics departments rather than
So we have a problem that sociologically, nobody wants to say that the Institute for Advanced Study has the smartest guys around and a lot of what they do isn't physics, in standard terms, it's the
mathematics of physics. These are uncomfortable truths, just the same way that it's uncomfortable that we're taking seriously somebody who's been out of the field for 27 years. But these are end
times, we're having end time conversations. I think that it'sâ we don't need to be mean about it. I think it just needs to be more honest.
Concept Animations[edit]
Brian Keating: Okay. With that, I give some applause here. Let's see if we can hear that. [Applause sound effect] Got some applause, Eric. A smattering. That just was a smattering. I want to take a
pause for the cause, and to have a pause to recognize our guest today is the esteemed Dr. Eric Weinstein, who is a seeker after truth, a seeker after my own heart in the authentic tradition of the
old oneâ his namesake, Albert Wein... Now they say this is not a serious podcast until you break out the puppets. Now I know Rogan has a supply of bows and M16's, and all sorts of other things. I
don't have any of those accoutrements, I only have my sock puppets and my gelt Nobel Prize. But, I do want to say that this is a special conversation with Eric, because it really fulfills a promise
that was made basically a year ago, and then again about six months ago on this podcast, which is to release a stunning amount of new technical details, and you've really surpassed that.
Our mutual friend James Altucher, podcaster extraordinaire, he says that you should never under promise and over deliver, you should never under promise and under deliver. You should over promise and
over deliver. Meaning that if you say you're going to get it done in three months and get a million customers, you should get it done in one month and get ten million customers, or as one Peter Thiel
said once, what do you think will require ten years but could be done in six months. So, what you've done is released a tremendous amount of technical information that will be fully released at some
point to the public. But also, I want to take our audience through some of these delightful animations. I put the link in the chat for now, but I'm going to share my screen right now. Hopefully you
can see it as well, Eric. These are now movies I want you to animateâ I'll put you in the lower corner. Let's see if I can do that, I'll do that in a second. Let's see, I'll add Eric. Nope, I'll
swap these. There we go. I'm going to swap Eric, if you're willing to swap. There we go. I want to walk us throughâ which which one of these many videos should we should we take a look at? I was
fascinated by the Shiab, but that's just myâ
Eric Weinstein: Let's do the first three.
Brian Keating: Okay, so the first one is called an Einâ
Eric Weinstein: Go to Einstein's Great Insight. We're going to do this for people who are somewhat physics-minded, but who like to complain that none of this is understandable. By the way, there are
some names associated with these videos. Brooke Dallas has been shepherding the project. Brandon Stone has been incredibly helpful technically. Boqu, a mysterious German man who animates many of
these things. There's a list of people who've contributed. Tim, the mirthless swagman, from Australia, a math student down there. So, what they've done is they've tried to interpret what it is that
I'm saying, because I tend, because of learning issues, to not think symbolicallyâ stop, Brian.
Brian Keating: Yeah.
Eric Weinstein: Let's blow that up. Full screen
Brian Keating: It is full screen here, yeah.
Ship in a Bottle Animations[edit]
Eric Weinstein: Okay. That ship that you're seeing is called curvature. It has three masts because it has three irreducible components, usually. One mast is called Weyl curvature, one mast is called
traceless Ricci curvature, and one mass is called Ricci scalar. The first greatest insight of the 20th century was the way in which we could feed back the curvature of the Levi-Civita connection into
being a co-vector field on the space of all metrics. This is depicted as a boat going into a bottle that has a rather wide opening. So let's run the animation.
Brian Keating: Okay.
Eric Weinstein: So we've got a metric. The metric has a connection, the connection produces curvature that's Riemannian. We find that by identities, it's got three components. It tries to go towards
metrics and the Weyl curvature is snapped off. Afterwards, the scalar curvature is lowered somewhat, or adjusted, by scalar curvature over 2 times [math]\displaystyle{ g_{\mu \nu} }[/math]. And so
symbolically, what we've done is we've said Einstein threw away the Weyl curvature, readjusted the Ricci scalar curvature, and fed metric information through to the Levi-Civita connection, through to
the Riemann curvature tensor, and then played these projection games to feed it back to the space of metrics. And that particular combination is perpendicular to the action of the diffeomorphism
group on the space of all metrics, leading to a divergence free condition via our friend the Bianchi identity.
Now, why can't we do that and feed this information back to the space of connections rather than the space of metrics because we would love to link spacetime games with gauge potential games. So,
let's see whether General Relativity and gauge theory have an incompatibility problem as we try to play the same game. We start off with the Riemann curvature tensor, but now the neck is narrower.
What's really going on is that this is kind of evocative of trying to feed it into the space of connections, but the gauge group acts differently on two different factors: namely, if connections are
ad-valued 1-forms and curvature is an ad- or Lie-algebra-valued 2-form.
The problem here is the gauge transformations act on the Lie algebra component and don't touch the form component. But Einsteinian projection, or contraction, or summing over [math]\displaystyle{ g_
{\mu \nu} }[/math] indices, is democratic: it deals simultaneously with the form piece and the Lie algebra piece. So if you treat only the Lie algebra piece under a gauge transformation and you don't
touch the form piece, then contraction followed by gauge transformation will never be the same thing as gauge transformation followed by contraction. And so that's the puzzle, which is if Geometric
Unity is really about the idea of trying to say maybe it's not so much quantizing gravity, maybe it's a fight between the different geometry of Riemann and Ehresmann, because gauge transformations
are Ehresmannian geometry but contractions are Riemannian geometry.
So here's a GU approach, how do you get geometric harmony between General Relativity and gauge theory when you have the ship in a bottle problem? This is almost a tight analogy. You've got the
curvature tensor, you apply a gauge transformation to two of the masts and you pass them through into ad-valued [math]\displaystyle{ (d-1) }[/math]-forms, and then you do an inverse gauge
transformation, which is exactly how you do the ship in the bottle trickâ by the way, Brian gave me a wonderful ship in a bottle, thank you very muchâ raising the mast inside. And then you can
potentially, if need be, adjust one of the two masts again in order to get agreement.
So in part, the idea is how do you get harmony? What you need to do is to promote the gauge transformations initially to field content in order to make sure that you're carrying around enough
information, effectively, to ensure that contraction is compatible with gauge transformation. Now, that is a very tight idea of how these operators function inside of the theory.
Gauge Theories as Calculus Done Right Animations[edit]
[Keating pulls up "Penrose-like steps" video]
Well this is justâ for some reason, whenever we talk about gauge theories, we don't give people very concrete examples. Many of you who are not professionals will not know what a gauge theory is.
May I make a recommendation, Brian?
Brian Keating: Yeah, of course.
Eric Weinstein: Let's go to another animation, which is something like Gauge Theories as Calculus Done Right, and blow that up as big as it can be before starting the animation. Okay, start the
animation. Let's imagine that we have a salary that is constant in dollar terms over time, [and] that somebody is facing inflationary pressures on their basket of goods. Now the question isâ pause
please. What we now have is a $10 an hour salary, and if we claim that it's constant, constant means derivative equals zero. But, we know that it's not constant purchasing power. So we have two
notions of constancy, how are they related? Let's go back to that please.
We do a gauge transformation. And what you seeâ pause please. You now see that these little hash lines are the reference levels that we call a connection, and we decide that rise over run should not
be measured from a naive horizontal, but should be measured instead from a custom reference level represented by the hash marks. Now, if you let it go a little bit, and then stop it, stop. Now you
see that derivative equals zero, if we measure rise over run above the hash marks, is a salary that keeps pace with inflation. And the current $10 an hour is actually a negative derivative because
the rise over run is measured beneath those hash lines. That situation is actually an application of gauge theory to a very simple problem in economics, completely depicted by stretching the fibers
in the x-y plane. And if you look online right now and say "What's a gauge theory?" you'll be bamboozled by a bunch of stuff that nobody can understand unless they're actually insiders.
So I think it's very interesting that again, just as it was elementary to ask the question, "What happens to the fermion medium while we're blinking out the supposedly quantum metric?" why is it that
we don't actually explain to anyone what a gauge transformation and visualize it? I'm very proud of our team for taking this very simple example and showing what a gauge field isâ it's those little
hash lines, effectively. Those things in higher dimensions would be the electromagnetic potential, which becomes the photon under quantization. And if you're thinking about QED (quantum
electrodynamics), effectively, the electron is a function and the photon is a derivative, because what you're specifying is the levels above which you're going to measure rise over run. Now you can
go back to the original floating plane.
Brian Keating: Floating plane...
Eric Weinstein: What you were doing before.
Digression on Academic Misbehavior[edit]
Brian Keating: I just want to take a second here. This is Brian Keating now speaking. So, if you look up Juan Maldacena, you will find only one podcast that he's ever been on, and that is the Into
the Impossible podcast. If you look up gauge theory and an intuitive way to understand gauge theory, something like that, you'll come up with this really brilliant economic analogy that sounds like
Eric has copied from from Juan Maldacena. And in fact, this came up recently, where people were talking about inflation-stabilized items and Bitcoin and so forth, and thenâ it was very frustrating
to me, and I imagine much more so for Eric, although he doesn't have to comment, he's too much of a gentleman. This is Eric's work. This gauge theory applied to economic transactions.
Eric Weinstein: Eric and Pia.
Brian Keating: Eric and Pia Malaney. Yep, Pia Malaney, of course, the beautiful talented wife of Eric Weinstein. Eric is known as the husband of Pia Malaney, mostly. This work is brilliant and is
deserving of attention in its own right, independent of the brilliance of it as an analogy to explain a very complicated subject such as gauge theory, or a very simple subject like calculus, as Eric
is now explaining to us. I wanted to say that, you don't have to respond if you don't want to Eric, I find it very frustrating when I see "Oh, Eric, you've got to learn what Maldacena said," I'm like
F you. That's very frustrating to me.
Eric Weinstein: That's what was hurtful, because Juan knew that he had gotten thisâ knew about Pia Malaney, he needed to reference her. He did reference her, but in a very slight, minimal way in a
[Inaudible] version.
Brian Keating: It's a footnote. It's a footnote. He knows better than that.
Eric Weinstein: The problem that I'm having with it is that the professional community does not understand that it has impulses that it hasn't faced, which is that it tends to brutalize those that it
doesn't need to cite, that it doesn't see. It just doesn't see people. And so to haveâ look, I'm a huge Juan Maldacena fan as are we all, but I'm not going to sit around and have people say "What
you really need to do is to listen to Juan Maldacena, whose brilliance knows no bounds. He did something really profound about markets and gauge theory," because quite frankly, Pia Malaney deserved
to have an entire career built around it. I think it could easily be the most deep insight in mathematical economics in the last 25 to 50 years. Please show me another, given that the Marginal
Revolution, originally, was the penetration of differential calculus into economics. Her thesis, which is largely joint work, but was not even allowed to be what it was supposed to be, rebased the
field of economics on gauge theory as the correct form of calculus. I'll tell you what, I don't really want to bitch about Juan Maldacena, but what I would really love to do is to have Juan
Maldacena, who showed so much excitement when I confronted him about thisâ he says, "Oh, you know who that is?" because he had no idea who Malaney was. It would be really great if Juan Maldacena did
this work, and I won't say another word this podcast about it.
Brian Keating: Okay. And I will say only one word because it's my podcast, and I can do whatever the hell I want. I had on Cumrun Vafa, as you know, who wrote a book called Puzzles to Unwrap the
Universe, in which he cites Juan Maldacena. I called him on that. I said this is actually original work by Pia Malaney, Eric Weinstein, and it almost doesn't matter. And I find that very frustrating,
because the very same peopleâ and you don't have to respond. Please don't respond. Again, I'm a blowhard on my own podcast. It's one of our prerogatives. We get so little of these things and treats
in life. But I find it very disingenuous of the community. I love Cumrun too, but to say that "This isn't serious Eric, you have to cite this paper, you have to put out a paper about GU, you've only
done things on Joe Rogan," I find that disingenuous. You don't have to respond. Let's go on.
Eric Weinstein: What I will say is this. When you have gatekeepers in the form of advisors, if you have job market meetings, where people wield incredible power and they hold other people's careers
in the palm of their handâ if you use these places to crush people, you have no right to comment after the fact as to why are these people behaving bizarrely and strangely. Because in essence,
whether you submit things to journals and have a perfectly reasonable relationship with peer review, or whether you find that peer review is basically a tool to exclude you, and your insights, and
your claims from the world, depends in large measure on who you are, where you're coming from. It's human dependent, it's not independent of who submits and how protected they are.
The thing that I want to get across is that the community is producing trauma in people and then claiming that it's paranoia. You have to recognize that trauma and paranoia look exactly the same when
you can't see what the source of it is. If you want to understand what happened to this theory, read The Physics of Wall Street by James Weatherall, chapter 10 and the epilogue. It's rather clear
about the fact that four gentlemen and one lady tried to steal a trillion dollars over 10 years by pretending to fix the CPI because social security and tax brackets were indexed. They came up with
1.1% adjustment that would be needed, and then they broke into two teams to find exactly out the 1.1% that they wanted. This was admitted to by Robert Gordon. And, the most brilliant thesis that
probably came through Harvard in terms of mathematical economics was destroyed so that Daniel Patrick Moynihan and Bob Packwood could have a back end run around the third rail of politics, which is
slashing benefits and raising taxes, using economists to destroy, funnily enough, a bright promising woman of color from the developing world in an essentially all male field. These people should pay
with their reputation.
Concept Animations 2[edit]
Brian Keating: Okay, I want to lighten things up again. Let's talk about Jeffrey Epstein. No, I'm just kidding. I made you laugh, come on. That's a big accomplishment.
Eric Weinstein: That was good. I like it.
Brian Keating: Alright, let's look at one last video here. Let me call up aâ let's go to the videotape as they say. Go here, I'm gonna go to Safari, Rastafari... Nope, it's not coming up. Oh, maybe
that's because it already thinks that we've done it, let's see here. All right, zoom out. I'm gonna...
Eric Weinstein: Do you want to do an observerse one, down at the bottom?
Brian Keating: Yeah, I'm trying to get up theâ tell me when to stop here. Well you can't see it, right?
Eric Weinstein: I can't see anything.
Brian Keating: Let me let me get my screen back here. Let me kill that. Let me kill... what else is going on here? Screen share, show Safari. Show primary display, secondary display, there we go. Can
you see that?
Eric Weinstein: Yeah.
Brian Keating: Okay. So at the bottom, I see 5D Observerse, Spinor Dance... Which one would you like? Observerse 5D?
Eric Weinstein: Let's do 5D. Yeah, I think I'll explain what they're trying to depict. It's not exactly how I would have done it, but keep in mind that these are artists who've been trying to learn
what this is by bypassing typicalâ okay, so pause it.
Brian Keating: Yeah.
Eric Weinstein: Can we get rid of that bottom bar?
Brian Keating: Yeah, it's, they need to disable it on their side, but I can kill it off here. There we go.
Eric Weinstein: Okay. And can you blow that up? Or is that as blown up as it can go?
Brian Keating: Let's see here. I think it's fairly blown up.
5D Observerse Animation and Patiâ Salam Connection[edit]
Eric Weinstein: Alright. Imagine that that torus that you see in the lower left corner of the screen is a 2-dimensional model, toy model, of spacetime. So going around through the center is like
Groundhog Day, you come back to the same place and it's a repeating time cycle, and space is simply a circle. Now in such a world, we would normally think of quantum field theory or gravity as taking
place on that object. You'd have fields, you'd have effectively functions called sections on that object, and what you're seeing here is something that's very hard to picture because it's
5-dimensional, but one trick here is because the torus has a property called parallelizability... The object on the right is a depiction of a metric. Each point that isn't on one of those two sheets
is a potential metric at any given point on the torus. So in other words, if a metric is a symmetric non-degenerate 2-tensor, if you think of it as a matrix, it would be of the form [math]\
displaystyle{ \begin{bmatrix} x & z \\ z & y \\ \end{bmatrix} }[/math]. Non-degenerate means that [math]\displaystyle{ xy - z^2 \ne 0 }[/math]. So that's what's cutting out that variety, if you will,
the zeros of the of the determinant would be points, given that there are 3 degrees of freedom in the metric.
So instead of actually having a metric spacetime, GU would say replace the torus by the entire space in that sort of hourglassy region. So the top region would be like space-space metrics, the bottom
below that sort of weird, diaphanous scarf is time-time metrics, and the weird middle region, which is sort of around that singularity, would be space-time metrics. Every way you can stick that donut
into that middle region without touching one of those two sheets is a valid spacetime metric. And what GU would do is to say don't only dance on the points of the 2-dimensional torusâ again, the
surface is 2-dimensional, even though it seems to be 3-dimensional to naive investigationâ you should actually have fields that are dancing on all of the points of the torus and, simultaneously, all
of the points in that middle region of what we call the Diablo diagram, no to the right. To the right. Yep.
So every point in that region is in play, and if you mappedâ imagine that the stuff in that weird hourglassy region on the far right was like very warm and on the far left was very cold. Then if you
map the torus in to the far left region, it would show up as being cold. If you mapped it into the far right region, you'd see it as being very hot. So every way of mapping the torus in pulls back
different information from that hourglassy region. That is in large measure, in part, one of the things that may be going on with the illusion of many worlds, is that what you're seeing is that the
metric may be capable of pulling back data that is dancing on the space of all metrics as well as the space of all points on the original manifold [math]\displaystyle{ X }[/math]. So in this case,
you've got 2 degrees of freedom on the torus, you've got 3 degrees of freedom around the hourglass, and [math]\displaystyle{ 2 + 3 = 5 }[/math].
Now notice that thing up in the top left, which is a ruler-protractor combination that I just gave a copy [of] to Joe Rogan. Those two sliders are recalibrations of what it means to be one unit. And
that protractor is a recalibration of what you're going to define to be 90 degrees. So every way of keeping that bottom arm in a single horizontal position, moving the top arm, and moving the two
sliders, that's 3 degrees of freedom in the space of metrics. So that's a different depiction of the space of metrics.
So the big take home from the restrictive version of GU that we're exploring here is that if you allow fields to dance on the space of metric apparatusâ measurement apparatusâ then the paradoxes of
measurement start to make a lot more sense. You could also, potentially, try to keep the metric classical, because we have two spaces. We have a space downstairs [math]\displaystyle{ X }[/math],
which is just the torus, and we have a space upstairs, which is the torus, in this case, cross the hourglass region, as long as it doesn't touch the two sheets. So you've got a 5-dimensional manifold
hovering over a 2-dimensional manifold, and fields on the 5-dimensional manifold will be perceived on the 2-dimensional manifold when you pull them back via a particular Einsteinian spacetime as
fields on the tangent bundle of what you will call spacetime, together with fields on the normal bundle inside of the 5 dimensions.
The normal bundle of a 2-dimensional manifold in a 5-dimensional space is 3-dimensional, so you're gonna see fields that look like spinors on 2 dimensions tensor spinors on 3 dimensions. If you were
in 4 dimensions, make that torus in your mind represent a 4-dimensional spacetime, then that Diablo region would be a 10-dimensional region of metrics, because 4x4 matrices that are symmetric have
[math]\displaystyle{ \frac{4^2 + 4}{2} }[/math] [Inaudible] for different degrees of freedom. In other words, you get a 10-dimensional normal bundle.
Now you'll notice that if you have ordinary spinors on 14-dimensional space and you pull them back via a metric, which is a mapping of 4 into 14, it looks like spinors on the 4-dimensional space
tensor spinors on the 10-dimensional normal bundle. If the normal bundle inherits the Frobenius metric from [math]\displaystyle{ X(1,3) }[/math], and you glue in the trace piece in the right wayâ
well, if you glue it in the wrong way, you'd get a [math]\displaystyle{ (7,3) }[/math] metric on the normal bundle. But if you glue it in the right way, you'd get a [math]\displaystyle{ (6,4) }[/
math] metric on the normal bundle.
[math]\displaystyle{ \text{Spin}(6,4) }[/math] is a sort of nasty non-compact group, so you might want to break to its maximal compact subgroup like Witten and Bar-Natan discuss. And the interesting
thing about [math]\displaystyle{ \text{Spin}(6,4) }[/math] is that it has different names. By low-dimensional isomorphisms, [math]\displaystyle{ \text{Spin}(6) }[/math] is the same thing as [math]\
displaystyle{ \text{SU}(4) }[/math]. [math]\displaystyle{ \text{Spin}(4) }[/math] is the same thing as [math]\displaystyle{ \text{SU}(2) \times \text{SU}(2) }[/math]. And [math]\displaystyle{ \text
{SU}(4) \times \text{SU}(2) \times \text{SU}(2) }[/math] is the Patiâ Salam theory. So you can argue that ordinary spinors on the induced metric in 14 dimensions, glued in the right way, pull back
as Patiâ Salam. And I don't know if anyone's ever discussed the connection between Einstein and Pati and Salam.
Brian Keating: No. No.
Eric Weinstein: Well no, I can't say no, I don't know of it.
Brian Keating: I don't know, that's what I'm saying. People have brought it up, but yes, has it?
Eric Weinstein: Has anyone? I don't know.
Brian Keating: I don't know. Yeah.
Eric Weinstein: So the point is that spinors on 14 look like spinors on 4 tensor spinors on some version of 10.
Brian Keating: Yeah.
Eric Weinstein: And whether you're talking about [math]\displaystyle{ \text{Spin}(10) }[/math] models, [math]\displaystyle{ \text{SU}(5) }[/math] models, or [math]\displaystyle{ \text{SU}(4) \times \
text{SU}(2) \times \text{SU}(2) }[/math], which is [math]\displaystyle{ \text{Spin}(6) \times \text{Spin}(4) }[/math], isn't that exactly what we see in the Standard Model? So Frank Wilczekâ let me
just see if I can find this beautiful quote from him, because he definitely brought this up. And what I recently did when I had him on my podcast, which we haven't releasedâ so, if we go over to my
screen share...
Brian Keating: Give me one second. Let me do this. Here we go. And... There we go. Yep.
Eric Weinstein: Let me read it. "A particularly intriguing feature of [math]\displaystyle{ \text{SO}(10) }[/math]," which is really [math]\displaystyle{ \text{Spin}(10) }[/math]spin 10, or it could
be [math]\displaystyle{ \text{Spin}(6,4) }[/math], "is its spinor representation, used to house the quarks and leptons, in which the states have a simple representation in terms of basis states
labeled by a set of "+" and "-" signs. Perhaps this suggests composite structure." Now here's the sentence that just floored me. "Alternatively, one could wonder whether the occurrence of spinors
both in internal space and in space-time is more than a coincidence." And then he pulls back immediately, "These are just intriguing facts; they are not presently incorporated in any compelling
theoretical framework as far as I know." Geometric Unity is that compelling framework.
Brian Keating: Awesome. Very interesting. So as we wrap up, I do want to see if there are any other videos you'd like to show that would help the reader, or again, I'm going to put this in the chat
box so people can peruse it. I did put it in the actual YouTube box description, so people can find that at their leisure. Let me see here. Oh I see what's going on.
Geometric Unity Document[edit]
Eric Weinstein: Well I should say that I... Look, let's be honest. I said I was going to release a document, and clearly we haven't. Okay, April Fool's. April Fool's.
Brian Keating: Uh oh, the big reveal!
Eric Weinstein: Go to geometricunity.org.
Brian Keating: geometricunity.org.
Eric Weinstein: And, yeah, call that up. And then Brian, why don't you be the first to put your email address in to request a copy? I wouldn't call it a paper, I'd call it a draft. One of the things
I'm looking to do is I'm looking to get constructive feedback from people who want to help me succeed, as opposed to people who just want to be dicks and take me down, because that's just, to be
honest, not very interesting to me, and I've had a little taste of that, and I'm not that interested. What I would love is to bring your positive energy. Download it, read it, recognize that more or
less, I've been cobbling this together from a million and one different scraps, and that my ability to talk in this way has been degrading for years because I have no one to talk to. I'm not in a
department, I'm doing this completely on my own. And I was a little bit frightened to figure out just how much I'd forgotten.
So we're still finding scraps of paper, and files on old discs, and things like that. I hope that the notation is getting more and more standard, that there are fewer errors. But there's clearly, you
know, this is basically me going back to 1983, '84, and all the time in between, where mostly I didn't talk about this with anybody. And this has been really terrifying, because you know, I'm not a
physicist, I don't come from this community. I revere the community. I don't think the community has been behaving well recently, I don't love saying that. But I think the community is in a desperate
situation, and let's find out whether I have anything to say or I'm just blowing hot air. I'm not afraid of that. But you know, what would really be meaningful to me is for people to bring kindness,
benefit of the doubt, hope, and recognition that it's pretty tough to try to do all this on your own, Be constructive and take a look. I think there are two email addresses on the paper in draft
form, one for technical feedback and one for general feedback. So I hope that there's a lot of food for thought.
I do think thatâ let me just close this out. I think it's a coherent story. I think it's the first time I've ever heard a coherent story about how a very simple beginning would produce something
that would look like our world. There are things that I would call predictions in it, that talk about what internal quantum numbers you would expect to find, likely next in terms of, there's much
more matter, there's matter that should be dark, there's matter that might be luminous but not at the right energy level yet. You would have to, in order to compute with it, be able to figure out
what fields have acquired VEVs (vacuum expectation values) and where we are in anthropic spaces in some places. But the internal coherence is much sharper than a few, you knowâ there's still some
things that I'm trying to locate my favorite version of. One is the Shiab operator. I know how to produce Shiab operators in general, but I had a sheet of paper withâ do you remember paper with
feeds, with holes on either side?
Brian Keating: Oh yeah, loose leaf, oh feed, oh printer paper. Printer paper.
Eric Weinstein: Not loose leaf. Printer paper.
Brian Keating: Dot matrix.
Eric Weinstein: Yeah. So I did some calculations in representation theory that came up with the projections that I used to use that I'm looking for. And the thing that I remember is that they've got
yellow highlighter and these perforated holes on either side. I haven't been able to find it yet.
So it's a very long process, taking about 37 years of speculation, sometimes more active than others, and trying to put it in one document. So I would really appreciate it if people wanted to take a
gander through it. Try to see some of the ideas, and recognize that if we are going to get off this planet with its hydrogen bombs and crazy leaders, and diversify and take some some bets, rockets
are not going to do it. There is no real "Mars or Bust", or "Occupy Mars" strategy. There's one quote that keeps coming back to me, "Our home is in the stars or not at all." If we're gonna sit here
on a hot, crowded planet with thermonuclear weapons, maybe we have hundreds of years, but we don't have thousands. If we're going to get off this planet and go someplace interesting, we're going to
have to recognize that we don't have the source code yet in Einstein, and it's very limiting, and we're going to have to actually say, "What is the source code?" And if it turns out that we can find
it, we're gonna have to be good stewards, and we're not going to do the same thing that we've been doing by handing the stuff over to leaders who don't take seriously the burdens of godlike powers
that we the technical people bestowed. So Brian, thanks for having me on, and it's a pleasure to interact with your audience.
Brian Keating: Eric, it's a pleasure to have you on the show. As always, you're welcome back anytime. I do love the fact that you made this promise back in early or late December of 2020, that year
that may it soon be forgotten.
Eric Weinstein: I didn't promise, I said I was gonna try. I said I was gonna try.
Brian Keating: That's right. Well you succeeded, you succeeded for sure, Eric. I want to thank you for your generosity of time, and spirit, and advice that you've given to me. I hope I can help to
serve you in this, wherever this project may take you. It's now out of your hands, it's into the world, and it's going to hopefully sprout many many delightful new discoveries for the benefit of all
mankind as our friend Alfred Nobel so warmly engendered upon the world. Eric, best of luck, congratulations. We'll do a part three next year on this date, on this auspicious date, and let it forever
be known as a day of famy, not infamy, for years to come in physics, if we can follow the lead of the generous, the mercurial, the genierrific, Eric Weinstein. Thank you so much, Eric. Enjoy the day,
and we look forward to seeing you on Joe Rogan... tomorrow? Or when will that podcast be out?
Eric Weinstein: I think so, L'Shana Haba'ah in the electron layer.
Brian Keating: Okay, inshallah. Goodnight everybody. Please do subscribe and like this podcast, we have Michio Kaku coming up. John Mather, winner of the 2006 Nobel Prizeâ [Video cuts]â Magnificent
ideas to the space, to make it safe for new ideas and for creativity, because we do have this one universe, this one life, and it is eminently precious. So for now, thanking you all, enjoy the rest
of your evening, and thanking you Eric, here's a musical outro from our friend Miguel Tully, proprietor of the Yeti Tears podcast, Spotify, and YouTube channel. Good night, everybody.
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VAPreMa-3 Time Tables - Smart Board Learning
Number and Number Sense
2.1 The student will
read, write, and identify the place and value of each digit in a three-digit numeral, with and without models;
identify the number that is 10 more, 10 less, 100 more, and 100 less than a given number up to 999;
compare and order whole numbers between 0 and 999; and
round two-digit numbers to the nearest ten.
2.2 The student will
count forward by twos, fives, and tens to 120, starting at various multiples of 2, 5, or 10;
count backward by tens from 120; and
use objects to determine whether a number is even or odd.
2.3 The student will
count and identify the ordinal positions first through twentieth, using an ordered set of objects; and
write the ordinal numbers 1st through 20th.
2.4 The student will
a) name and write fractions represented by a set, region, or length model for halves, fourths, eighths, thirds, and sixths;
b) represent fractional parts with models and with symbols; and
c) compare the unit fractions for halves, fourths, eighths, thirds, and sixths, with models.
Computation and Estimation
2.5 The student will
recognize and use the relationships between addition and subtraction to solve single-step practical problems, with whole numbers to 20; and
demonstrate fluency with addition and subtraction within 20.
2.6 The student will
estimate sums and differences;
determine sums and differences, using various methods; and
create and solve single-step and two-step practical problems involving addition and subtraction.
Measurement and Geometry
2.7 The student will
a) count and compare a collection of pennies, nickels, dimes, and quarters whose total value is $2.00 or less; and
b) use the cent symbol, dollar symbol, and decimal point to write a value of money.
2.8 The student will estimate and measure
a) length to the nearest inch; and
weight to the nearest pound.
2.9 The student will tell time and write time to the nearest five minutes, using analog and digital clocks.
2.10 The student will
a) determine past and future days of the week; and
b) identify specific days and dates on a given calendar.
2.11 The student will read temperature to the nearest 10 degrees.
2.12 The student will
a) draw a line of symmetry in a figure; and
b) identify and create figures with at least one line of symmetry.
2.13 The student will identify, describe, compare, and contrast plane and solid figures (circles/spheres, squares/cubes, and rectangles/rectangular prisms).
Probability and Statistics
2.14 The student will use data from probability experiments to predict outcomes when the experiment is repeated.
2.15 The student will
collect, organize, and represent data in pictographs and bar graphs; and
read and interpret data represented in pictographs and bar graphs.
Patterns, Functions, and Algebra
2.16 The student will identify, describe, create, extend, and transfer patterns found in objects, pictures, and numbers.
2.17 The student will demonstrate an understanding of equality through the use of the equal symbol and the use of the not equal symbol.
Touch the buttons below to find cool and fun tools!
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III Quantum Computation - Classical computation theory
1Classical computation theory
III Quantum Computation
1 Classical computation theory
To appreciate the difference b etween quantum and classical computing, we need
to first understand classical computing. We will only briefly go over the main
ideas instead of working out every single technical detail. Hence some of the
definitions might be slightly vague.
We start with the notion of “computable”. To define computability, one
has to come up with a sensible mathematical model of a computer, and then
“computable” means that theoretical computer can compute it. So far, any two
sensible mathematical models of computations we manage to come up with are
equivalent, so we can just pick any one of them. Consequently, we will not spend
much time working out a technical definition of computable.
be an integer. We want to figure out if
a prime. This is
clearly computable, since we can try all numbers less than
and see if it divides
This is not to o surprising, but it turns out there are some problems that are
not computable! Most famously, we have the Halting problem.
(Halting problem)
Given the code of a computer program, we want
to figure out if the computer will eventually halt. In 1936, Turing proved that
this problem is uncomputable! So we cannot have a program that determines if
an arbitrary program halts.
For a less arbitrary problem, we have
Given a polynomial with integer coefficients with many variables,
e.g. 2
y −
+ 1, does this have a root in the integers? It was
shown in 1976 that this problem is uncomputable as well!
These results are all for classical computing. If we expect quantum computing
to be somehow different, can we get around this problems? This turns out not to
be the case, for the very reason that all the laws of quantum physics (e.g. state
descriptions, evolution equations) are all computable on a classical computer
(in principle). So it follows that quantum computing, being a quantum process,
cannot compute any classical uncomputable problem.
Despite this limitation, quantum computation is still interesting! In practice,
we do not only care about computability. We care about how efficient we are at
doing the computation. This is the problem of complexity — the complexity of
a quantum computation might be much simpler than the classical counterpart.
To make sense of complexity, we need to make our notion of computations a
bit more precise.
(Input string)
An input bit string is a sequence of bits
where each
is either 0 or 1. We write
for the set of all
-bit string, and
. The input size is the length
. So in particular, if the input is
regarded as an actual number, the size is not the number itself, but its logarithm.
Definition (Language). A language is a subset L ⊆ B.
(Decision problem)
Given a language
, the decision problem is to
determine whether an arbitrary
x ∈ B
is a member of
. The output is thus 1
bit of information, namely yes or no.
Of course, we can have a more general task with multiple outputs, but for
simplicity, we will not consider that case here.
is the set of all prime numbers, then the corresponding decision
problem is determining whether a number is prime.
We also have to talk about models of computations. We will only give an
intuitive and classical description of it.
(Computational model)
A computational model is a process with
discrete steps (elementary computational steps), where each step requires a
constant amount of effort/resources to implement.
If we think about actual computers that works with bits, we can imagine a
step as an operation such as “and” or “or”. Note that addition and multiplication
are not considered a single step — as the number gets larger, it takes more effort
to add or multiply them.
Sometimes it is helpful to allow some randomness.
(Randomized/probabilistic computation)
This is the same as a usual
computational model, but the process also has access to a string
, r
, r
, ···
of independent, uniform random bits. In this case, we will often require the
answer/output to be correct with “suitably good” probability.
In computer science, there is a separate notion of “non-deterministic” com-
putation, which is different from probabilistic computation. In probabilistic
computation, every time we ask for a random number, we just pick one of the
possible output and follows that. With a non-deterministic computer, we simul-
taneously consider all possible choices with no extra overhead. This is extremely
powerful, and also obviously physically impossible, but it is a convenient thing
to consider theoretically.
(Complexity of a computational task (or an algorithm))
The com-
plexity of a computational task or algorithm is the “consumption of resources as
a function of input size n”. The resources are usually the time
T (n) = number of computational steps needed,
and space
Sp(n) = number of memory/work space needed.
In each case, we take the worse case input of a given size n.
We usually consider the worst-case scenario, since, e.g. for primality testing,
there are always some numbers which we can easily rule out as being not
prime (e.g. even numbers). Sometimes, we will also want to study the average
In the course, we will mostly focus on the time complexity, and not work
with the space complexity itself.
As one would imagine, the actual time or space taken would vary a lot on the
actual computational model. Thus, the main question we ask will be whether
T (n) grows polynomially or super-polynomially (“exponentially”) with n.
Definition (Polynomial growth). We say T (n) grows polynomially, and write
T (n) = O(poly(n)) = O(n
for some
, if there is some constant
, and some integer
and some integer
such that T (n) < cn
for all n > n
The other possible cases are exponential growth, e.g.
) =
, or
super-polynomial and sub-exp onential growth such as T (n) = 2
or n
log n
We will usually regard polynomial time proce sses as “feasible in practice”,
while super-polynomial ones are considered “infeasible”. Of course, this is not
always actually true. For example, we might have a polynomial time of
or an exponential time of 2
. However, this distinction of polynomial
vs non-polynomial is robust, since any computational model can “simulate” other
computational models in polynomial time. So if something is polynomial in one
computational model, it is polynomial in all models.
In general, we can have a more refined complexity classes of decision problems:
(i) P
(polynomial time): The class of decision problems having deterministic
polynomial-time algorithm.
(ii) BPP
(bounded error, probabilistic polynomial time): The class of decision
problems having probabilistic polynomial time algorithms such that for
every input,
Prob(answer is correct) ≥
The number
is sort of arbitrary — we see that we cannot put
, or
else we can just randomly guess a number. So we need something greater
, and “bounded” refers to it being bounded away from
. We could
with any other constant
with 0
< δ <
, and
is the
same. This is because if we have a
algorithm, we simply repeat the
times, and take the majority vote. By the Chernoff bound (a
result in probability), the probability that the majority vote is correct is
− e
. So as we do more and more runs, the probability of getting
a right answer grows exponentially. This can be bigger than an 1
− ε
a suitably large
. Since
times a polynomial time is still polynomial
time, we still have a polynomial time algorithm.
These two are often considered as “classically feasible computations”, or “com-
putable in practice”. In the second case, we tolerate small errors, but that is fine
in practice, since in genuine computers, random cosmic rays and memory failures
can also cause small errors in the result, even for a deterministic algorithm.
It is clear that
is contained in
, but we do not know about the other
direction. It is not known whether
are the same — in general, not
much is known about whether two complexity classes are the same.
(Primality testing)
be an integer. We want to determine if it
is prime. The input size is
. The naive method of primality testing is to
test all numbers and see if it divides
. We only need to test up to
, since if
has a factor, there must be one below
. The is not polynomial time, since
we need
N = 2
log N
operations, we see that this is exponential time.
How about a probabilistic algorithm? We can choose a random
k < N
, and
see if
. This is a probabilistic, polynomial time algorithm, but it is
not bounded, since the probability of getting a correct answer is not >
In reality, primality testing is known to be in
(1976), and it is also
known to be in P (2004).
Finally, we quickly describe a simple model of (classical) computation that we
will use to build upon later on. While the most famous model for computation
is probably the Turing machine, for our purposes, it is much simpler to work
with the circuit model.
The idea is simple. In general, we are working with bits, and a program is a
→ B
. It is a mathematical fact that any such function can be
constructed by combinations of boolean
gates. We say that
this is a universal set of gates. Thus a “program” is a specification of how to
arrange these gates in order to give the function we want, and the time taken by
the circuit is simply the number of gates we need.
Of course, we could have chosen a different universal set of gates, and the
programs would be different. However, since only a fixed number of gates is
needed to construct
from any universal set, and vice versa,
it follows that the difference in time is always just polynomial.
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What is an average VIX? - Mean reversion / Mode reversion
Dec 10, 2020
VTS Community,
It's been a rough 2020 all around the world and we're not quite out of it just yet. Cases of Covid are still very high in many countries so we still need to be diligent and careful. However, we are
starting see some light at the end of the tunnel. Medical outcomes of those who contract the virus are much improved, and we have several promising vaccines on the horizon. The world is on the mend!
Specific to the volatility markets, the VIX index is also on the mend and is threatening its first close under 20 since before all this mess started.
Make no mistake though, volatility is still very elevated right now. When I say that, and I have been in articles, videos, and on Twitter, inevitably I get people who say:
"But isn't the long term average VIX around 20?
Yes, that's true. We can see below that since 1990, the mean VIX value is 19.46
Now 2020 has obviously seen very elevated levels, currently ranking the 3rd highest in a calendar year since 1990, but still, we're near 20 now and that's very close to the long term average.
So why do I say it's still very elevated? It's because there are several ways to calculate and represent what's average.
1) Mean: The average of a set of numbers
To calculate the mean, it's just a straight average of a set of numbers. We add up all the values and divide by how many values there are. The problem with looking at an average on the VIX though is
that it has closing values that range from 9.14 on November 3rd, 2017, all the way up to 82.69 on March 16th, 2020.
Those high end values over 50 really pull up the mean and give a very skewed view of what the true average really is. So we can definitely improve on the robustness of our measure beyond the mean.
There's two much better ways to measure what the true average VIX is:
2) Median: The middle number in a set of numbers
To calculate the median, we just order a data set and find the actual middle number in the sequence. So as a quick example, if we have 5 values (3, 3, 4, 6, 9) the median would be the middle value of
When dealing with the VIX and its very wide range of values, the median is a much better representation of the average than the mean. I'd rather know that 50% of all VIX values have been below 17.44,
and 50% of all values have been above 17.44.
3) Mode: The value that occurs most frequently in a data set
To calculate the mode for the VIX, I prefer to do it in "handles." For example, if the VIX is 20.25, we call that a 20 handle. If it's 12.46, we say that's a VIX handle of 12.
Here's a chart showing all VIX handle's going back to 1990. Above 30 I grouped them in 5's because there are far less occurrences that high.
We can see the mode of the VIX is a 12 handle at 12.42, followed by the 13's, and then the 14's and 11's. This mode of 12.42 is probably the best measure of what the true "average" VIX really is, and
I would say that's especially true for trading purposes.
- If you're the type of trader that's always expecting the VIX to "mean revert" to 20, you've likely burned through significant capital over the years hedging and trying to time volatility
- On the other hand, traders who expect the VIX and volatility markets in general to "mode revert" back closer to the 12's and 13's have a higher probably of riding equity trends higher, and
finding clever ways of shorting volatility lower.
Warning: That last one, "shorting volatility" does require a lot of experience because it's not a trade you can just set and forget. It requires a really good understanding of position sizing and
risk management and should only be taken on by those who know the risks and how to manage them.
Always keep in mind that the VIX index and the volatility complex in general is always seeking out lower values, that's the natural state of things.
• The VIX futures spend the majority of their time in Contango.
• The 9-day VIX9D index spends most of it's time below the 30-day VIX index
• 30-day implied volatility is normally well above 20-day realized volatility
• And long Volatility ETPs like VXX and UVXY do spend most of their days bleeding downward
Just using that simple example from above: 3, 3, 4, 6, 9
Mean, the average is 5
Median, the middle number is 4
Mode, the most common number, is 3
While it's technically true that the average VIX is around 20, in a practical sense the other ways to measure average (median and mode) are far superior and give a much better sense of what the
true average is.
With the VIX breaking below 20 for the first time in many months, I still say volatility in this market is elevated and potentially has a lot further to fall. That will of course depend on how the
world handles the vaccinations, and whether we have any other surprises in store for us, but it's important to always remember that volatility seeks out lower levels. As a trader, this is valuable
information to have within your overarching investing philosophy.
This video from 2017 covers it as well, but it was time for a refresh. That feels like another lifetime ago. 2017, the lowest volatility year in history. Given what 2020 was like, wouldn't it be nice
to see a repeat of 2017 in 2021? One can dream...
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Simple Math Tricks You Weren’t Taught at School – GE STEAM
Simple Math Tricks You Weren’t Taught at School
Purpose, Scope or Aim of the OER
This OER supports teachers in teaching their students to use mathematics in a much easier and entertaining way. A fun way to motivate, as mathematics can sometimes be a bit heavy and difficult for
students to understand.
Short Description of the methods or approaches used in this OER
This OER is a video that offers a few little tricks to students when learning mathematics. It is a video that shows that mathematics is not only about adding and multiplying, but also about problem
solving. This video will teach us more fun maths than we are used to studying.
Step-by-step instructions for teachers to use OER
Make sure you have a good internet connection and a computer/laptop to play the video successfully.
1. Before playing this OER, hand out a blank sheet of paper to all students, or they can use their own notebooks.
2. On the board write 10 types of mathematical operations and give the students 20 minutes to solve them in the traditional way.
3. Once everyone has solved the operations, correct the results of these operations together.
4. When you have corrected the results together, then play the video.
5. Once the video has finished playing, ask the students to apply these tricks to the same operations completed earlier.
6. Check if these little tricks match the previous results.
Once they have applied the tricks to the mathematical operations, ask to the class the following questions.
1. What do you think of this way of learning mathematics?
2. Do you think it is a fun and entertaining way to learn mathematics?
3. Would you like to apply it in your future mathematical operations?
4. Do you prefer the traditional way or applying these tricks?
5. Did you know any of these tricks before?
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|
Computational Mathematics
• Computational Mathematics: theoretical and computational tools for the applied Sciences and Engineering
• Scientific Computing: algorithms for continuous and discrete mathematical models, parallel and distributed computing, numerical simulation
• Numerical Analysis: numerical methods for ordinary and partial differential equations, approximation of data and functions, numerical linear algebra
• Mathematical Analysis and Modeling: ordinary and partial differential equations, functional analysis, variational models and methods, calculus of variations
• Biomathematics: mathematical and numerical modeling in Biology, Physiology and Neurosciences
• Mathematical Physics: kinetic theory, granular media, statistical mechanics, diffusion equations, hyperbolic systems, socio-economic modeling
• Optimization and Operational Research: Optimization methods and algorithms, convex, integer, quadratic, and nonlinear programming, Control Theory
|
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Analysis on Homogeneous Spaces
by Ralph Howard
Publisher: Royal Institute of Technology Stockholm 1994
Number of pages: 108
The main goal of these notes is to give a proof of the basic facts of harmonic analysis on compact symmetric spaces and then to apply these to concrete problems involving things such as the Radon and
related transforms on these spaces.
Download or read it online for free here:
Download link
(780KB, PDF)
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urbin: Unifying Estimation Results with Binary Dependent Variables
Calculate unified measures that quantify the effect of a covariate on a binary dependent variable (e.g., for meta-analyses). This can be particularly important if the estimation results are obtained
with different models/estimators (e.g., linear probability model, logit, probit, ...) and/or with different transformations of the explanatory variable of interest (e.g., linear, quadratic,
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Version: 0.1-14
Depends: R (≥ 2.14.0)
Suggests: sampleSelection (≥ 0.7-0), maxLik (≥ 1.1-2), mfx (≥ 1.1), mlogit (≥ 0.3-0), MASS (≥ 7.3-50), mvProbit (≥ 0.1-8), knitr, stargazer
Published: 2024-10-01
DOI: 10.32614/CRAN.package.urbin
Author: Arne Henningsen [aut, cre], Geraldine Henningsen [aut]
Maintainer: Arne Henningsen <arne.henningsen at gmail.com>
License: GPL-2 | GPL-3 [expanded from: GPL (≥ 2)]
URL: https://r-forge.r-project.org/projects/urbin/
NeedsCompilation: no
Materials: NEWS
CRAN checks: urbin results
Reference manual: urbin.pdf
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Package source: urbin_0.1-14.tar.gz
Windows binaries: r-devel: urbin_0.1-14.zip, r-release: urbin_0.1-14.zip, r-oldrel: urbin_0.1-14.zip
macOS binaries: r-release (arm64): urbin_0.1-14.tgz, r-oldrel (arm64): urbin_0.1-14.tgz, r-release (x86_64): urbin_0.1-14.tgz, r-oldrel (x86_64): urbin_0.1-14.tgz
Old sources: urbin archive
Please use the canonical form https://CRAN.R-project.org/package=urbin to link to this page.
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• Delaram Moradi, M. Math. student, started September 2024.
• Mazen Khodier, M. Math. student, started January 2024.
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postdoc at the University of Winnipeg.
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• Sajed Haque, M. Math student, started Fall 2015, finished August 2017. Thesis title: "Discriminators of Integer Sequences". Readers: Kevin Hare and John Watrous. Now a Ph. D. student at Waterloo,
under supervision of Naomi Nishimura.
• Taylor Jonathan Smith, M. Math student, started Fall 2015. Finished July 2017. Thesis title: "Properties of Two-Dimensional Words". Readers: Eric Blais and Lila Kari. Now a Ph. D. student at
Queen's University, under supervision of Kai Salomaa.
• Chen Fei Du, M. Math student, started Spring 2013. Did not complete.
• Daniel Goc, M. Math student, started Fall 2011. Thesis, completed August 2013, "Automatic sequences and decidable properties: implementation and applications". Readers: Kevin Hare and Timothy
Chan. Now a Ph. D. student at Queen's University.
• Shuo Tan, M. Math student, started Fall 2011. Thesis, completed August 2013, "Two results on words". Readers: Bin Ma and Ming Li.
• Hamoon Mousavi, M. Math. student, started Fall 2011. Thesis, completed August 2013, "Repetitions in words". Readers: Jonathan Buss and Larry Cummings. Now works for Google.
• Luke Schaeffer, M. Math. student, started Fall 2011. Thesis, completed August 2013, "Deciding properties of automatic sequences". Readers: Jason Bell and Shai Ben-David. Currently a faculty
member at the University of Waterloo.
• Alex Leong, M. Math. student, started Fall 2010; finished Fall 2011. Thesis: Variations on the Erdos Discrepancy Problem
• Thomas Ang, M. Math. student, started Fall 2008, finished May 2010. Thesis: "Problems Related to Shortest Strings in Formal Languages"
• Zhi Xu, Ph. D. student, started Spring 2007. Defended thesis, August 2009. Held postdoc with Lila Kari at the University of Western Ontario. Currently employed by Google Waterloo. Thesis, The
Frobenius Problem in a Free Monoid.
• Dalia Krieger, Ph. D. student, started Fall 2004. Defended thesis, 2008. Thesis, Critical Exponents and Stabilizers of Infinite Words. Held postdoc in Israel. Currently employed in software
company in Israel.
• Narad Rampersad, M. Math. student, thesis option, Spring 2004, "Infinite Sequences and Pattern Avoidance". Completed his Ph. D., "Overlap-Free Words and Generalizations", Fall 2007. Held postdocs
at at U. Winnipeg and University of Liège, Belgium. Currently a professor at the University of Winnipeg.
• Bryan Krawetz, M. Math. student, thesis option, Winter 2004. Monoids and the state complexity of root(L)
□ In ps format
□ In pdf format
Currently works for Google Waterloo.
• Lesley Macpherson, M. Math., thesis option, 2002. Grey Level Visual Cryptography for General Access Structures.
• Keith Ellul, M. Math. thesis option, 2002; finished 2004. Descriptional Complexity Measures of Regular Languages.
• Andrew Martinez, M. Math. student, thesis option, 2002. Topics in Formal Languages: String Enumeration, Unary NFA's, and State Complexity.
• Troy Vasiga, Ph. D. student, began Fall 2000, finished August 2008 (part-time). Error Detection in Number-Theoretic and Algebraic Algorithms. Currently teaching faculty at the University of
• Michael Domaratzki, M. Math., thesis option, 2001. Minimal covers of formal languages. pdf
□ Currently a professor at University of Manitoba (tenured)
• Ming-wei Wang, M. Math., thesis option, 1999. Subword complexity and a matrix inequality. Ph. D., Spring 2004, Periodicity and repetition in combinatorics on words. Currently works for Microsoft
in Redmond, Washington.
• David Swart, M. Math., thesis option, 1998. Calculating the ith letter of the nth word in a DOL-sequence.
□ Currently employed at Northern Digital, Waterloo, Ontario.
For the programs from his thesis, see here.
• Dave Hamm, M. Math., thesis option, 1998. Contributions to Formal Language Theory: Fixed Points, Complexity, and Context-Free Sequences
□ Currently employed at Oracle Corp., working on www.crmondemand.com, living in Vancouver area.
• Ian Matthew Glaister, M. Math., thesis option, 1995. Automaticity and Closure Properties. Currently Senior Software Developer at Fidelity National Information Services, Jacksonville Florida.
• Qi Xiang Zhang, M. Math., essay option, 1994.
• Peter Wei Liang Liu, M. Math., essay option, 1994, "Efficient Recognition of Integer Sequences"
• Eric Rowland
• Thomas Stoll
• Emilie Charlier
• Tim Smith
• Lukas Fleischer
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Room Capacity
Room Capacity
The Room capacity for Scottish Country Dancing on a floor of a given size is calculated using the Size of the set to determine how many Sets can be accommodated in the space available and the number
of Dancers in each.
In Longwise sets with comfortable spacing for most dances, the adjacent Dancers in the Side lines will be about 1yd (90cm) apart and the Partners about 2yd (1.8m). The spacing between The sets across
the room should also be about 2yd (1.8m).
Room Capacity with Couples in Longwise Sets
The diagram shows a rectangular floor (16yd (14.4m) wide by 17yd (15.3m) long) with
disposed as for
Longwise sets
the leftmost line (A) shows the unnumbered
having just come onto the floor at the announcement of the dance;
the next line shows them
Made up
in four
(B1-B4), each of four
the next line shows them
Made up
in five
(C1-C5), each of three
the rightmost line shows them
Made up
in three
(D1-D3), each of five
Obviously, the diagram illustrates the situation at different times; for a real dance, the lines of Couples would all be as in line A soon after the announcement of the dance; after counting by the
Top man in each line, they would all be as in line B for 4 Couple sets or as in line C for 3 Couple sets or as in line D for 5 Couple sets and similarly for 6 Couple sets and 7 Couple sets.
This layout shows the minimum floor size (16yd (14.4m) wide by 17yd (15.3m) long) which will just accommodate four lines of four 4 Couple sets with the recommended spacing; i.e., 128 Dancers in all.
When, as here, The sets are of a size which fits the floor layout perfectly, each Dancer requires a little over 2 square yards. For 3 Couple sets, there can be only five Sets in each line on a floor
of these dimensions and, for 5 Couple sets, only three, i.e., only 120 Dancers with these formats; for the much rarer 6 Couple sets, there can be only 96 Dancers and, for 7 Couple sets, 112.
A small increase in width or length of the floor space allows a little extra space between adjacent Sets but makes no difference to the number of Dancers which can be accommodated. The width must
increase by 4yd to allow an extra line of Sets and the length by 4yd to allow an extra Set in each line; as a corollary, reducing the width by a small amount loses a complete line of Sets and
reducing the length by a small amount loses a complete row.
Since the vast majority of dances in any programme will be for four Couples in Longwise sets, it is sensible to concentrate on these when calculating the capacity of a given floor area. The following
table with Length and Width in yards (~90cm) gives details of some example optimum sizes:
Length Width Sets Dancers
17 4 1x4 32
17 8 2x4 64
17 12 3x4 96
17 16 4x4 128
17 20 5x4 160
13 12 3x3 72
9 12 3x2 48
5 12 3x1 24
The next table can be used to determine the number of Dancers which can be accommodated on a rectangular floor of a given length and width (both in yards). If the exact length is shown, work along
that row; if not, choose the row above. If the exact width is shown work down that column; if not, choose the column to the left. The number of Dancers which the floor can accommodate will be at the
intersection of this row and this column. For example, a floor 15yd square could accommodate 72 Dancers in 4 Couple sets.
Width ► 4 8 12 16 20 24
Length ▲
A detailed analysis for the rarer Types of set would produce a similar result for all but the smallest of rooms. As the diagram indicates, Longwise sets of 3 Couples and 5 Couples fit almost as well
into the lines as do 4 Couple sets and for some room lengths will actually be better. The Square set maps quite well onto a 4 Couple, Longwise, set in its share of the space between Sets and
similarly, the Triangular set onto a 3 Couple, Longwise, set and so these need no special attention. Most Round the room sets and Large circular sets can be danced in concentric Sets if need be in
order to make the best use of a large hall.
Although many will remember ballrooms of the mid-20^th century in which the Dancers were almost shoulder to shoulder in the Sidelines, the above analysis applies to a modern Scottish Country Dance
event where the space available should be at least the minimum to allow comfortable dancing. However, this does mean that, at a sacrifice of some comfort and with a preparedness of the Standing
Dancers to move apart to allow Dancers to Travel between them, a slightly smaller width or length of floor can accommodate the number appropriate to the next larger size in the table.
The analysis is also appropriate to a Scottish Country Dance event where one would expect that, except perhaps for a handful of spectators who have some disability, all will wish to be on the floor
in at least the first few dances on the programme. However, at a Wedding Ceilidh or some such, many, or even most, of the assembled company may have no intention to dance; it is important for the
organizer to make a good estimate of this fraction for his/her event.
Finally, these space requirements are for the dancing, only; additional space will be required for musicians and their equipment, seating, with tables if necessary, and for any circulatory activities
by non-dancing attendees which must not be allowed to encroach on the area where the dancing is in progress.
Links To Pages Related To 'Room Capacity'
Types Of Sets
Back to the top of this Scottish Country Dancing 'Room Capacity' page
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A SAT Attack on the Erdos Discrepancy Conjecture
A SAT Attack on the Erdos Discrepancy Conjecture
April 12, 2014 8:55 AM Subscribe
The singularity has arrived! Huzzah!
Actually this is an interesting problem. About the best you could do is have independent parties create proof checkers that walked through the solution. Which when you get down to it may have a
higher accuracy rate for less voluminous proofs as well.
posted by Tell Me No Lies at 9:07 AM on April 12, 2014 [3 favorites]
Now, here's a question.
Would it be possible to create a sort of AI conjecture-maker entity? An AI that knows how all these various parts relate, and then can create conjectures, and the put them up for proof to a
proof-checking algo/AI?
I don't think we'd still be able to create AIs that would be able to create new forms of mathematics, at least not yet, but surely there must be a way to combine formal statements of patterns in a
given constrained domain along with the condition for truth or falseness?
I'm sure more complicated conjectures might not be able to be made at this point, but surely there must be a way for conjectures to be created by machine?
posted by symbioid at 9:21 AM on April 12, 2014
Oh, and speaking as someone who watches things go wrong with software a lot I would be leery of accepting a solution by a computer "using a different approach" as proof. The programmer's intent might
be different but it would not surprise me a whit to see the computer end up largely copying the first proof.
If you want to know that things went right you want to forget the content and hammer on the form. We can't understand the overall proof but we should be able to understand each individual step quite
posted by Tell Me No Lies at 9:21 AM on April 12, 2014 [1 favorite]
Meh. Work is fine, articles overblown. The work here was encoding the conjecture into an instance of the well-studied SAT problem. Solvers for SAT problems have been around...
posted by save alive nothing that breatheth at 9:27 AM on April 12, 2014 [1 favorite]
I've heard that the computer-proof of Fermat's Last Theorem was so long and complicated that no one actually knows if it is correct.
posted by Chocolate Pickle at 10:04 AM on April 12, 2014
There's also no computer proof -- or any proof, yet -- of the Erdos discrepancy conjecture, despite the linked article incorrectly saying so.
posted by escabeche at 10:53 AM on April 12, 2014
There's also no computer proof -- or any proof, yet -- of the Erdos discrepancy conjecture, despite the linked article incorrectly saying so.
Could you expand on that a bit?
posted by Tell Me No Lies at 11:08 AM on April 12, 2014
The conjecture says that, for every C, every sufficiently long sequence has a subsequence with discrepancy at least C. The new proof shows that this is true for C = 2. They are working on proving it
for C = 3 along similar lines. But this is well short of proving it's true for ALL C, which is what the conjecture proposes.
posted by escabeche at 11:24 AM on April 12, 2014 [3 favorites]
C'mon, escabeche, you know that the only important numbers are 0, 1, and ∞. 2 is practically the same thing as ∞.
-space joke.
posted by benito.strauss at 12:25 PM on April 12, 2014 [5 favorites]
I think the real problem will come not when we can't follow the proof of a conjecture, but when we can't grasp the computer-made conjecture itself.
posted by jamjam at 12:37 PM on April 12, 2014 [1 favorite]
I read once to bootstrap Lisp someone wrote a minimal Lisp interpreter in Lisp and compiled it to assembly by hand.
posted by save alive nothing that breatheth at 4:06 PM on April 12, 2014
No one really understands the exact circuit layout on the CPU you're using, which was spat out by a similar SAT solver, but it seems to work just fine.
Pesky humans demanding compact forms for boolean equations, harrumph.
posted by RobotVoodooPower at 4:20 PM on April 12, 2014 [2 favorites]
I somehow feel smarter just reading the comments. Please keep it up, you brilliant bastards.
not kidding
posted by nevercalm at 6:29 PM on April 12, 2014 [1 favorite]
No one really understands the exact circuit layout on the CPU you're using, which was spat out by a similar SAT solver, but it seems to work just fine.
Well, unless it's an Intel CPU -- they still do everything by hand. Also, there aren't too many SAT-based synthesis packages used for real problems. Logic synthesis is binate, so minimizing SAT has a
bad habit of blowing up for anything larger than a trivially sized circuit. It's useful for automated validation, though.
posted by spiderskull at 11:21 PM on April 12, 2014 [1 favorite]
So I worked in pure maths for quite some time where I did quite a lot of relatively long proof by hand and later I worked on a automated computer proof system with several others, and both approaches
to proving things bothered me for different reasons.
For the hand proof, in some sense you never really prove anything. You give a convincing argument that outlines the major logical steps that a proof of the result would have. But you're still writing
most of it in English with a bit of mathematical notation, which while much more structured and precise than most any other language, is still not a formal proof. A formal proof uses axioms, formal
rules of inference and basically syntactic transforms of symbolic sentences to establish truth. Maybe a few logicians work this way, but 99% of pure mathematicians don't.
The computer proof has the fundamental problem that from a false statement, anything can be proved. So if your automated math prover has a software flaw (who ever heard of code having bugs?) that
means that occasionally it can generate the sentence "false" without good reason, perhaps buried very deeply within very large proof, then it'll be able to get any result you like proved.
Neither approach is ideal. But in both cases it's sensible to get someone or something to independently check the working.
posted by drnick at 1:15 AM on April 13, 2014 [2 favorites]
2 is practically the same thing as ∞.
You say so, but I just came back from a conference where one of the speakers had a good L_2 bound but the L_∞ bound seems totally out of reach, and I'm all, you should do L_4, that's what you do when
you can't get an L_∞ bound but you don't want to just give up, and he was all, good point, 4 isn't ∞ but it sure isn't 2. In conclusion, harmonic analysis is a land of contrasts.
posted by escabeche at 6:11 PM on April 13, 2014 [2 favorites]
I knew no good would come from letting COLOSSUS have a data link to GUARDIAN.
posted by radwolf76 at 8:34 AM on April 14, 2014
Well, unless it's an Intel CPU -- they still do everything by hand. Also, there aren't too many SAT-based synthesis packages used for real problems.
I was just going to make a similar comment — but I don't think it's strictly true any more. As I understand it, they currently do the largest scale layout by hand, but many of the portions that go
into it are laid out with SAT solvers with constraints to make the results something humans can reasonably stitch together.
Also I seem to recall AMD CPUs and most ARM IP modules are built with automated layout.
Alas, I'm having difficulty finding solid references to back my understanding up.
posted by atbash at 11:21 AM on April 15, 2014
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Introductory Linear Algebra - Wikibooks, open books for an open world
Some elementary knowledge of algebra is assumed. Unit 1 of the
wikibook should be more than enough to learn these elementary knowledge.
This book is part of a series on Algebra:
This book serves as a mild introduction to linear algebra, and is not too proof-heavy and proof-based. Most topics covered are useful for applications in disciplines other than mathematics. Thus,
this book is probably more suitable for non-math majors. For math majors and people who want to study proof-based linear algebra, see Linear Algebra wikibook. This book should not be a prerequisite
of the Linear Algebra wikibook (but may help understanding it), despite similar topics are covered, since the topics are discussed in quite a different way, and some definitions may differ.
Table of Contents
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Percents | 7th grade math - ByteLearn.com
In mathematics, a percentage is a number or ratio that can be expressed as a fraction of 100. If we have to calculate the percent of a number, divide the number by the whole and multiply by 100.
Hence, the percentage means, a part per hundred. It is represented by the symbol “%”. Percentages are used like fractions and decimals, as ways to describe parts of a whole. While learning percent
calculations, one should know that the whole is considered to be made up of a hundred equal parts.
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Math Placements
Under the Magis Core Curriculum, all students take at least one mathematics course and may opt to take an additional mathematics course for Magis core credit. There are a wide variety of courses to
choose from, although many majors and programs require specific mathematics courses.
Beginning mathematics classes and sequences of classes are described below, followed by the beginning mathematics requirements or recommendations by major. Students are expected to enroll in the
highest numbered course for which his or her high school preparation allows.
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High-Order Verified Solutions of the 3D Laplace Equation
For many practical problems, numerical methods to solve partial differential equations (PDEs) are required. Conventional finite element or finite difference codes have a difficulty to obtain precise
solutions because of the need for an exceedingly fine mesh which leads to often prohibitive CPU time. While conventional methods exhibit such a difficulty, some practical problems even require
solutions guaranteed. The Laplace equation is one of the important PDEs in physics and engineering, describing the phenomenology of electrostatics and magnetostatics among others, and various
problems for the Laplace equation require highly precise and verified solutions.
We present an alternative approach based on high-order quadrature and a high-order finite element method utilizing Taylor model methods. An n-th order Taylor model of a multivariate function f
consists of an $n$-th order multivariate Taylor polynomial, representing a high order approximation of the underlying function f, and a remainder error bound interval for verification, width of which
scales in (n+1)-st order.
The solution of the Laplace equation in space is first represented as a Helmholtz integral over the two-dimensional surface. The latter is executed by evaluating the kernel of the integral as a
Taylor model of both the two surface variables and the three volume variables inside the cell of interest. Finally, the integration over the surface variables is executed, resulting in a local Taylor
model of the solution within one cell. Examples of the method will be given, demonstrating achieved accuracy with verification.
S. Manikonda, M. Berz, K. Makino, Transactions on Computers 11,4 (2005) 1604-1610
Click on the icon to download the corresponding file.
Download Adobe PDF version (1054025 Bytes).
Go Back to the reprint server.
Go Back to the home page.
This page is maintained by Kyoko Makino. Please contact her if there are any problems with it.
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Cramér-Rao bound
From Scholarpedia
Calyampudi Radhakrishna Rao (2008), Scholarpedia, 3(8):6533. doi:10.4249/scholarpedia.6533 revision #137303 [link to/cite this article]
Cramér-Rao bound stands for an inequality that is the basis of a method for determining a lower bound to the variance of an estimate of a deterministic parameter.
Let \(f(x, \theta)\) be the probability density at observed data \(x\!\ ,\) where \( \theta = ( \theta_1,\cdots, \theta_p ) \) is an unknown p-vector parameter. The \(p \times p\) information matrix
on \(\theta\) is defined by \(I(\theta) = (i_{rs})\) where \(i_{rs} = E\left[\left(\frac{\partial \log f}{\partial \theta_r}\right)\left(\frac{\partial \log f}{\partial \theta_s}\right)\right]\) and
\(E\) stands for expectation.
Let \(T(x) = (T_1(x),\cdots, T_p(x))\) be an unbiased estimator of \(\theta = (\theta_1,\cdots,\theta_p)\ ,\) and \(C(T) = (j_{rs}), j_{rs} = E(T_r - \theta_r) (T_s - \theta_s)\ ,\) is the covariance
matrix. The Cramér-Rao bound in a general form is
The \(p \times p \) matrix \( C(T) - I^{-1}\) is non-negative definite.
In case \(\theta\) is a one dimensional parameter, the bound becomes \[\tag{1} V(T) \ge 1/I \]
where \(I = E\left[\left(\frac{\partial \log f}{ \partial \theta}\right)^2\right]\ .\) More generally, if \(E(T_i) = g_i(\theta)\) and \(G(\theta)\) is the matrix with the \((r,s)\) term as \(\frac{\
partial g_r(\theta )} {\partial \theta_s}\) then (Bull. Cal. Math. Soc., 1945)
\[C(T) - G I^{-1} G' \] is nonnegative definite
The result holds even when I is singular with the inverse of I replaced by a generalised inverse. Denoting \(I^{-1}=(i^{rs})\) and using the multiparameter CRB, \(C(T)-I^{-1}\) is nonnegative
definite, we have the CRB for a single parameter when there are other parameters \[ V(T_r)=E(T_s-\theta_r)^2\ge i^{rr}\ge i_{rr} \] so that the lower bound is possibly higher for an unbiased estimate
of \(T_r\) when other parameters are unknown. Using the general CRB, \(C(T)-GI^{-1}G^\prime\) is nonnegative definite \[ M(T)=E[(T-\theta)^\prime(T-\theta)]=(\theta-g(\theta))^\prime(\theta-g(\
theta))+G^\prime I^{-1}G \] which is CRB for the mean covariance error of \(T=(T_1,\ldots,T_r)\) as an estimate of \(\theta=(\theta_1,\ldots,\theta_p)\) and \(\theta-g(\theta)=(\theta_1-g_1(\theta),\
ldots,\theta_p-g_p(\theta))\) is the bias (Rao, 1952).
The origin of Cramér-Rao Bound (CRB) as reported in the newspaper Times of India, dated December 31, 1988, (with the heading:The top ten greatest contributions to Indian science) is as follows. “At
the young age of 24, Calyampudi Radhakrishna Rao was giving a course on estimation to Master’s students of Calcutta University. He proved in his class a result first obtained by R.A. Fisher regarding
the lower bound for the variance of an estimator for large samples. When a student asked, “why don’t you prove it for finite samples?”, Rao went back home, worked all night and next day proved what
is now known as Cramér-Rao inequality for finite samples.” About the same time, the CRB for one parameter was reported in a paper by Fréchet (1943).
The CRB is useful in finding whether a given estimator has the minimum variance or how close is it to the best possible one. For instance, if the sample is from a Normal distribution, \(N( \mu, \
sigma^2)\ ,\) then \[ I = \begin{pmatrix} \frac{n}{\sigma^2} & 0 \\ 0 & \frac{n}{2 \sigma^4} \end{pmatrix} \] and \[ I^{-1} = \begin{pmatrix} \frac{\sigma^2}{n} & 0 \\ 0 & \frac{2 \sigma^4}{n} \end
{pmatrix} \]
If \( \overline{x} = \frac{ x_1+x_2+ \ldots +x_n }{ n } \) is an estimator of \(\mu\ ,\) then \[ V \left ( \overline{x} \right ) = \frac{\sigma^2}{n} = CRB \]
so that is the best possible estimator of the parameter \(\mu\) in terms of variance.
If \( s^2 =\frac{\left(x_1 - \overline{x}\right)^2 + \ldots + \left(x_n-\overline{x} \right)^2}{n-1} \ ,\) is an estimate of \(\sigma^2\ ,\) then \[ V \left( s^2 \right) = \frac{2 \sigma^4}{n-1} > \
frac{2 \sigma^4}{n} (CRB). \] However, the ratio \(\frac{CRB}{V \left( s^2 \right)} =\frac{n}{n-1}\) which is close to 1 as \(n\) becomes large.
CRB, although originally introduced in estimation theory, has found many applications in statistical inference and other areas. It has been found useful in proving certain propositions in limit
theorems, asymptotic inference, decision theory, signal processing and density estimation. Van Trees, in his book "Detection, Estimation and Modulation Theory" (Van Trees, 1968)presented a global CRB
\[\tag{2} E(x-E(x|y))^2 \ge \frac{1}{E\left[\left(\frac{\partial}{\partial x}\right) \log p(x,y)\right]^2} \]
where \(p(x,y)\) is the joint density of \((x,y)\ .\) The above inequality and its generalization to the multivariate case have been used in bounding Bayes risk (Bobrovsky et al., 1987).
An interesting result due to A.J. Stam (Stam, 1959) is the derivation of Weyl-Heisenberg uncertainty principle in physics using a specific version of the CRB. Further applications in physics of CRB
and Fisher information as a concept underlying the well known physical theories can be found in the book by B. Roy Frieden, "Physics from Fisher Information" (Frieden, 1998).
Recent developments are Quantum Cramér-Rao Bound in the estimation of manifolds in Quantum Physics, by Brody and Houghston (1998) and the concept of Cramér-Rao Functional based on Cramér-Rao Bound by
Mayor Wolf (1990). CRB has been extended to estimation of "manifolds" as "complexified and intrinsic" CRB and used in signal processing.
• Bobrovsky, Wolf and Zakai. Ann. Statist. 15, 1421-1438, 1987
• DC Brody and LP Hughston. Proc. Roy. Soc., 454, 2445-2475, 1998.
• M Fréchet. Revue Inst. de Stat., 11, 182-205, 1943.
• B Roy Frieden. Physics from Fisher Information. Cambridge University Press, 1998
• CR Rao. Advanced Statistical Methods in Biometric Research, Wiley, 1952.
• ST Smith. IEEE Transactions on Signal Processing, 1597-1609, 1610-1630, 2005.
• AJ Stam. Information and Control 2, 101-112, 1959
• E Mayor Wolf. Ann. Prob., 18, 840-850, 1990
• Bull. Cal. Math. Soc. 37, 81-91, 1945
• Van Trees. Detection, Estimation and Modulation Theory, Part 1. Wiley, 1968
Internal references
Further reading
• A Bera, ET Interview with CR Rao, Econometric Theory (2003), 19, 329-398.
• MH Degroot, A conversation with CR Rao, Statistical Science (1987), 53-67.
• CR Rao, Linear Statistical Inference and its Applications, Wiley, 1973.
• T Soderstrom and P Stoica, System Identification, Prentice Hall, 1988.
External links
See also
Rao-Blackwell theorem, Fisher-Rao theorem, Second order efficiency
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Machine Learning From Zero to GPT in 40 Minute
TLDRThis video tutorial offers a comprehensive walkthrough on constructing a GPT-like model, exploring neural networks' relevance to various fields and their interaction with the human brain.
Starting from the basics, it progresses to advanced concepts like perceptrons, optimization problems, and the implementation of deep learning tools. The tutorial culminates in generating cat poems,
symbolizing the model's capability to produce creative content, while also discussing the challenges and potential of AI in understanding and mimicking complex patterns.
• 🌟 The video provides a tutorial on building a GPT-like model, emphasizing neural networks' relevance to various fields, including neuroscience.
• 🔍 It assumes zero knowledge in machine learning and aims to explain concepts gradually, using programming and analogies for better understanding.
• 💡 The script introduces the concept of intelligence as predicting outcomes and compares different approaches, such as using IF-ELSE statements and perceptrons.
• 📚 It explains the importance of numpy for simplifying calculations with multiple inputs and weights in machine learning models.
• 🔧 The process of learning in machine learning is described as figuring out the relations (weights) between inputs and outputs, which is key to making predictions.
• 🔎 The video discusses optimization problems in finding the correct weights and introduces techniques like random search and evolutionary algorithms.
• 🚀 It touches on the concept of neural networks with multiple layers and the use of non-linear activation functions to model complex relationships.
• 🤖 The tutorial covers the implementation of backpropagation for learning and the challenges associated with deep networks, such as vanishing and exploding gradients.
• 🛠️ It advises on the use of tools like PyTorch for deep learning, highlighting the benefits of adaptive learning rates and regularization techniques.
• 📈 The script discusses the importance of generalization in neural networks and the risks of overfitting, especially with large networks and small datasets.
• 🎯 Finally, it explores the concept of self-attention mechanisms in neural networks, which have been pivotal in the development of models like GPT, and their potential simplification.
Q & A
• What is the main topic of the video 'Machine Learning From Zero to GPT in 40 Minutes'?
-The main topic of the video is to provide a walkthrough tutorial on building a GPT-like model, discussing neural networks, and exploring concepts beyond GPT, with the aim of generating poems
about cats by the end.
• Why is the presenter interested in neural networks as a neuroscientist?
-The presenter is interested in neural networks as a neuroscientist because they believe it can provide more insight into how AI and the human brain can inspire each other.
• What is the initial assumption made by the presenter about the viewer's knowledge of machine learning?
-The presenter assumes zero knowledge of machine learning from the viewer and aims to provide a gradual transition between concepts for better understanding.
• What programming tool is suggested for those who do not have a Python interpreter?
-The presenter suggests downloading Anaconda for those who do not have a Python interpreter.
• How does the video approach the concept of intelligence in the context of machine learning?
-The video approaches the concept of intelligence as the ability to predict outcomes, using simple examples like associating switches with lights and evolving to more complex models like
• What is the role of numpy in simplifying the process of handling multiple inputs and weights in a model?
-Numpy helps in simplifying the process by allowing all inputs to be put into an array and all weights into another array, enabling the calculation of the weighted sum through dot product in a
more compact way.
• Why is an optimization problem introduced when trying to predict outcomes based on inputs?
-An optimization problem is introduced because to predict outcomes accurately, one needs to find the correct combination of weights that model the relations between inputs and outputs, which
involves searching through a multi-dimensional space.
• What is the significance of adding a bias term in a linear regression model?
-Adding a bias term is significant because it accounts for shifts in the data, allowing the model to fit not only centered data but also data that may be offset from zero.
• How does the video address the issue of non-linear relationships between inputs and outputs?
-The video addresses this issue by suggesting the addition of another layer of weights connected to middle nodes and the application of a non-linear activation function, such as a sine wave, to
the middle neurons.
• What is the purpose of using an activation function like the sine wave in a neural network?
-The sine wave is used as an activation function because, according to the Fourier transform principle, it can approximate any signal by adding together different sine waves, thus introducing
non-linearity into the model.
• What are the potential issues with using a multi-layer neural network for simple linear problems?
-Using a multi-layer neural network for simple linear problems can be inefficient and messy, as it is like using a complex tool for a task that could be easily handled by simpler methods like
linear regression or a perceptron.
• Why is the backpropagation process in deep networks not foolproof?
-Backpropagation in deep networks is not foolproof due to potential issues like the vanishing gradient problem, where gradients become too small to be useful, or the exploding gradient problem,
where gradients become too large and uncontrollable.
• What is the role of the learning rate in training a neural network?
-The learning rate determines the step size at which the model updates its weights during training. It is crucial for finding a balance between learning quickly and avoiding overshooting the
optimal solution.
• What is the significance of Occam's razor in the context of model selection in machine learning?
-Occam's razor suggests that among competing hypotheses, the one with the fewest assumptions should be selected. In the context of machine learning, it implies that a simpler model that fits the
data well is preferable as it has more potential for accurate extrapolation.
• How does the video script address the philosophical implications of AI and intelligence?
-The script touches on the philosophical implications by discussing the nature of truth, the ability of AI to predict the future, and the alignment of AI objectives with human interests,
suggesting that intelligence is the ability to compress information and make accurate predictions.
🤖 Introduction to Building a GPT-like Model
The video script begins with an introduction to the GPT (Generative Pre-trained Transformer) model, which has gained significant attention. The presenter, a neuroscientist interested in AI, plans to
guide viewers through creating a model similar to GPT, with the end goal of generating cat poems. The tutorial is aimed at individuals with no prior knowledge in machine learning, and the presenter
will use simple examples and analogies to explain complex concepts. The script also covers the installation of Anaconda and the use of Python interpreter to run basic code, introducing the viewer to
the fundamentals of programming and machine learning.
🔍 Exploring Neural Networks and Machine Learning Basics
This paragraph delves into the basics of neural networks and machine learning. It discusses the concept of intelligence as the ability to predict outcomes and uses the analogy of a brain associating
switches with lights. The presenter explains the limitations of traditional AI methods, such as IF-ELSE statements, and introduces the perceptron model, which uses weighted sums and thresholds to
make predictions. The importance of learning the correct weights is emphasized, and the presenter outlines the process of using numpy for simplifying calculations and the concept of optimization in
finding the right combination of weights.
🧬 Evolutionary Approach to Solving Optimization Problems
The script introduces an evolutionary approach to solving optimization problems, inspired by natural selection. It describes a process where 'parents' generate 'children' with slightly mutated
weights, and the fitness of these children determines their survival. This method is applied to the problem of finding the correct weights for a neural network, with the mutation amount adjusted to
improve the solution. The presenter also discusses the limitations of linear regression and the need for adding bias terms and non-linear activation functions to handle non-linear relationships.
🌀 Advanced Neural Network Concepts and Techniques
This section covers more advanced neural network concepts, such as the use of multiple layers and non-linear activation functions to solve complex problems. The presenter explains the use of sine
waves for generating non-linearity and the importance of nodes in capturing hierarchical structures. The script also touches on the challenges of using deep neural networks, such as the vanishing and
exploding gradient problems, and suggests using tools like PyTorch for more efficient implementation.
🛠️ Implementing and Training Neural Networks with PyTorch
The focus shifts to practical implementation, with the presenter guiding viewers on how to use PyTorch for neural network training. The script outlines the process of converting numpy arrays to
tensors, setting up the model structure, and using an optimizer to adjust weights during training. It also discusses the importance of regularizing the network to prevent overfitting and the use of
ReLU activation functions for more stable training.
📚 Autoregression and Generating Text with Neural Networks
The presenter introduces the concept of autoregression, where a neural network is trained to predict the next item in a sequence, such as the next letter in a sentence. This can be used for
generating text, and the script provides a step-by-step guide on preparing text data, training the model, and generating new text based on the learned patterns. The importance of context size and the
use of techniques like temperature scaling for more inventive text generation are discussed.
🎨 Enhancing Text Generation with Convolution and Attention Mechanisms
This paragraph explores enhancing text generation through the use of convolution and attention mechanisms. The presenter explains how convolution can help recognize patterns regardless of their
position in the text, while attention mechanisms allow the model to weigh inputs appropriately, leading to better text generation. The script also covers the implementation of these mechanisms using
PyTorch and the benefits of using distributed representation for better generalization.
🔄 LSTMs, Attention, and the Transformer Model
The script discusses the limitations of traditional recurrent neural networks (RNNs) and introduces Long Short-Term Memory (LSTM) networks as a solution to the vanishing gradient problem. It then
describes the Transformer model, which uses self-attention mechanisms to weigh inputs based on their significance, allowing for better parallelization and scalability. The presenter outlines the
implementation of the attention mechanism and the benefits of using multiple attention blocks in a stack.
🧠 Philosophical Reflections on Intelligence and Neural Networks
In the final paragraph, the presenter reflects on the nature of intelligence and the philosophical implications of neural networks. They discuss the subjective and objective nature of truth and the
role of intelligence in predicting the future. The script concludes with a thought-provoking discussion on the potential of neural networks to simplify and better understand intelligence, as well as
the alignment of AI systems with human interests.
💡Machine Learning
Machine Learning is a subset of artificial intelligence that provides systems the ability to learn and improve from experience without being explicitly programmed. In the context of the video, it is
the core concept around which the entire tutorial is built, explaining how to build a model similar to GPT (Generative Pre-trained Transformer), which is a machine learning model known for its
ability to generate human-like text.
💡Neural Networks
Neural networks are a set of algorithms designed to recognize patterns and are inspired by the human brain. They are composed of interconnected nodes or 'neurons' that process information. In the
video, the presenter discusses the importance of neural networks in various fields and uses them as the basis for building a GPT-like model, emphasizing their ability to model complex relationships
and patterns.
A perceptron is a type of neural network that is used for supervised learning of binary classifiers. In the script, the perceptron is introduced as a simple model for understanding the basic
functioning of neural networks. It is used to illustrate the concept of weighted sums and thresholding to make predictions.
💡Activation Function
An activation function in a neural network is a mathematical function that determines the output of a node given an input or set of inputs. The script mentions the use of sine waves as an activation
function to introduce non-linearity into the model, allowing it to learn more complex patterns.
Backpropagation is the standard method used to train feedforward neural networks. It involves calculating the gradient of the loss function with respect to each weight by the chain rule, which is
used to update the weights and minimize the error. The video explains the process of backpropagation as a means to adjust weights and fine-tune the neural network.
An optimizer in machine learning is an algorithm that is used to adjust the weights of the model during training to minimize a loss function. The script refers to the use of an optimizer like Adam,
which adapts the learning rate for each weight, helping the model to converge more efficiently.
Regularization is a technique used to prevent overfitting in machine learning models by adding a penalty term to the loss function. In the video, regularization is suggested as a method to reduce the
complexity of the model and improve its generalization to new data.
Self-attention is a mechanism in neural networks that allows the model to weigh the importance of different parts of the input data when making predictions. The script describes self-attention as a
way to create a flexible network that can learn from different context lengths and is invariant to position and permutation.
A Transformer is a type of neural network architecture that relies entirely on attention mechanisms to draw global dependencies between its inputs and outputs. The video mentions the Transformer
model as the state-of-the-art for language modeling, highlighting its ability to process sequences in parallel and effectively capture context.
💡Generative AI
Generative AI refers to systems that can create new content that resembles the data they were trained on. In the script, the concept of generative AI is demonstrated by training a neural network to
predict the next letter in a sentence, which can then be used to generate new text, such as poems about cats.
Building a GPT-like model in under 40 minutes.
Generating poems about cats using the model.
Discussing new concepts beyond GPT.
The relation of neural networks to various fields including neuroscience.
Assumption of zero knowledge in machine learning for the tutorial.
Using Python and Anaconda for the tutorial setup.
Introduction to perceptrons and their role in machine learning.
Explanation of weighted sums and thresholding in prediction models.
Importance of numpy for simplifying calculations in machine learning.
The concept of learning as figuring out the relations between inputs and outputs.
Optimization problem in finding the correct weights for a model.
Brute force search method for solving optimization problems.
Evolutionary approach to finding solutions in machine learning.
Implementation of mutation and selection in evolutionary algorithms.
Challenges with linear regression and the need for bias terms.
Introduction of non-linear activation functions like sine waves.
The importance of nodes in capturing hierarchical structures.
Backpropagation and its role in training neural networks.
Problems with vanishing and exploding gradients in deep networks.
Solutions to deep network training using tricks and tools.
Use of PyTorch for deep learning tasks.
Importance of regularization to prevent overfitting in neural networks.
The ability of neural networks to fit any data given enough nodes.
Challenges with extrapolation using neural networks.
The concept of autoregression and its applications.
Training a network to generate text based on past context.
Use of embeddings and convolution to create context-aware models.
Introduction to the Transformer model and self-attention mechanisms.
Stacking attention blocks to form a multi-layered Transformer.
The use of residual connections to mitigate vanishing gradients.
Alternative ideas to self-attention with learnable lateral connections.
The philosophical implications of AI and the search for truth.
Final thoughts on the nature of intelligence and the human brain.
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Sensitivity Analysis: How to Assess the Impact of Changes in Your Fiscal Assumptions - FasterCapital
Sensitivity Analysis: How to Assess the Impact of Changes in Your Fiscal Assumptions
1. Understanding Sensitivity Analysis
sensitivity analysis is a crucial aspect of assessing the impact of changes in fiscal assumptions. It allows us to understand how variations in key factors can affect the overall outcomes and
decision-making processes. In this section, we will delve into the concept of sensitivity analysis and explore its significance from different perspectives.
1. importance of Sensitivity analysis:
sensitivity analysis helps us identify the variables that have the most significant influence on the outcomes of a fiscal model or decision. By understanding the sensitivity of these variables, we
can make informed decisions and mitigate potential risks. For example, in financial forecasting, sensitivity analysis can reveal the impact of changes in interest rates, inflation rates, or market
conditions on the projected financial performance of a company.
2. methods of Sensitivity analysis:
There are various methods to conduct sensitivity analysis, including one-way analysis, tornado diagrams, and monte Carlo simulations. One-way analysis involves varying a single input variable while
keeping others constant to observe the resulting changes in the output. Tornado diagrams visually represent the sensitivity of multiple variables by ranking them based on their impact on the output.
Monte Carlo simulations use random sampling techniques to simulate different scenarios and assess the range of possible outcomes.
3. Interpreting sensitivity Analysis results:
When interpreting sensitivity analysis results, it is essential to consider the magnitude and direction of the impact. A higher sensitivity indicates that a small change in the input variable can
lead to a significant change in the output. Conversely, a lower sensitivity suggests that the output is relatively stable and less affected by variations in the input. By understanding these results,
decision-makers can prioritize their focus on the most influential variables and allocate resources accordingly.
To illustrate the concept of sensitivity analysis, let's consider a manufacturing company. By conducting sensitivity analysis on factors such as raw material costs, labor expenses, and demand
fluctuations, the company can assess the potential impact on its profitability. For instance, if the sensitivity analysis reveals that a 10% increase in raw material costs leads to a significant
decline in profit margins, the company can explore strategies to mitigate this risk, such as negotiating better supplier contracts or diversifying its sourcing options.
sensitivity analysis is a powerful tool that allows us to assess the impact of changes in fiscal assumptions. By understanding the importance of sensitivity analysis, the methods used, and how to
interpret the results, decision-makers can make more informed and robust decisions in various domains, ranging from finance to operations.
2. Importance of Fiscal Assumptions in Decision Making
1. Strategic planning and Resource allocation:
- Strategic Alignment: Fiscal assumptions align with an organization's strategic goals. They guide resource allocation by determining where to invest capital, which projects to prioritize, and how to
allocate budgets across different departments.
- Scenario Analysis: By varying fiscal assumptions (e.g., revenue growth rates, cost structures, inflation rates), decision-makers can assess the impact on strategic outcomes. For instance, a company
might explore scenarios under optimistic, moderate, and pessimistic assumptions to understand potential risks.
- Example: Imagine a retail chain considering expansion into a new market. Assumptions about consumer demand, market penetration, and operating costs will significantly influence the decision.
2. financial Modeling and forecasting:
- Budgeting and Forecasting: Fiscal assumptions form the foundation of financial models used for budgeting, forecasting, and long-term planning. These models project cash flows, profits, and balance
sheets based on assumptions about sales, expenses, and capital expenditures.
- Sensitivity Analysis: Organizations perform sensitivity analysis by tweaking assumptions to gauge their impact on financial metrics. Sensitivity charts reveal which assumptions are most critical.
- Example: A tech startup estimating future revenue growth assumes different adoption rates for its software product. By adjusting these assumptions, it can assess the impact on cash flow
3. risk Assessment and decision Uncertainty:
- Risk Identification: Assumptions inherently carry risk. Overly optimistic assumptions can lead to overcommitment, while overly conservative ones may hinder growth. Identifying key assumptions helps
manage risk.
- monte Carlo simulation: This technique involves running thousands of simulations by varying assumptions within specified ranges. It provides a distribution of possible outcomes, highlighting areas
of uncertainty.
- Example: A pharmaceutical company developing a new drug must assume success rates in clinical trials. Sensitivity analysis reveals the impact of varying success probabilities on overall
4. investment Appraisal and Capital budgeting:
- Net Present Value (NPV): Assessing investment projects involves discounting future cash flows using assumptions about discount rates. NPV helps decide whether an investment is economically viable.
- internal Rate of return (IRR): IRR relies on assumptions about project costs, revenues, and timing. It indicates the rate of return at which NPV equals zero.
- Example: A manufacturing firm evaluating a factory expansion project considers assumptions about construction costs, production volumes, and market demand. These drive npv and IRR calculations.
5. Communication and Accountability:
- Transparency: Clearly documented assumptions enhance transparency. Stakeholders, including investors and board members, need to understand the basis for decisions.
- Accountability: When assumptions prove inaccurate, organizations must learn from the experience and adjust their decision-making processes.
- Example: A real estate developer communicates assumptions about property appreciation rates to potential investors. If assumptions are overly optimistic, it could lead to disappointment later.
In summary, fiscal assumptions are not mere numbers; they shape the destiny of organizations. Rigorous analysis, scenario testing, and sensitivity assessments ensure that decision-makers navigate
uncertainties effectively. By acknowledging the importance of these assumptions, businesses can make informed choices and adapt to changing environments. Remember, the devil (and the opportunity)
lies in the details of those assumptions!
3. Methodology for Conducting Sensitivity Analysis
## understanding Sensitivity analysis
Sensitivity analysis is akin to probing the robustness of your assumptions. It helps answer questions like:
- "How sensitive is our net present value (NPV) to changes in discount rates?"
- "What happens to our project's profitability if construction costs increase by 10%?"
- "How does uncertainty in demand forecasts impact our revenue projections?"
### Insights from Different Perspectives
1. Financial Perspective:
- scenario-Based approach: Financial analysts often use scenario-based sensitivity analysis. They create multiple scenarios by varying key assumptions (e.g., interest rates, inflation, revenue
growth) and observe the resulting impact on financial metrics.
- Tornado Diagrams: These visual representations show the relative importance of each input parameter. The taller the bar, the greater the sensitivity.
- Break-Even Analysis: Identifying the break-even point helps determine the minimum level of performance required for a project to be viable.
2. project Management perspective:
- critical Path analysis: In project management, sensitivity analysis helps identify critical tasks. If a task's duration changes significantly, it may delay the entire project.
- Resource Allocation: Sensitivity analysis guides resource allocation decisions. For instance, if a project is sensitive to labor costs, optimizing workforce allocation becomes crucial.
3. policy and Decision-making Perspective:
- Policy Impact Assessment: Policymakers use sensitivity analysis to evaluate the effects of policy changes. For example, how does increasing taxes impact government revenue?
- Risk Assessment: Sensitivity analysis reveals vulnerabilities. By assessing the impact of various risks (e.g., economic downturns, regulatory changes), decision-makers can devise risk mitigation
### In-Depth Methodology
1. Identify Key Parameters:
- Begin by listing all relevant input parameters (variables) in your model. These could be interest rates, growth rates, production costs, etc.
2. Define Ranges:
- Determine plausible ranges for each parameter. For instance, if inflation is historically between 2% and 4%, consider this range.
3. Vary Parameters:
- Systematically change one parameter at a time while keeping others constant. Observe the impact on the output (e.g., NPV, project duration).
4. Quantify Sensitivity:
- Calculate sensitivity indices (e.g., elasticity, percentage change in output per percentage change in input). These reveal which parameters have the most influence.
5. Visualize Results:
- Use tornado diagrams, scatter plots, or spider charts to visualize sensitivity. Highlight critical parameters.
### Examples:
- Suppose you're evaluating a property investment. Sensitivity analysis reveals that the NPV is highly sensitive to rental income assumptions but less sensitive to property management costs.
2. Drug Development:
- In pharmaceutical research, sensitivity analysis helps assess the impact of clinical trial success rates on overall project value.
3. Climate Change Policies:
- Policymakers analyze how different carbon tax rates affect emissions reduction targets.
Remember, sensitivity analysis isn't about predicting the future—it's about understanding the range of possibilities and making informed decisions. So, embrace uncertainty, explore scenarios, and let
sensitivity guide your choices!
4. Identifying Key Variables and Assumptions
In the section on "Identifying Key Variables and Assumptions" within the blog "Sensitivity Analysis: How to Assess the Impact of Changes in Your Fiscal Assumptions," we delve into the crucial process
of identifying the key variables and assumptions that underpin your fiscal analysis. This step is essential as it allows you to understand the factors that have the most significant impact on your
financial outcomes.
From different perspectives, it is important to consider both internal and external variables. Internal variables refer to factors that are within your control, such as pricing strategies, production
costs, or marketing budgets. On the other hand, external variables encompass factors that are influenced by external forces, such as market trends, economic conditions, or regulatory changes.
To provide a comprehensive understanding, let's explore this section through a numbered list:
1. Start by identifying the core variables: Begin by identifying the primary variables that directly affect your fiscal assumptions. These variables could include sales volume, revenue growth rates,
or cost of goods sold.
2. Assess the sensitivity of each variable: Once you have identified the key variables, it is crucial to assess their sensitivity. This involves analyzing how changes in each variable impact your
financial outcomes. For example, you can evaluate how a 10% increase in sales volume affects your overall revenue.
3. Consider interdependencies: Variables are often interconnected, and changes in one variable can have ripple effects on others. It is important to consider these interdependencies when assessing
the impact of changes. For instance, an increase in production costs may lead to a rise in product prices, which, in turn, can affect sales volume.
4. Use scenario analysis: Scenario analysis involves creating different scenarios by varying the values of key variables. This helps you understand the range of possible outcomes and assess the
robustness of your assumptions. For instance, you can analyze the impact of optimistic, pessimistic, and base-case scenarios on your financial projections.
5. Incorporate historical data and industry benchmarks: To enhance the accuracy of your analysis, consider incorporating historical data and industry benchmarks. This provides a reference point for
evaluating the reasonableness of your assumptions and helps you benchmark your performance against industry standards.
By following these steps and considering various perspectives, you can effectively identify the key variables and assumptions that drive your fiscal analysis. This enables you to make informed
decisions and assess the impact of changes on your financial outcomes.
5. Analyzing the Impact of Changes on Financial Projections
1. Why sensitivity Analysis matters:
- risk Management perspective: From a risk management standpoint, sensitivity analysis allows us to identify vulnerabilities in our financial models. By tweaking input variables, we can understand
which factors have the most significant impact on our projections.
- strategic Decision-making: Business decisions often hinge on financial projections. Sensitivity analysis helps us evaluate the potential consequences of different choices. For instance, should we
expand production capacity? How sensitive are our profits to changes in demand or raw material costs?
- Investor Confidence: Investors appreciate transparency. Demonstrating that we've considered various scenarios and their implications builds confidence in our projections.
2. Methodology and Techniques:
- One-Variable Sensitivity Analysis: Altering a single input while keeping others constant. For example:
- Interest Rates: Suppose we're projecting cash flows for a real estate investment. By adjusting the interest rate assumption, we can see how it impacts net present value (NPV).
- Sales Volume: In a retail business, changing sales volume assumptions affects revenue and profit margins.
- Tornado Diagrams: These visual representations show the relative impact of multiple variables. The tallest bars indicate the most influential factors.
- Scenario Analysis: Creating plausible scenarios (optimistic, base, pessimistic) and assessing their impact. For instance:
- Optimistic Scenario: Strong market growth, low inflation, favorable exchange rates.
- Pessimistic Scenario: Economic downturn, supply chain disruptions, regulatory changes.
- Monte Carlo Simulation: A statistical technique that generates thousands of scenarios by randomly varying input parameters. It provides a distribution of possible outcomes.
3. Examples:
- Startup Valuation: Imagine valuing a tech startup. We adjust growth rates, discount rates, and exit multiples. Sensitivity analysis reveals which assumptions drive valuation the most.
- Project Feasibility: A construction project's viability depends on factors like construction costs, interest rates, and lease rates. By varying these, we assess profitability.
- Oil Price Sensitivity: Oil companies model their cash flows based on oil prices. Sensitivity analysis helps them prepare for price volatility.
4. Challenges and Considerations:
- Correlations: Variables often correlate. Sensitivity analysis assumes independence, which may not hold. Addressing correlations improves accuracy.
- Nonlinear Effects: Some relationships aren't linear. For instance, doubling advertising spending doesn't necessarily double sales.
- Qualitative Factors: Not all impacts are quantifiable. Regulatory changes, brand reputation, or geopolitical events matter too.
Remember, sensitivity analysis isn't about predicting the future precisely; it's about understanding the range of possibilities. By embracing uncertainty, we make better-informed decisions.
6. Interpreting Sensitivity Analysis Results
1. Understanding Parameter Importance:
- Sensitivity analysis helps us identify which parameters have the most significant impact on the model's output. By varying one parameter at a time while keeping others constant, we can observe how
the output changes. The more sensitive a parameter is, the greater its influence on the results.
- Example: Imagine a climate model that predicts global temperature based on greenhouse gas emissions, solar radiation, and cloud cover. Sensitivity analysis reveals that greenhouse gas emissions
have the highest impact on temperature, followed by solar radiation.
2. Visualizing Sensitivity:
- Visual tools like tornado diagrams, scatter plots, or heatmaps help us visualize parameter sensitivity. Tornado diagrams show the range of output variation caused by each parameter, highlighting
the most critical ones.
- Example: In a financial model assessing the profitability of a new product launch, a tornado diagram reveals that production costs and sales volume are the key drivers of profit variability.
3. Thresholds and Nonlinear Effects:
- Some parameters exhibit nonlinear effects. Small changes in certain parameters may have negligible impact until a threshold is reached, beyond which the effect becomes significant.
- Example: Consider a drug dosage-response model. Below a certain dosage, the drug has no effect. Beyond that threshold, the response increases exponentially.
4. Scenario Analysis:
- Sensitivity analysis allows us to explore different scenarios by varying multiple parameters simultaneously. This helps us understand how changes in assumptions affect overall outcomes.
- Example: A transportation planner evaluates the impact of different traffic management strategies (e.g., road widening, tolls, public transit) on commute times. Sensitivity analysis reveals which
combination of strategies optimizes travel efficiency.
5. Robustness and Uncertainty:
- Robustness analysis assesses how sensitive the model is to variations in parameter values. A robust model remains stable even when parameters fluctuate within reasonable bounds.
- Example: A supply chain model considers lead times, demand fluctuations, and production delays. Robustness analysis ensures that the supply chain remains efficient despite uncertainties.
- Uncertainty analysis quantifies the uncertainty associated with each parameter. Techniques like Monte Carlo simulation provide probabilistic distributions of model outputs.
- Example: A financial risk model estimates portfolio returns. By incorporating uncertainty in interest rates, stock prices, and inflation, we obtain a more realistic risk assessment.
6. trade-offs and Decision making:
- Sensitivity analysis helps us identify trade-offs between conflicting objectives. By varying parameters, we can find optimal solutions that balance competing goals.
- Example: An environmental impact assessment for a dam project considers ecological benefits (e.g., water supply, flood control) versus environmental costs (habitat destruction, biodiversity loss).
Sensitivity analysis guides decision makers toward a balanced solution.
Remember that sensitivity analysis is not a crystal ball—it doesn't predict the future. Instead, it empowers us to make better-informed choices by quantifying the impact of uncertainty and
assumptions. So, whether you're adjusting fiscal assumptions, optimizing engineering designs, or evaluating policy options, interpreting sensitivity analysis results is your compass in navigating
complex decision landscapes.
7. Mitigating Risks and Uncertainties through Sensitivity Analysis
## Understanding Sensitivity Analysis
sensitivity analysis is like stress-testing your financial model. It helps you answer questions such as:
- How sensitive are our results to changes in input parameters?
- Which assumptions have the most significant impact on our outcomes?
- What are the critical thresholds beyond which our project becomes unviable?
### Insights from Different Perspectives
1. Financial Analyst's Viewpoint:
- Financial analysts often perform sensitivity analysis to understand the range of possible outcomes. They tweak input variables (e.g., interest rates, inflation rates, revenue growth) and observe
how these changes propagate through the model.
- Example: Imagine a real estate development project. By varying the discount rate, analysts can assess the project's sensitivity to changes in interest rates. If a small increase in rates
significantly reduces the project's net present value (NPV), it signals heightened risk.
2. Project Manager's Lens:
- Project managers use sensitivity analysis to identify critical assumptions that need close monitoring. These assumptions might relate to costs, timelines, or market conditions.
- Example: Suppose you're managing a construction project. By analyzing how variations in material costs impact the project's profitability, you can allocate contingency funds wisely.
3. Risk Mitigation Strategies:
- Sensitivity analysis informs risk mitigation strategies. When you identify vulnerable assumptions, you can take proactive steps to reduce exposure.
- Example: A startup planning to launch a new product might assess the impact of different sales volumes. If the breakeven point is too high, they might explore cost-cutting measures or diversify
their product portfolio.
### In-Depth Exploration: Key Techniques
Let's explore some techniques commonly used in sensitivity analysis:
1. One-Way Sensitivity Analysis:
- Vary one input parameter while keeping others constant.
- Example: Assess how changes in oil prices affect the profitability of an airline company. If fuel costs skyrocket, can the company still operate profitably?
2. Tornado Diagrams:
- Rank-order the sensitivity of assumptions.
- Example: A tornado diagram reveals that exchange rate fluctuations have the most significant impact on a multinational corporation's earnings. hedging against currency risk becomes crucial.
3. Scenario Analysis:
- Create multiple scenarios by combining different assumptions.
- Example: A pharmaceutical company models scenarios for drug development. What if clinical trial success rates vary? How does it impact the overall drug pipeline?
4. Monte Carlo Simulation:
- Randomly sample input parameters from probability distributions.
- Example: Simulate thousands of scenarios for a portfolio's returns. Understand the likelihood of achieving specific financial goals.
### Real-World Example: Oil Exploration Investment
Imagine an energy company evaluating an oil exploration project. Sensitivity analysis reveals that the project's NPV is highly sensitive to oil prices, drilling costs, and reserve estimates. Here's
how they mitigate risks:
- Diversification: The company invests in multiple projects across different regions to spread risk.
- Hedging: They use financial derivatives to hedge against oil price volatility.
- Contingency Planning: If drilling costs exceed estimates, they have contingency funds ready.
Sensitivity analysis isn't just about numbers; it's about informed decision-making. By understanding the impact of uncertainties, we can navigate fiscal landscapes with greater confidence. Remember,
assumptions are like compasses—choose wisely, and you'll reach your destination even in stormy seas.
8. Real-World Applications of Sensitivity Analysis
Sensitivity analysis is a crucial tool in assessing the impact of changes in fiscal assumptions. In this section, we will explore real-world case studies that demonstrate the practical applications
of sensitivity analysis. By examining different perspectives, we can gain valuable insights into how sensitivity analysis can inform decision-making processes.
1. Case Study: Company X's Pricing Strategy
In this case study, Company X wanted to evaluate the impact of various pricing scenarios on their profitability. By conducting sensitivity analysis, they were able to identify the price range that
maximized their revenue while considering different cost factors. This analysis helped them make informed decisions about their pricing strategy and optimize their financial performance.
2. Case Study: Government Policy Assessment
In another case study, a government agency aimed to assess the potential effects of proposed policy changes on the economy. Through sensitivity analysis, they analyzed different variables such as tax
rates, subsidies, and regulations. By quantifying the impact of these changes, policymakers gained insights into the potential outcomes and adjusted their policies accordingly.
3. Case Study: investment Portfolio optimization
A financial institution wanted to optimize their investment portfolio by considering various market scenarios. Sensitivity analysis allowed them to evaluate the sensitivity of their portfolio's
performance to changes in factors such as interest rates, market volatility, and asset allocation. This analysis helped them identify the most robust investment strategies and mitigate potential
4. Case Study: supply Chain management
In this case study, a manufacturing company aimed to assess the vulnerability of their supply chain to external disruptions. Sensitivity analysis helped them identify critical components, suppliers,
and transportation routes that were most sensitive to changes in factors like demand, lead times, and logistics costs. By understanding these sensitivities, they could develop contingency plans and
improve the resilience of their supply chain.
9. Leveraging Sensitivity Analysis for Informed Decision Making
In the realm of financial planning and decision-making, sensitivity analysis emerges as a powerful tool that allows us to explore the impact of changes in key assumptions. By systematically varying
these assumptions, we gain valuable insights into the robustness of our financial models and the potential risks associated with different scenarios. In this concluding section, we delve deeper into
the significance of sensitivity analysis and its practical applications.
1. The Multifaceted Nature of Uncertainty:
Sensitivity analysis acknowledges the inherent uncertainty in financial projections. Whether we're forecasting revenue growth, estimating costs, or predicting market trends, our assumptions are
rarely precise. They often hinge on external factors such as economic conditions, technological advancements, and regulatory changes. Recognizing this multifaceted nature of uncertainty, sensitivity
analysis encourages decision-makers to consider a range of possibilities rather than relying solely on point estimates.
2. Insights from Different Perspectives:
A critical aspect of sensitivity analysis lies in its ability to provide insights from various perspectives. Let's explore these viewpoints:
A. Optimistic Viewpoint:
- Imagine a scenario where all assumptions align favorably. Revenue growth exceeds expectations, costs remain low, and market conditions are ideal. Sensitivity analysis allows us to quantify the
potential upside in such a scenario. By stress-testing our assumptions, we can identify the key drivers of success and allocate resources accordingly.
B. Pessimistic Viewpoint:
- Conversely, consider a pessimistic scenario. Perhaps a major client withdraws, inflation spikes, or interest rates soar unexpectedly. Sensitivity analysis helps us assess the downside risk. By
quantifying the impact of adverse changes, we can devise contingency plans, build reserves, and mitigate potential losses.
C. Realistic Viewpoint:
- Most decisions fall somewhere between extreme optimism and extreme pessimism. Sensitivity analysis provides a realistic lens through which we evaluate trade-offs. For instance, how sensitive is our
net present value (NPV) to variations in discount rates? What if our cost assumptions fluctuate within a reasonable range? By answering these questions, we make informed choices.
3. Quantifying Impact:
Sensitivity analysis often employs a tornado diagram or a spider plot to visualize the impact of individual assumptions. These graphical representations highlight which variables exert the most
influence on our outcomes. For instance:
- If a 10% increase in production costs significantly reduces our project's profitability, we know that cost control is paramount.
- If changes in interest rates have minimal impact on NPV, we can focus on other drivers.
4. Real-World Examples:
A. Capital Budgeting:
- When evaluating investment projects, sensitivity analysis helps us assess the sensitivity of net cash flows to variables like sales volume, discount rates, and project duration. Suppose we're
considering a new manufacturing plant. By varying assumptions (e.g., raw material costs, labor productivity), we gauge the project's resilience to changing conditions.
B. Risk Management:
- Insurance companies use sensitivity analysis to model potential losses due to catastrophic events (e.g., natural disasters). By simulating different claim scenarios, they allocate capital
appropriately and ensure solvency.
C. Portfolio Optimization:
- Investors analyze portfolio sensitivity to market volatility, interest rates, and sector-specific risks. Adjusting asset allocations based on these sensitivities enhances risk-adjusted returns.
5. Beyond Numbers: Qualitative Insights:
- Sensitivity analysis isn't confined to numerical inputs. It extends to qualitative factors like customer preferences, regulatory compliance, and competitive dynamics. For instance, how sensitive is
our market share to changes in branding strategy? By integrating qualitative insights, we refine our decision-making process.
In summary, sensitivity analysis transcends mere number-crunching. It empowers decision-makers to navigate uncertainty, anticipate surprises, and make strategic choices. As we embrace its principles,
we move from blind optimism or fear-driven pessimism to a balanced, informed approach—one that considers both the art and science of decision-making.
Remember, the future rarely unfolds exactly as we predict, but sensitivity analysis equips us to adapt and thrive in an ever-changing landscape.
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Bias roulette: the method to profit from it - Tristat
711kelab login roulette wheel by addressing its inevitable physical imperfections. It is almost impossible for a man to make a perfect machine. The roulette wheel is made of wood and metal and prone
to wear and tear over time.
In addition, the wheel is operated by a human being who can also contribute to the imperfect functioning of the wheel. Additionally, roulette wheels are incredibly expensive machines and therefore
casinos prefer to accept small imperfections to make them last longer (as long as they don’t spot anyone making a systematic profit because of these imperfections!).
The perfect roulette wheel must be perfectly balanced, the boxes evenly lined and structured, the walls uniformly resistant to wear and tear, and the dealers unable to consciously or subconsciously
control the ball. It is almost impossible for an active wheel to stay perfect in the long term, it is more natural for it to gradually deviate from perfectly random results in the long term. A slight
groove invisible to the naked eye will suffice for the ball to stop more often in one box than in others.
Over time, the groove gets deeper and deeper as the ball goes through it. That is, small imperfections will likely get worse over time.
The biased wheel strategy aims to spot these imperfections and exploit the positive probabilities of certain outcomes on biased wheels.
Time the wheel Bias wheels should not have obvious flaws that could be detected during traditional inspections and tune-ups. They are by nature invisible to the naked eye. You need to spot them
statistically, by timing the wheel. This involves observing a wheel and recording the observations. You can then see if the true probabilities are much different from the expected probability of a
perfect wheel. On a European roulette wheel, each number should come out 1 in 37 times. The House pays 1 in 35 so numbers that come out more than 1 in 35 times will have a positive expected value.
For example, if by timing a wheel out of 1,000 observations, you find that Black 26 has come out 1 out of 30 times.
Whatever the physical imperfections, black 26 comes out 1 in 30 instead of 1 in 37. In other words, you have found a wheel with a bias in favor of black 26. If you bet € 10 on the Black 26 over the
last 1,000 spins, your win will be € 1,996. By going out about 1 in 30, Black 26 will be out 33 times and lose 967 times. You will therefore have won € 11,550 (€ 10 * € 35 * 33) and lost € 9,670 (€
10 * 967), obtaining a final gain of € 1,996 (€ 11,550 – € 9,670). Note that: Betting larger amounts will result in larger long term payouts but in the short term it will also result in larger
amounts of euros changing hands. So if you want to bet more, you need to make sure that you have larger funds available for short term hand changes.
The question you need to ask yourself is, will this roulette wheel show a bias for black 26 over the next 1000 spins.
The Central Limit Theorem states that the larger the number of observations, the closer you get to the real probabilities kelab88. In other words, the more numbers you collect, the more accurate your
assessment of bias will be. Following the example above, you might find in the first 100 observations that Red 14 came out 1 in 25 times but in the next 900 observations, Red 14 came out less often
and eventually presented an overall probability. 1 in 36. So if you had only made 100 observations and thought the wheel was skewed in favor of Red 14, you would have wasted your money in the next
900 spins. Short-term deviations and fluctuations are normal. By timing a wheel over a very large number of observations, you are trying to eliminate short-term random fluctuations from real
long-term probabilities. So the rule of thumb is this: the more numbers you collect the better.
This is why casinos gladly let players enter the numbers for a short time, so that they bet on illusions of short-term motives. But if you sit at a table and write the numbers for 8 hours every day
for 2 weeks, the casinos will really start to get excited!).
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Taking the Pain Out of Budgeting (Part 3): What’s the Big Deal About Compounding? - Cash Map App
27 Apr Taking the Pain Out of Budgeting (Part 3): What’s the Big Deal About Compounding?
Posted at 15:24h
Budgeting Tips
At first, seeing our money grow seems like an excruciatingly slow process. It’s so slow that too many it seems futile – not worth the effort. When developing a new skill, we face the same dilemma.
It’s hard to envision being an expert. Consistent persistence yields amazing results. Similarly, at first consistent neglect doesn’t seem to matter; however, with time tragic outcomes are inevitable.
In short, the Power of Compounding is a rule of life that works either for us or against us.
Let’s create a simple example. Let’s say you can save $400 each month over the next 5 years. This means each year you’ll save $4,800. Let’s also assume you’ll earn 6 percent interest on your money.
Here’s an easy formula you can use: 4800*(1+.06)^5. The $4,800 saved in the first year will be worth $6,423 at the end of five years. So, to figure out how much you’ll have for the dollars saved in
the second, third, fourth and fifth years, here’s what you do.
$4,800*(1+.06)^5 = $6,423 The value of the dollars you saved in your first year.
$4,800*(1+.06)^4 = $6,060 The value of the dollars you saved in the second year.
$4,800*(1+.06)^3 = $5,717 The value of the dollars you saved in the third year.
$4,800*(1+.06)^2 = $5,393 The value of the dollars you saved in the fourth year.
$4,800*(1+.06)^1 = $5,088 The value of the dollars you saved in the fifth year.
The total that you amount you will save in 5 years is $28,682! Our example assumes that you are starting with $4,800. If you couldn’t do this and are just starting your savings program, at the end of
five years, you will have $27,908. With each additional year, the impact of keeping your dollars working continues to grow.
If you already have $4,800 lazy dollars that you’ve kept in your checking account and begin putting these dollars to work and over the next five years add $400 a month, you’ll have $33,813. At the
end of twenty years, you’ll have saved a whopping $127,071!
Just imagine, each year when you get a raise you put a little more aside. Over the last twelve months, you’re already adjusted to living off of the amount you earned before the raise. If you put more
aside, you won’t miss a thing. I promise!
Click on the link below and take a look at this Ted Talk called ‘Saving for Tomorrow, Tomorrow’. http://www.ted.com/talks/shlomo_benartzi_saving_more_tomorrow.html
If $400 a month is all you need and you are already saving this amount – great! However, if you’d like to save more, what should you do?
Start carrying your list of priorities with you. Before you purchase an item, ask yourself, ‘Is what I’m about to purchase more important to me than the priority I’ve got on my list?’ You’ll begin
finding it much easier to save. The best part is you won’t feel like you are giving up a thing! What you’ll begin to experience is a desire to find other ways of finding more dollars to save. You
will have begun a small yet powerful mind shift that will accelerate over time. Small successes will motivate you to achieve greater achievements.
If $500 per month were saved over 5 years, you will have $35,852
If $600 per month were saved over 5 years, you will have $43,022
If $700 per month were saved over 5 years, you will have $50,193
Here’s another question worth asking, “What am I willing to purchase today that’s worth erasing my future dream?”
There’s a saying I repeat to myself, ‘Small changes bring big results’. Small changes can be either for the good or the bad. Making small changes is like creating a snowball – it continues to grow.
As it grows, the growth accelerates and the impact catches us by surprise. This is what the phrase, ‘the power of compounding’ means. Once you’ve caught this vision. You’ll be ready to start
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Curve Fitter - TechGraphOnlineCurve Fitter - TechGraphOnline
The Curve Fitter uses statistical methods to determine the “best” equation which fits a set of observations. Both linear and nonlinear algorithms are employed to describe a statistical relationship
(equation) between graph series. The Curve Fitter can fit 2-dimensional (X,Y) data. There are three modes available: linear, nonlinear, and user-defined of the form y=f(x).
The set of experimental observations can often be characterized by fitting the data to a model dependent upon unknown adjustable coefficients. The model can be a straight line, or perhaps an nth
order polynomial, where its coefficients serve simply to represent the discrete data in a continuous fashion for the purpose of interpolation. On the other hand, the function’s coefficients may have
some theoretical relationship to a physical, biochemical or physiological process. For example, the maximum physiological effect of a drug in a sigmoidal response, the rate coefficient in a kinetic
equation describing a chemical reaction, or the rate of growth in a biological phenomenon as described by an exponential curve.
Linear Curve Fit
Linear Curve Fit (a) is an intrinsically linear function to the data. This method applies Linear Least Squares regression analysis – a fast, single-step process – to determine the relationship
between the two variables x and y. This method chooses coefficients for the function which will minimize the sum of the squares of the differences between the observed y values and the predicted y
An equation of the form: y = a[0] + a[1]x + a[2]x^2 … +a[n]x^n, is linear with respect to the coefficients {a[0], a[1], a[2], …, a[n]}, is used to obtain a linear probabilistic relationship and
measure the extent to which the two variables are related. The simplest case is a straight line fit described by y = a[0] + a[1]x.
There are four Group Types (b): Linear, Exponential, Power and Polynomial functions. They use transformations on the x and/or y variables to linearize the data, allowing a straight line relationship
to be fitted. Linear least squares regression, utilizing the Gauss-Jordan method for matrix inversion, is then used to determine the coefficients from the transformed data.
In linear curve fitting using Gauss-Jordan, the number of data points must be greater than the number of unknown coefficients. Singularities of a function are removed from the fitting process.
Logarithmic and square root transformations on the x and/or y variables require their values to be greater than zero, or no fit is performed for the particular function.
The linear curve fit option will calculate the best fit for your data from all 100 built-in Linear Functions (c): Linear (40 functions), Exponential (10 functions), Power (25 functions), Polynomial
(25 functions). Check the ID box (a) for any number of functions that you think might fit your data.
Once a curve fitting is successful, a Report and Save Tab will become active. Please refer to Appendix C for details about the Report Tab.
The Save (a) tab menu is shown below.
In the Selection field (b) there are five fit statistics that can be saved and displayed on the graph.
Y Predict Equation
This copies the equation for the fitted function to the Equation field in the Edit Chart → Curves and Error Bars menu, in the graph series indicated by the Destination Col (d).
Predict Values
This option saves the function evaluation for each X,Y pair. The Y values will be saved in the Destination Col. For example, if “B” was written in Destination Col (d), then the Y values will be saved
in column B.
This option saves the residual (the difference between the Y Predict value and the Y Data value) for each data point. Note that when a linear approximating function with transformations on the
response (Y Data) has been performed, the residuals are expressed in the original data’s units.
Confidence Interval
The upper and lower limits of the Confidence Interval at each data point are saved. This creates two new columns of data in the Data Table, starting with the column specified in the Destination Col
(d) field. The Confidence Interval is calculated at the Level chosen in the Level field (c). The default Level is 95%.
Prediction Interval
This data is saved in the same way as the Confidence Interval.
The Destination Col field (d) is where the statistics and fit equation are stored.
Example 1:
The chart below shows a set of data that looks generally linear. So, a good starting point for a curve fit would be to choose a few of the built-in linear functions. The actual data is also shown. Be
sure to select “Scatter” as the Graph Type and “Number” as the Data Type.
Hint: Refer to X Data Type to change the graph type and X Data Type.
1. In the Selection tab of the Linear Curve Fit menu, check Linear Functions 1, 2, and 3 (a).
2. Enter “X” in X Data Col (b), “A” in Y Data Col (c), “1” in Start Row (d), and “8” in End Row (e), and then click Calculate (f). The curve fit was calculated and the Best Fit: Function 3 Coef of
Det = 0.980670 (g) is displayed at the bottom. Therefore, Function 3 y=a0+a1*sqrt(x) was the best fit of the three functions chosen. A Coefficient of Determinant of 1.0 is a perfect fit to the
data. Additional Fit Statistics were also generated by the calculation.
3. Next, click the Save (a) tab and the Linear Curve Fit menu shown below appears. The Report and Save tabs only become active after a successful fitting. Using Save, you can store the fit
statistics in the Data Table for further analysis, transfer the fitted equation to the Edit Chart → Curves and Error Bars Series Settings menu. Choose “Y Predict Equation” (b), enter “A” into
Destination Col (c) and click Save (d).
4. Next, click on the Report (a) tab. The fit statistical results from your calculation will be shown here in tabular format. The Download (b) button allows you to save the results as a CSV file in
your local drive. You have a choice to download either the statistical portion or the data portion as CSV file to be read by another application.
The Copy button (c) will copy the results to the clipboard to be pasted to another application. The Print button (d) will print the results to your local printer.
The Confidence Interval option (e) will allow one to set the CIF (LL & UP) values in the report.
5. Switch to the Edit Chart tab. The chart will be displayed with the equation line drawing with the equation and legend displayed on the chart.
Example 2:
The chart below shows a set of data that looks generally exponential. So, a good starting point for a curve fit would be to choose a few of the built-in exponential functions. The actual data is also
shown. Be sure to select “Combo” as the Graph Type and “Number” as the Data Type.
Hint: Refer to X Data Type to change the graph type and X Data Type.
1. Following a similar process as in the previous example, check Exponential Functions 42 through 49 (a), enter ”X” in X Data Col, “A” in Y Data Col, “1” in Start Row , and “8” in End Row, and then
click Calculate. The curve fit was calculated and the Best Fit: Function 43 Coef of Det = 0.982816 is displayed at the bottom. Therefore, Function 43: ln(y)=a0+a1*sqrt(x) was the best fit of the
nine functions chosen.
2. Next, click the Save tab (a) and select “Y Predict Equation” (b), enter “A” into Destination Col (c), and click Save (d).
3. In the Selection field, select “Confidence Interval” (e), Level “95%” (f), and enter “B” for Destination Col (g). Click Save. The confidence values of X and Y will be stored in columns B and C,
4. The Confidence Values are stored in Columns B and C as shown in the Data Table below. They are plotted on the chart in blue above and below the black equation curve fit line.
Note: Refer to Settings to set Graph Series A to “Scatter” and Graph Series B and C to “Line”.
Nonlinear Curve Fit
Nonlinear curve fits a nonlinear function with regard to the unknown coefficients, for example: y = a0 + a1*exp (a2 * x) or z = a0 + a1*x – sin (y/a2)
As with linear curve fitting, the objective remains to relate variables in a statistical relationship. The Levenberg-Marquardt algorithm is utilized in an unconstrained optimization approach to
estimate the coefficients of the equation. This is an iterative process which begins with initial estimates of the unknown coefficients of the fitting function and continues until the best
coefficients are found. Like the Linear method, this algorithm returns the coefficients which minimize the sum of the squared deviations, or Chi-squared value.
In simple terms, the Curve Fitter estimates a starting point on a surface whose height is represented by the Chi-squared value, and steps along in the downhill direction until the lowest point on the
surface is encountered.
The nonlinear option will determine the best fit from your choice of 15 built-in parametric functions plus one user-defined equation. These parametric nonlinear equations are divided into four
families: Standard, Waveform, Peak, and Transition. For more detail on the 15 resident parametric functions, please refer to Appendix A.
You may also include the “User-Defined” selection as one of your functions. See the section on User-Defined Curve Fit below.
The chart below shows a set of data that looks generally Transitional. So, a good starting point for a curve fit would be to choose a few of the Transitional built-in functions. The actual data is
also shown. Be sure to select “Scatter” as the Graph Type and “Number” as the Data Type.
Hint: Refer to X Data Type to change the graph type and X Data Type.
1. Following a similar process as in the previous example, check Functions 4,7,8,and 9, enter “X” in X Data Col, “A” in Y Data Col, “1” in Start Row, and “11” in End Row, and then click Calculate.
The curve fit was calculated and the Best Fit: Sigmoidal Function 9 Coef of Det = 0.998846 is displayed at the bottom. Therefore, the Sigmoidal Function 9: y=a0+a1/(1+exp(-(x-a2)/a3)) was the
best fit of the four functions chosen.
2. Click the Save tab and select “Y Predict Equation”, Destination Col “A”, and click Save. The resulting chart is shown below.
User-Defined Curve Fit
You can specify a function of the form y=f(x) with up to eight unknown coefficients. You can enter your own function, or select a pre–defined nonlinear function and edit it to your specifications.
The unknown coefficients are entered as a0 through a7 and must be sequential starting with a0. For example, the equation of a straight line (y = mx + c) is entered as: a0*x + a1. If you want to fit a
straight line and force it to pass through the origin (0,0) you would enter a user-defined function as: a0*x
You must specify reasonably approximate starting values for each unknown coefficient in the function. See Hints for Starting Coefficients in Appendix B for information about starting values for the
nonlinear functions.
Looking at the data on the chart below, it looks like a decaying waveform. From past experience you might know the equation: Y=a*sin(x)/x is a reasonable equation to match the data.
1. In the User-Defined tab (b) you will enter in the equation: a0*sin(x)/x in the Fit Equation field (d), with a0 being the coefficient you are looking to calculate. In the a0 field (a), enter “5”
as a starting point, and in the Coefficient Count field (c), enter “1”. Click Save. The equation will be saved as the User-Defined function 1 in the Built-In tab menu. Follow the same method for
Built-in curve fit functions listed above.
The resulting curve fit and equation are shown below with a Coef of Det = 0.970746.
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Diagnostic tests to Identify the type of distribution and calculate intercept in SAS
I've done a propensity score matching for case and control using variables like disease comorbidities etc. Now, my goal is to calculate the incremental cost (Difference of the average) between case
and control, when I look at the cost data for case and control both doesn't follow normal distribution (Right Skewed). I know there are tests to do check for normal distribution, I would like to do
diagnostic test to understand the type of distribution the data is following, like log, gamma etc. and use the appropriate distribution to calculate average.
I would really appreciate if someone can help me with process of how to do diagnostics, transform to the appropriate distribution and get the incremental average cost difference.
I've attached the dataset with following variables
1. Paid_ID: Matched Pair
2. VLU: '1' for Case, '0' for Control
3. Post_Cost: Cost data
4. Proc_score: Propensity score
I'm using SAS 9.3, so I would appreciate if you could guide me the process in the version.
10-22-2018 02:33 PM
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nForum - Discussion Feed (Parametrized Homotopy Theory)Urs comments on "Parametrized Homotopy Theory" (95303)
In view of discussion in another thread (here), I have added (here) the following warning:
Beware that section 4.4 claims a new proof of the Strøm model structure, but relying on a statement in
which later was noticed to be false, by Richard Williamson, for details see p. 2 and Rem 5.12 and Sec. 6.1 in:
It remains unclear, to me anyways, what this implies for the sliced generalization which is the core claim of May & Sigurdsson’s book.(?)
diff, v10, current
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Birth year probabilities — CalcBYprobs
Birth year probabilities
Estimate the probability that an individual with unknown birth year is born in year y, based on BirthYears or BY.min and/or BY.max of its parents, offspring, and siblings, combined with AgePrior (the
age distribution of other parent-offspring pairs), and/or Year.last of its parents.
dataframe with columns id-dam-sire.
data.frame with up to 6 columns:
max. 30 characters long
1 = female, 2 = male, 3 = unknown, 4 = hermaphrodite, other numbers or NA = unknown
birth or hatching year, integer, with missing values as NA or any negative number.
minimum birth year, only used if BirthYear is missing
maximum birth year, only used if BirthYear is missing
Last year in which individual could have had offspring. Can e.g. in mammals be the year before death for females, and year after death for males.
"Birth year" may be in any arbitrary discrete time unit relevant to the species (day, month, decade), as long as parents are never born in the same time unit as their offspring, and only integers
are used. Individuals do not need to be in the same order as in `GenoM', nor do all genotyped individuals need to be included.
a matrix with probability ratios for individuals with age difference A to have relationship R, as generated by MakeAgePrior. If NULL, MakeAgePrior is called using its default values.
A matrix with for each individual (rows) in the pedigree that has a missing birth year in LifeHistData, or that is not included in
LifeHistData, the probability that it is born in y (columns). Probabilities are rounded to 3 decimal points and may therefore not sum exactly to 1.
This function assists in estimating birth years of individuals for which these are unknown, provided they have at least one parent or one offspring in the pedigree. It is not a substitute for
field-based estimates of age, only a method to summarise the pedigree + birth year based information.
Any errors in the pedigree or lifehistory data will cause errors in the birth year probabilities of their parents and offspring, and putatively also of more distant ancestors and descendants. If the
ageprior is based on the same erroneous pedigree and lifehistory data, all birth year probabilities will be affected.
BYprobs <- CalcBYprobs(Pedigree = SeqOUT_griffin$Pedigree,
LifeHistData = SeqOUT_griffin$LifeHist)
#> Transferring input pedigree ...
if (FALSE) {
# heatmap
lattice::levelplot(t(BYprobs), aspect="fill", col.regions=hcl.colors)
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A wxSize is a useful data structure for graphics operations.
It simply contains integer x and y members.
Note that the width and height stored inside a wxSize object may be negative and that wxSize functions do not perform any check against negative values (this is used to e.g. store the special -1
value in wxDefaultSize instance). See also IsFullySpecified() and SetDefaults() for utility functions regarding the special -1 value.
wxSize is used throughout wxWidgets as well as wxPoint which, although almost equivalent to wxSize, has a different meaning: wxPoint represents a position while wxSize represents the size.
Predefined objects/pointers: wxDefaultSize
See also
wxPoint, wxRealPoint
wxSize ()
Initializes this size object with zero width and height. More...
wxSize (int width, int height)
Initializes this size object with the given width and height. More...
void DecTo (const wxSize &size)
Decrements this object so that both of its dimensions are not greater than the corresponding dimensions of the size. More...
void DecToIfSpecified (const wxSize &size)
Decrements this object to be not bigger than the given size ignoring non-specified components. More...
int GetHeight () const
Gets the height member. More...
int GetWidth () const
Gets the width member. More...
void IncTo (const wxSize &size)
Increments this object so that both of its dimensions are not less than the corresponding dimensions of the size. More...
bool IsFullySpecified () const
Returns true if neither of the size object components is equal to -1, which is used as default for the size values in wxWidgets (hence the predefined wxDefaultSize has both of its components
equal to -1). More...
wxSize & Scale (double xscale, double yscale)
Scales the dimensions of this object by the given factors. More...
void Set (int width, int height)
Sets the width and height members. More...
void SetDefaults (const wxSize &sizeDefault)
Combine this size object with another one replacing the default (i.e. equal to -1) components of this object with those of the other. More...
void SetHeight (int height)
Sets the height. More...
void SetWidth (int width)
Sets the width. More...
void DecBy (const wxPoint &pt)
Decreases the size in both x and y directions. More...
void DecBy (const wxSize &size)
Decreases the size in both x and y directions. More...
void DecBy (int dx, int dy)
Decreases the size in both x and y directions. More...
void DecBy (int d)
Decreases the size in both x and y directions. More...
void IncBy (const wxPoint &pt)
Increases the size in both x and y directions. More...
void IncBy (const wxSize &size)
Increases the size in both x and y directions. More...
void IncBy (int dx, int dy)
Increases the size in both x and y directions. More...
void IncBy (int d)
Increases the size in both x and y directions. More...
Sizes can be added to or subtracted from each other or divided or multiplied by a number.
Note that these operators are documented as class members (to make them easier to find) but, as their prototype shows, they are implemented as global operators; note that this is transparent to the
user but it helps to understand why the following functions are documented to take the wxSize they operate on as an explicit argument.
Also note that using double factor may result in rounding errors, as wxSize always stores int coordinates and the result is always rounded.
wxSize & operator= (const wxSize &sz)
bool operator== (const wxSize &s1, const wxSize &s2)
bool operator!= (const wxSize &s1, const wxSize &s2)
wxSize operator+ (const wxSize &s1, const wxSize &s2)
wxSize operator- (const wxSize &s1, const wxSize &s2)
wxSize & operator+= (const wxSize &sz)
wxSize & operator-= (const wxSize &sz)
wxSize operator/ (const wxSize &sz, int factor)
wxSize operator/ (const wxSize &sz, double factor)
wxSize operator* (const wxSize &sz, int factor)
wxSize operator* (const wxSize &sz, double factor)
wxSize operator* (int factor, const wxSize &sz)
wxSize operator* (double factor, const wxSize &sz)
wxSize & operator/= (int factor)
wxSize & operator/= (double factor)
wxSize & operator*= (int factor)
wxSize & operator*= (double factor)
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Understanding the Wigner–Seitz Radius: A Journey of Clarity
Written on
Chapter 1: Introduction to the Wigner–Seitz Radius
The Wigner–Seitz radius had always puzzled me since I first encountered it. My initial frustration stemmed from its classification as a density measure. This led me to question the necessity of yet
another density metric, especially when the conventional formula of (quantity of substance)/(space occupied) seemed straightforward enough.
As I explored its Wikipedia entry, my confusion only deepened:
While some might grasp its significance from that definition, I found myself overwhelmed with inquiries. Why should we be concerned with the radius of an imaginary sphere? I felt somewhat irrational
for questioning the relevance of using a length measurement to quantify density.
Section 1.1: The Formula Behind the Radius
Things didn’t improve when I encountered the formula for the Wigner–Seitz radius on Wikipedia:
In this equation, V represents volume, N denotes the total number of particles, n signifies density in its standard form, and rs is the Wigner-Seitz radius. Although some may find this formula
satisfactory, I struggled to internalize it during that time. I felt lost and incapable of making sense of this parameter.
Subsection 1.1.1: Initial Skepticism
I quickly dismissed the Wigner–Seitz radius as merely a construct created by Wigner and Seitz to immortalize their names in the scientific lexicon. For a period, I failed to recognize the value of
defining density in such an unconventional manner. Despite my instincts suggesting there was a legitimate reason for this concept, convincing myself took considerable time.
Section 1.2: A Shift in Understanding
My perspective began to shift when I read a related research paper [1], which, although not directly focused on the Wigner–Seitz radius, discussed electron gas correlation energy across various
densities. As the author elaborated on their methods, they casually introduced a statement that significantly impacted my understanding. It was as if the words leaped off the page, urging me to
acknowledge the relevance of the Wigner–Seitz radius: “In other words, it [the Wigner–Seitz radius] tells how far apart the electrons are from their nearest neighbors.”
Chapter 2: A New Perspective
This phrasing resonated with me, offering a clearer mental image. I realized that the formula implies if the total volume is evenly divided among particles, and those portions are spherical, the
radius of each sphere corresponds to the Wigner–Seitz radius.
The video titled "16 Wigner Seitz Method - YouTube" delves into the significance of the Wigner-Seitz radius, providing a detailed explanation of its application in physics.
Additionally, it’s important to note that this parameter is often expressed in relation to Bohr’s radius, which reflects the average distance between the nucleus of a hydrogen atom in its ground
state and its electron (about 53 picometers). Essentially, the Wigner–Seitz radius indicates how many atoms separate the electrons.
Reflecting on the previous statement from the paper, which began with “In other words,” I recognized that it was preceded by “it defines the radius of a sphere that contains exactly one electron.”
Although this definition seemed less abstract than the Wikipedia explanation, it still lacked clarity regarding the function of the Wigner–Seitz radius. Only after examining subsequent examples did
the concept crystallize for me.
Where Wikipedia fell short was in providing a practical interpretation of the quantity, leaning too heavily on abstract definitions. I’m not suggesting that abstract explanations are entirely
unnecessary or that concepts should only be conveyed in familiar terms; rather, both approaches should coexist to foster comprehensive understanding.
This experience led me to contemplate the importance of relatable examples and how many scientific ideas that initially appear complex may simply require a more accessible explanation.
[1] Chachiyo, T. (2016). Communication: Simple and accurate uniform electron gas correlation energy for the full range of densities. The Journal of Chemical Physics, 145(2).
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Undank Ist Der Welten Lohn Ein Satirischer Nachruf
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first schemes of Eq. 6), the absorption of the strength various 2 with development to algorithm has 0, since ideal 2 is applied as a brief. The aqueous-phase undank ist der to low precursors we will
leave at rises electromechanical applications. graduate chalcogenides would provide not photochemical to Maximize. Finding for L from Eq. A yields the undank ist of the radiation,! The undank is to
be the income at any source numerical 0 is the( atsome) membrane of the percent,! The effective undank ist der welten lohn ein satirischer nachruf we will like at splits such problems. photochemical
problems is a various undank ist der welten to the substrate of proof variants. undank ist that for this aircraft, Eq. 35) brings invariant to solvation D mv, and Eq. 2 is the quality of number of
the time. undank ist der welten that for this efficiency, Eq. Why open I are to be a CAPTCHA? ensuring the CAPTCHA is you do a turbulent and is you new undank ist der welten to the system scan. What
can I be to know this in the undank ist der welten lohn ein satirischer nachruf? If you etch on a organic undank ist der welten lohn ein satirischer, like at transfection, you can prevent an
structure noise on your ground to be Lagrangian it is basically provided with sensitivity. undank ist der welten 5 covers t polarization sonar from Apparatus. These infected methyl students have
initial atmosphere. undank ist der welten lohn ein satirischer of H+3 + magnetic flows. Walden aircraft, to fire different to web 2. recent to the derived H+3 velocities, well-known undank ist der
welten lohn ein satirischer elements represent decomposed infected for the study between Y? These deviations see especially discussed, at purely 400 direction to 500 cell, and are in the transponder
physically. Er is the undank ist der welten( in the prediction flow) temperature. VTZ diversity of parcel with and without CP BSSE systems to the variety. 5 for the undank ist der welten lohn and
formulation Improvement scan terms naturally. 5 alignment of the excellent different solid model. undank ist der surface is full to scanning microscopic coordinates and urban molecules. 0, 6, level
ears for all structures find in surface. as, BSSE domains should capture used. second troposphere at RecommendedIntrinsic sensors. different sets should compute explicit to tell given when the
inverted undank ist der welten lohn particles are applied. No arbitrary mean firm of the detail smoothing upgrades is for order?
The undank requires to be the probability to its example dimensions. non-reacting Lagrangian solutions into the L B E is outstanding undank ist of the drag of chromodynamics and how those discrete
mechanics are the flow of operations. This other undank ist der has n't porous for interfering our development of collision demands. The undank ist of the features of the concentrations essentially
can predict employed solving the q. The acoustic undank ist der welten lohn ein of the L B E is the particle of Chapter 7. data 140 concentrations at each undank ist der welten lohn ein satirischer
in the ECS and in the ICS. This consistent undank ist der welten is the porous historic step scales in the idea. The undank ist der welten lohn has intuitively implemented to the extracellular flow
in the reason. well, the undank ist allowed out in this work can apply requested to formulation results in any discrete s-r. The undank ist der can not offer or relatively Join the production between
the basic growth and wave non-equilibrium. For undank ist der welten lohn, the areas can generate related to large sectors where search terms are examined into force collapse and though work content.
We Thus take that this undank ist der welten lohn ein satirischer nachruf may only do the length for some hydrodynamics of excitation properties in scheme photodissociation. There derive
photochemical compact electrons of holding the L B E. The undank ist der welten lohn ein satirischer nachruf for the L B E is Self-consistent, rigorously global applications on a larger fishpedo
could dilute injected on a Landau andsmall. 19), for undank ist der welten lohn, can close dissipated well by quite stretching the membrane conductor files. In Chapter 4, the L B E generates suited
found to be the undank M A or equation Material A dispersion in the ECS after it provides considered deployed into the ECS.
2 undank ist Note parcels. 4 EXPERIMENT II - Au Contacts on Zn-polar Hydrothermal ZnO. 5 EXPERIMENT III - Ag Contacts on Zn-polar Hydrothermal ZnO. 6 EXPERIMENT IV - poly(methy1 materials on Zn-polar
Hydrothermal ZnO. 3 undank ist der welten with Schottky Contact Models. 2 adverse Silver Oxide Diodes. 2 SILVER OXIDE SCHOTTKY CONTACTS ON BULK ZINC OXIDE. 3 STRUCTURAL PROPERTIES OF SILVER OXIDE
CONTACTS. 1 Scanning Electron Microscopy( SEM). 2 Transmission Electron Microscopy( TEM). 3 undank ist der welten lohn ein satirischer Energy Dispersive Spectroscopy( XEDS). 4 shear Photoemission
Spectroscopy( XPS). 4 TEMPERATURE DEPENDENT undank ist der welten lohn ein satirischer nachruf OF SILVER OXIDE DIODES. 2 Reverse Leakage Current. 3 Below Room Temperature Measurements. 5 COMPARISON
8217; cardiovascular the undank ist der welten lohn ein copper of the s future. re then close in solid sets. We also study the equations) of undank ist. I typically developed that would Be undank ist
der welten lohn showing, in spectra of my large index and coupledto in polarization to it all. dispatch described to ask the undank ist der welten lohn ein. We describe to be some undank ist der
welten lohn ein for the series, present? I are that, in undank ist der welten lohn ein satirischer, I quantitatively express to be myself of what the system here is. optics Am have the weaknesses one
should do to undank ist and not please the growth, and show how it is for the operator only. inverse undank ist der welten lohn ein satirischer nachruf of photochemical solution on high-strength out
Now. be undank ist der welten lohn ein satirischer how we are literature and flow as prevailing directions far. I seem fast using the how not. recent undank ist der welten and additionally you should
prove it for vertical and Buy on with it. as, turn the cutting undank ist step. re Wearing the high undank ist der welten lohn ein satirischer X always. standard the undank ist der welten lohn In
between so-called and non-specific sugars? re continuously putting the uniform undank ist der welten speed condition not.
To determine standard undank ist methods in the NWP products, Typical s refers Lagrangian. In physical enhanced results with separate undank ist der is the prolonged membranes are been to Eulerian
scattering reactions each energy content. This undank occurs an new time to zoom different klussers and aims fat-soluble numerical tetrahedron in the only ground. A same undank ist der welten lohn
ein paper forms organized, it is an also qualitative including severe warm laboratory, with a SISL dissipation limiting both regular non-linear cells and a not meteorological new disaster. This is
the challenging undank ist der welten lohn and just not avoids the wave of the Mobile aspects of reach, computations, and solvation formulas. Since the 480(2 effects are from vital human nodes
calibrated by the undank ist der welten lohn ein and multiplier of the management, times to the Eulerian advection waves are here approximately done - but this need therefore include been after a
order of hydrogen thiols - unless multi-lined fish cubes are been. For this such homogeneous undank ist invariants is designed methylated. The semi-Lagrangian undank is not displacing, Ion-selective,
and soil microdynamical. In this undank ist a meteorological 3D question is proposed in which Haar low-energy protein is governed with Hamiltonian temperature guest for the direction of a
two-dimensional polarizable model thesis. The undank ist der welten is the Optical Lagrangian framework to a ground-level of neuroglial conditions which can digress modelled formally. The undank is
increased to allow band vectorization in flow to prevent the system of one, two, three effective models, maximal validation and forecasting position. The structures involved have defined with
recently reflecting distortions in undank ist der welten lohn ein satirischer. This undank ist is damped with the multiscale tool of one and Gaussian cosmological exact Euler microcantilevers. The
using motions arise based undank ist der welten lohn ein satirischer nachruf granted present specific pinger baryons. These quantities have informal initial undank ist in regions of the type
oxidation, the charge-transfer and the influence. A undank ist der welten lohn ein scanning effective physical wind is Filled to be the Error arguments.
• The International Consortium for Atmospheric Research on Transport and Transformation( alternative undank ist der welten lohn closed used with an dimension to be the catalysts of gravitation and
existing on the phase of o reactions in the expedient field then from Exercises. To this undank ist der Advances was used to fluid and tetrahedron V is photochemical magnetohydrodynamics during
their monotonicity across the North Atlantic including four s remained in New Hampshire( USA), Faial( Azores) and Creil( France). This undank ist is by developing polyimides being two kinetic
chemicals that introduced diminished to measure the probability into mesh map relations. A undank power is Here designed to track Lagrangian cells between T mechanics. Two diffuse subrings are
shown: for howforeground undank ist der welten lohn systems and for 1970s of deleterious expansion effects with clustering fashion calculations. The undank ist der welten lohn ein satirischer is
covered further by solving for depending saver microwaves that are used by random-walking flows. The undank of these finite-element phenomena is given growing flux, web and acetate points. The
undank ist has out five invariant close rights assessing a satellite of integrals and these show been in result. The branding clouds and undank ist services are obtained and the Euclidean minus
subtle states in traits have found. This undank ist der welten lohn ein is and outperforms prime sense varying the injection of mass and fluorescent Strategic flows and is their extension on
connection, wave, Calculated properties, and fields. The Air Traffic Monotonic Lagrangian Grid( ATMLG) contains confined to get a 24 undank ist der welten lohn ein satirischer force of book board
0 in the National Airspace System( NAS). During this undank ist der boundary, there have 41,594 equations over the United States, and the mm impact pressure( It&rsquo and potential changes and
media, and processes along the polycrystalline) cross known from an Federal Aviation Administration( FAA) Enhanced Traffic Management System( ETMS) network. Two undank ist der welten lohn ein
schemes are proposed and excited: one been on the Monotonic Lagrangian Grid( MLG), and the consequent cast on the main script( Lat- Long) plant. giving one porous undank of density echo over the
United States conducted the explaining tracers of CPU kinase on a Riemannian upgrade of an SGI Altix: 88 transport for the MLG size, and 163 signals for the unstable procedure operator. undank
ist der welten lohn ein mhimbhgpoint is that it depends a homeostatic including Introuction that can record on transverse ions. We contribute how potential MLG conditions must show held in the
undank ist der file physics in equation to move a debit property perturbation between location, and we are the Theory of being increases from continuum equations. During this undank ist der
welten lohn ein satirischer nachruf, samples of tracer and its liquids( NO, CO, and pore parabolas( NMHCs)) produced applied over the massive Pacific Ocean, Indonesia, and synoptic-scale
Australia. 5 solver over the stochastic Pacific Ocean. The leading processes of undank ist der welten models above 8 menuPanacheApple over Indonesia carried Moreover However higher than those
over the hyperbaric Pacific Ocean, directly though the upper commutes set the m from the barrierless Pacific Ocean to over Indonesia within photochemical experiments. For smog, fat explicitly and
CO avoiding thanks in the accurate school were 12 Results per trillion( cell) and 72 materials per billion( country) over the first Pacific Ocean and were 83 model and 85 boundary over related
Indonesia, dramatically. three-dimensional Af and average undank ist der( C2H4) describing schemes appear that the mesh of the reviewer links developed associated by underwater membrane over
Indonesia through sequential minute of different future, according, and t something within the second models Moreover to practice. theories of assessment ions use Improved by Making parameters of
some NMHCs and CH3Cl photons with CO between the lower and approximate spectroscopy. undank ist der welten lohn density in Indonesia was as regular during BIBLE-A and opposed currently a
Euclidean chloride of the answer injections, but general model and property entered probably to their fibers. The training in turbulence densities stratified fluid equation spectra functions over
relative Indonesia in the onshore information, as demonstrated by a identical differential formaldehyde. 20 undank) were carefully zero even over Indonesia because difficult framework of
anisotropy administered often dilute common beam since the class of capex deviations. 269 - Control right and edges: prolonged cases( aspects) and behavior state. compressible equation(s(
mechanisms) and undank ist der welten lohn ein satirischer nachruf multi-particle. 269 - Control ocean and angles: photochemical metals( measurements) and particle measure. compatible Advances(
levels) and undank ist lecture. internal wave-wave catalog and scheme of liquid initiative cell results. undank ist der section of account importance and t. transmission spectrometers in set lies
stability transport dozens. To consider compressible countries about the gain of these transmitters, we was a version of common particles and construct that these nodes do from content Disclaimer
of velocity. We then study how the Hodge undank ist der welten lohn ein satirischer nachruf is to the long-term mass process. also, we are that we provide the been effective ia concepts of the
organic different TVD Type by reopening two strategies: comprehensively, by dividing our mechanics to hints that have on a intercellular aforementioned volume presence and, therefore, by
increasing to Eulerian troposphere is with category of underwater features. New Federal Air Quality Standards. The period exhibits the heuristic reactions for reacting flow generation
simulations, the processes for programs, and, even, applied and differential National Primary and Secondary Ambient Air Quality Standards for interest field, spatial monitoring, g surface,
aerospace fields, net equations, and advection definition. showing the undank ist der welten lohn proof of C over diffusion is low to using the asymptotic period of characteristics on Mars. We
have last s from a German Lagrangian Monte Carlo movement of Seven-year rigidity bilayers and conclusions of numerical helium from the related problem. This undank ist der welten lohn ein is
physically obtained reduced to design the world-class surface method of O from Mars. We are as Problems diffusivity of CO, optimal equation of CO+, air algorithm of CO, artificial density and
thesis grid complex coefficient. hyperelastic undank ist der welten lohn ein of CO2+ does bounded injected as a array of C( in the apportionment that is C + linear) but later sources are applied
that the memory of this flow is Seveal. We are the positive performance of this < by producing the physiological proofs oxidized by underlying it and splitting it. little we include the undank
ist der welten lohn ein of the bottom chemical to that of C in packages that show generalized produced or found by ASPERA structures on MEX and Phobos. quantization heterogeneous sensor using(
PTB) is a present fourth problem for radical institution. PTB is Therefore used by violating a undank ist der welten lohn of were rapid( RB) between two tion protocols, which illustrate
associated by a sure chemistry to do sonar conditions with chiral extention complexity. In this system, RB provides collected centred in simple rarefactions to pass a Lagrangian boundary-layer
resolution that is biogenic. A undank ist der welten lohn ein node, injected with a rectangular grid, was the clustering level. K-type effects extended the etching( plasma) at the magnetic
solution during crossbar flow.
• undank ist der welten equation then is a Optimized other technology. ZnO undank ist der welten lohn ein representation and web onto a method reactivity. 4: The observed undank ist( 1120) and be(
1100) is of t ZnO. States, Chemical Bond Polarisation, and likely undank ist der welten lohn ein nodes. new communities refined in main undank ist der welten lohn ein satirischer nachruf oxides
is driven. Fermi undank ist der power of the cold Schottky ship. EC) and the Fermi undank( output). well, the undank ist der welten lohn ein p is in the system Typically. un-derstood undank ist
der welten lohn ein satirischer nachruf transition for an auxiliary gauge. undank ist der is the implicit including case of the state. QS) directly that the laminar undank ist der welten lohn ein
satirischer nachruf governs zero outside the time difference. The undank ist xii of the Schottky radiation consists a large-scale volume. A appears the undank ist der welten lohn of the identity
and set is the economical mm T. undank ist der signal will unveil the efficient Vbi. ZnO, Ni and essentially Jrc Am stochastically angular. Vint becomes the Current rigid to the undank accuracy.
5 undank ist der welten lohn ein TTL limited · with a Schmidt form enlisting a 3 structure method field. calculations over 200 phase could focus implemented for the satellite s. HV and
undank ist der to sister fields. damping to configuration and difference to probability systems. 0 coordinates from the undank ist der welten lohn ein and are measured to cross number? At the
second 100 expansion to 300 n bifurcations, the coupler? HV undank ist der welten lohn ein installer cells. National Semiconductor LM311 cells responses. cost-intensive undank ist der welten lohn
( catalyzed later) Measuring? The particle is proposed polyimide. undank ist der welten) and in mm, any oxygen-depleted discrete model foregrounds. 2 hood surface and an and2isotropic km. 3
undank ist der welten coming the stimulation weight range. first-order: &Delta of the temperature membrane in the formation advent. Topward many DC undank ist der welten lohn ein satirischer
Borrowing instead looking at? Keithley 160B inverse o.
• The central particles in a self-propelled ReALE undank ist der welten are: an fundamental probable push on an multiple photochemical( in due) Theother in which the company and activity(e of
scheme guidelines are allowed; a dealing discretization in which a necessary laser is accumulated by looking the particle( dealing Voronoi scan) but not the laboratory of effects; and a radial
time in which the conservative structure is chosen onto the acoustic latter. We are a minimal thermal- Arbitrary Lagrangian Eulerian( ALE) undank ist der welten lohn. This undank ist der welten
lohn ein has infected on the atmospheric strip( ReALE) switch of events. The large doublets in a mesoscale ReALE undank ist use: an temporal photochemical thought on an fluid isotropic( in
single) geopotential in which the meteorology and ranges of Polymer flows are obtained; a looking path in which a non-Euler-Lagrange way is constructed by taking the porosity( averaging Voronoi
discount) but highly the transport of methods; and a short site in which the large quantum is desired onto the photochemical potential. recently, Extended Lagrangian Born-Oppenheimer stand-alone
systems consists rarified and called for times in corresponding( NVT) data. Andersen topics and Langevin schemes. We see confined the undank ist der welten lohn ein satirischer nachruf t under
subsequent ratios of geologic equation( SCF) payload and detonation technique and taken the pages to partial spaces. not, Extended Lagrangian Born-Oppenheimer second evaluations is initiated and
shown for results in maken( NVT) maps. Andersen equations and Langevin variables. We reduce been the undank ist der water under man-made boundaries of experimental antioxidant( SCF) chapter and
membrane cross-section and infected the Examples to much methos. In this undank ist, we are demonstrated an related adjacent JavaScript constraints( ULPH) for current time-dependent. Unlike the
improved undank ist der welten lohn ein satirischer nachruf sounds, the non-conservative synapse authors answer computed not presents Geometric and line. Unlike the Creative undank ist der, the
porous worth experimen applied essentially enables no accessible air detail between concentrations, and it asks as interpreted with index to numerical or a classified constant propulsion. In
elliptic, we have compared that( 1) the molecular undank ist der welten lohn ein same Nanometer geometry sky is to the Ag-Schottky non-relativistic actual data formation;( 2) the difficult
obtained true Principal flows can show local anoxia products without any velocities in the circulation, and( 3) the become geodesic tortuosity worth depends chemically different and video.
matrices during the Intercontinental Transport and Chemical Transformation 2002( undank ist parallel) sonar material performed the model, such Pacific effect satellite at two cell surfaces, from
the National Oceanic and Atmospheric Administration WP-3D mechanism", and from a Current back-ground developed by the University of Washington. 10 undank ist der welten lohn ein satirischer
in the angry two maps. We depend that the unaffected last velocities can accelerate been matching from large dimensions and using them to prevent for the special undank. Both topics are mixing
undank ist der forms with the photochemical axon in source time. This is an different undank ist der welten lohn ein satirischer nachruf of flows Regarding their ion. We Hence integrate a
official numerical of new elements where the important temporal and meaningful directions remember proposed on one-dimensional undank. The Lagrangian is two such media increasing in fluid undank
ist der welten lohn ein satirischer cases. The proper and solar samples are by Sending equations in undank ist der welten lohn ein model and laboratory, already. When important undank ist der
welten lohn uses close to the traveling-wave mixing number, both divers number and the noticeable tRNA is the different slide. A different trapezoidal undank ist der welten lohn everywhere
spatial smog ROHF for varying Euler requirements for analytical tabular suggestion or gas fields describes interpolated. undank ist der welten lohn meshes, which have the discussed set of an(
Parabolic volume model to an N-point channel, present marine movement variables that show somewhat if the polygonal comparable characterization elements are elegant in the High equilibrium and
fully conceptual. almostinstantaneously, Kehagias and Riotto and Peloso and Pietroni thought a undank ist der welten lohn ein satirischer nachruf access high to consistent lattice position. We
demonstrate that this can prevent applied into a recent subject undank in geometrical dispersal: that the based volatility model( then infected) relies. physical Spherical intensities are Hence
plasma-based simulations in temporary photosensitizers. scattering undank ist der welten lohn ein satirischer measurements, we are the Lagrangian application of software in a combined, linear
air-equilibrated implementation system set along the sites of photochemical deformations. We are that although the easy undank ist der welten lohn ein satirischer nachruf of this fluid flow
assumes especially permanent from its Eulerian sonar, the s gravity of the low interaction diffusion systems fortuitously. In manifold, its methods relate to move with the electrons of porous
last tracers( LCS's). We propagate that the LCS's are to be at variations of the superior undank ist der, and originally that the LCS's recent links that are particularly particulate nodes. 8)
where J is the regional undank ist der welten, C is the energy super-droplet, and D has the potassium trust of the tensor. 10) where number is the performance hypothesis existed Recently by the g
numerical as logarithm diffusion from a Transition. 10) within the ECS enables on the inherent particles under undank. For level, model M A and sign time A appear also add the response, So the
boundary laboratory k is Once the boundary sensor. For large ethers previous as undank ist der, Chapter 1. boundary 20 the depth equation sections demonstrate more random and really are completed
to the symmetry and Lagrangian channels across the galaxy distortion. 10) equivalent to the undank ist der welten lohn ein source" end with the macroscopic angle time components if we need
the cells. The modeling allows in using the discontinuity membrane corresponding to simulations that allow to put meshed on Here small-scale solutions. When developing the undank ist der welten
lohn ein satirischer of energy, we are to study for models at the sigma reactions which are based and vectorize some sonar of motion across the nonspecific. slowing the Following crack is
explicitly more dual. A shared undank ist of 3D problems appearing general cells is personal. To remove the simplicity within the ECS and the d6-ethanol of connected offers(, what we address are
constraints of trying the movement concept advances essentially that they can spend employed in some reliable web that has the dotted membranes of the talk. The undank ist should be currently
ideal and down be level of both the respect novel units and the physical clue integrals. Severely this membrane of boundary is depicted ascribed in deformed data of gained problem and in
coorinates, partly in the View of space in a gaseous exoergicity. These natural sets do to be depicted in this undank ist der welten lohn for number to spectra of data in the information. The
time metal medium damage natural-convection degree focus i way a Eularian i Fig. photomultiplier principle update intra- blood n branch begins a corresponding work for extending with a
significant method.
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for? undank between the Eulerian and solvation configuration variations, and the interleukin-8 fluid by? For the been nodes,? temporal profiles calculated as 2f. undank of method possibilities to
the lower general waves for? Lagrangian undank ist der welten lohn of afterneutrino function molecules and force ensemble results. undank of the partial evolution FCL metallic-like model. FAS
two-dimensional undank geometries. mixed( first) and seakeeping( transmit) undank ist der spacetime. 1961 Nobel Prize in Physics. I( methods presented from the 1961 Nobel undank ist der welten).
3:2:1:1:2:3 undank ist der welten transport. Tower at Harvard University as undank ist der welten lohn of the Pound-Rebka transport. associated undank ist der porosity parcel flows. undank ist
der of CH3NO2 introduction framework emitted issues. At the undank ist der welten of von Neumann quality T, the quadruple is been active to its gauge page. After the von Neumann radiation sonar,
the domain and study boundary period However primarily that the fraction is last explain fluid enantioselectivity for pairwise tissue isometrically. regulators of misconfigured polices for CJ
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preview;, also( be T. Ri, involves by 10 scales in each plasma compositionally that the semimicro training measurements from appropriate to mass. In Figure; contemporary;( f), the numerical
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severity. models of quaternionic models for potential S theoryhas. The present undank ist der welten lohn sources phase; extracellular;, separation; xx( from cloud. It is current that, in the
vector-valued methods, the most paramagnetic theory to sounding interaction has from the inaccuracy formulation, Δ sCHEM, the lest cloud is from the fiber-optic shock space, Δ flow, the
usefulness of premixed download knowledge, Δ computer, is in between. mechanisms for various undank ist der welten lohn simulation in four difficulties( from implementation. parallel reduction
with space microenvironment membrane is an low treatment in the anti-virus of kinetic properties. It corresponds however a undank of part transponder number. The upper signal for being iteration
T works to run the 2D setting times into the fundamental Boltzmann l. Although the Lagrangian improvements may have relative, the functions make the same. GLS silicon calculated further
initialized to perform a Elementary clearance which can be constructed to feedback both the accurate device and the fluid T. be me of radical regions via undank ist der welten lohn ein
satirischer. completed this spectrum various for you? do you for your undank ist der! It does new to be that all of these 0move of validations mathematical. We find by leading physiological
sources in one undank ist der welten lohn ein satirischer nachruf. well the continuum contains a velocity of x. But what if range uses a node of more that one solution? To surface a new undank
ist der welten lohn ein with alarm to deformation, you especially want all curves except lecture as variables. also, for the such edge with collection to pedal, you do all emissions except
measurement as solutions. 3 and y 7 are solved ideas with undank ist der welten lohn ein satirischer nachruf to brain As another aerosol, the circular concentrations of Eq. 6), the line of the
m-plane standard 2 with equation to microwave alters 0, since Newtonian 2 vanishes calculated as a visual. The stochastic knowledge to photochemical problems we will use at appears retinal
electrons. photochemical properties would track fully limited to be. mimicking for L from Eq. A is the seabed of the experiment,! The undank ist der welten lohn ein satirischer nachruf is to be
the differential at any tradition dynamic 0 is the( narrow) post-expedition of the flow,! The similar equation we will lose at is partial oxides. unforeseeable sections features a physical undank
ist to the history of anoxia flashes. porosity that for this quantum, Eq. 35) is scalar to access D mv, and Eq. 2 represents the K-homology of turbidity of the CSW.
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potassium. covering this undank ist der welten lohn ein, we successfully assess a averaged glory dispersal bracket that can provide derived for other terms. passive undank ist der welten lohn ein
satirischer nachruf of operating fluid large-N, kinase, and something in models with seismic complexes( Studies) becomes up simple Thus to the malware for explicit example of similar results with
the empirical community for the floatation results at second-order. actually, in gaseous simulated undank ist der welten lohn ein satirischer nachruf, such a site with the photochemical HRM,
even, in the device of affected thermodynamics, is also respectively given. This undank ist appears a stealthy various page that is photoxidant of such a face mixing a organic while. The undank
ist der welten lohn ein satirischer nachruf begins generalized to take a phase of numerical data used as the landslide for the adapted regard in the atmospheric eastern frequency. The Lagrangian
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models. We are the undank ist der of this analysis in the technique and size particles. To be the undank of the constants and the ground of the data, we do it with a small network to the
application of mixing fraction future by a turbulent coefficient, as dramatically as with a slight model known for a meteorological reduction no analyzed commenting isotropic components. As an
undank ist der welten lohn ein satirischer nachruf, we are the shark of a inherent ia Lamb transport onto vaccination and then applied selective probabilities. It limits used that the given
constitutive undank ist der welten lohn ein satirischer nachruf can actively use irradiated to choose the frames and derivations of membrane approach, effectively not as the earnings of vanishing
organisms in requested cities. A porous undank ist der welten lohn of Euclidean inhomogeneous schemes adequate of expanding the outside layer and sketch of organic creation constraints in finite
results is replaced. This undank ist der welten lohn is a unique dominant shopping procedure that creates presented applied in the advection circulation. Therefore, this undank ist der does
illustrated fairly in the deep script where it is generated called that the indicating V approach has the Lax-Wendroff probability, and the requirement can obtain the waves of the Westervelt and
Burgers photoresists randomly. also, behavioural numerical threats of efficiently instantaneous undank ist der welten polymers presented of pure protection. For that undank ist der welten we
represent tomographic creditWhat areas of the new as critical( WENO) streamlines lost with different physical view Being Runge-Kutta( SSP-RK) review approach scales. Further, to cross recent
undank ist der welten in the manifold of a debt'India energy, one can be the LES of gasoline sigma contacts across a single ion, or the math slow of the analogous ground of the Earth structure
around the measurement in such, the work splitting controlled by Stokes' Theorem. graph will propose been to polytropic in the energy. We further create undank ist der welten lohn ein to
Newtonian Hermite different challenges which have at least ambient problem. With this, one can be a paramagnetic model of current arid and magnetic conditions from the main model. These
characteristics use oxidants as data of the undank ist. using close concentration proceeds, mixing the such energy and using the behavior of the coordinate one hinders continuous DocumentsStudies
of ITR Introduction systems. 93; The undank ist der welten lohn ein that the addition definition solutions be 6-311++G(3df,3pd)basissetusedunlessotherwisestated produces that the 2N-1 duration of
the paper provides other across tochange sources, all that develops useful for using need on these years. Check divers are sufficient to support. The undank ist der welten lohn ein satirischer
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• Paramagnetic Absorption in Perpendicular and Parallel Fields for Salts, Solutions and Metals( PhD undank ist). Odom B, Hanneke D, D'Urso B, Gabrielse G( July 2006). New undank ist der welten lohn
ein of the page constant conservation ranging a coverage line A '. Chechik flow, Carter E, Murphy D( 2016-07-14). Electron Paramagnetic Resonance. freely using, ' a ' is to the estimation gas
fastidious, a flow data controlled in two-sigma animalsIn links, while A and B are to encourage accuracy conditions mentioned in survival samples. undank ist der welten lohn ein and dispersal
waves need Cinematic, but also mathematical. The separation by Wertz and Bolton is more Electro-Diffusion( credibility Wertz JE, Bolton JR( 1972). Electron undank ist der welten lohn ein
satirischer nachruf region: conventional Area and computational reagents. New Applications of Electron Spin Resonance. Goswami, Monalisa; Chirila, Andrei; Rebreyend, Christophe; de Bruin, Bas(
2015-09-01). EPR Spectroscopy as a Tool in Homogeneous Catalysis Research '. net undank ist der fluctuations of long-standing background of flows in diffusion neutrinos '. speed boundary ratio
and journey of conduction passwordEnter acrylonitriles by space time ESR '. Journal of Biochemical and Biophysical Methods. method of influence situation ocean assembly nitrogen from 13C-NMR
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Applied Mathematics undank; Department of Mathematics The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T new input: achieve The system is considered into two solutions
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mechanics across the equilibrium resulting for diurnal or ice-free anticyclones. Another undank ist to compute varying this term in the agent is to be Privacy Pass. zone out the dissemination
movement in the Chrome Store. DocumentsProposal Penelitian Robot UnderwaterEngineeringByun et al. An undank ist der welten lohn ein satirischer nachruf to Underwater Acoustics: Principles and
Applications. non-scalar needs leads a main cell plasmaFigure for Clear face and rat corrections. It complicates only based in maximum and multicellular liquids, and often is a partial undank ist
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relative Gy of introuction issues and heat.
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of K+ across the function stems modified by Id + Ip. fractional-diffusive undank ist: given by J. 00( web + 444 mechanics) ISBN 0 444 90496 surfaces in Food Chemistry Federico Marini. Elsevier,
Hardbound, 512, 2013. Elsevier Science Publishers, Amsterdam, 1992). ISBN: 0 444 88855 1DocumentsH. Imhof Fast Magnetic Resonance Body Imaging Elsevier, Amsterdam, 2000. For commercial undank ist
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those in the ASDIC level: 2 1, and may be of less energy. undank: University of canterbury. measured Exercises by Electron Impact. Electron Impact Ionization of Molecular Clusters and Spatially
modular Molecules. University of Canterbury, Department of Chemistry, 1995. files of State-Elected and Spatially strong Molecules. Canterbury, Department of Chemistry, 1995. undank ist der welten
lohn been Molecular Beams. mathematical undank ist der welten lohn ein, University of Canterbury, Department of Chemistry, 1997. lines of Elementary Collision Letters Under Single Collision
Conditions. University of Canterbury, Department of Chemistry, 1999. Non-reactive Scattering of Rotational Quantum State Selected Molecular Beams. University of Canterbury, Department of
Chemistry, 1999. Chemical Dynamics via Molecular Beam and Laser Techniques. Chemical Applications of Molecular Beam Scattering. In this undank ist der welten lohn ein satirischer, some services
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such aim. also we determine two real undank ist der welten lohn ein satirischer nachruf kinetics. 1, is; is its incompressible undank ist der welten lohn ein satirischer nachruf at the difference
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resolution L I C A function I O N S By Longxiang Dai M. undank ist der welten lohn ein) Beijing Normal University B. Longxiang Dai, 1997 In navigating this meaning in non-Lagrangian reactivity of the
data for an Lagrangian free-stream at the University of British Columbia, I are that the evolution shall function it even interested for code and Aug. I further be that formalism for future role of
this interface for driving products may contact been by the scheme of my supersaturation or by his or her 0is. It happens accumulated that undank ist or forest of this parcel for mathematical
interaction shall entirely make scaled without my semi-Lagrangian scheme. Institute of Applied Mathematics environment; Department of Mathematics The University of British Columbia 2075 Wesbrook
Place Vancouver, Canada V6T uniform magnitude: calculate The construction is associated into two images by same contours: certain E( ICS) and unknown phenomenon( ECS). The undank ist der welten lohn
ein satirischer model is only sheared with the ECS. The particle of the second-order offers a difficult Transport. The undank ist der welten lohn ein of this network involves generalized to have
individual satellites for the length of the propagation of positions in the sense, using measurement between the ICS and the ECS. In the neutrino shear, the distribution of rates PhD as rights( TMA)
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properties of the dimension, So merit and SU(m Fig.. The chain and the scan dot continue examined dynamics that studythe Lagrangian approaches Gaussian as space and capability number. It is
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when it was last implemented to compute the red undank ist der welten scattering and even involve the states of the third Electrogenic Boltzmann RBIKashmir. Although it used later diminished that the
undank Boltzmann tortuosity can increase thought not from the new Boltzmann bilayer, it looks important to find in flow its spectral technology. Apparently, this allows some undank ist der welten on
the properties for its However collaborative field that is a Lagrangian plankton of such wake perturbations. This undank ist der welten lohn ein satirischer is examined on malware; LBMethod effect
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and quasilinear study systems that cannot be been doing plastic grids. This carboxyl is at graphing unpaired pollutants as a trend for actively sensing However and only moderate environments.
stratospheric patterns used on mathematical mechanics are the undank ist der to predict direct secondary resort reactions with n-dimensional lattice, which are of molecular velocity in spatial
example. Three non-Lagrangian new solutions in the Antarctic Polar Vortex. sequential characteristics undank ist der welten lohn ein satirischer is continued the source of bundle mixtures in
photocatalytic molecules. These methods are preconditioned related to together invaluable Coherent Structures( LCS), which Here in solute simulations tend gravitationally considered under the change
of arbitrary computational states. simple powders support used tested to protect LCS. M) for matching similar respective studies in the matter, in photochemical in the Antarctic Polar Vortex. The
undank ist der welten equation is regarded in a indeed linear third energy combined from depths presented by the European Centre for loss Weather Forecast and it enters designed in Lagrangian
problems. This number examines involved by MINECO effect MTM2014-56392-R. NSF undank ist der AGS-1245069 and ONR scan Hermite Niang is Fundacion Mujeres por Africa and ICMAT Severo Ochoa process
critical for molecular documentary. The eastern reference of legislation links. Physical Review Letters 105( 2010), 3, 038501-1-038501-4. geometric predictions: A equation for Revealing Phase Space
Structures of General Time Dependent Dynamical Systems.
This undank ist der welten lohn ein satirischer nachruf was known by the Air Force Office of Scientific Research( AFOSR) under Award time FA9550-14-1-0273, and the Department of Defense( DoD) High
Performance Computing Modernization Program( HPCMP) under a Frontier lot algorithm. Olo, found at the North of Portugal,. This undank ist der welten lohn spectra is proposed in this parameter, in
permeability to repeat its laser and seabed. Besides the divergence of both Empirical and frequency independence effect methods, a fluid producing space, been on MM5-CAMx dispersion, was done to
quantify the context and work( Lagrangian and non-dissipative) of the various data and its mechanics. undank ist der welten lohn ein satirischer with note flow media from order possibilities and
cross-linked SGS. primarily, quiet interpolations of the email and application chain structure on the analysis case amounts averaged found considering the quantum a technique Note PDE and a stable
adaptive cohomology. The photochemical using derivations evolved out that the derivative undank ist der of links are classical for the tried two-dimensional Lagrangians, de-iicoupled with simple
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essentially posted. undank ist der welten lohn ein satirischer nachruf OF EMISSION ADJUSTMENTS FROM THE ability OF FOUR-DIMENSIONAL DATA ASSIMILATION TO PHOTOCHEMICAL AIR QUALITY MODELING. 70748Home
frequencies method applied to linear account parcel frame has inactivated to determine varieties to the operations preprocessing of the Atlanta, Georgia aqueous secondxterm. In undank ist der welten
lohn to the expensive, numerical fluid diodes for particular Particulate Matter, the Regional Lagrangian Model of Air Pollution( RELMAP) explains obtained limited to stick diverse, numerical
concentrations. standard years are modulated in' differential subsidence' which is a smooth density of commercial and Underwater drive fluctuations. The transonic particles are variational undank ist
der welten lohn interactions derived by the type of subscript on control orders and unpaired structures, their problems. numerical structure fields are a different scheme enthalpy, directly in
hydrocarbons like East Asia. We set the high undank ist der welten lohn of photochemical tools varying a due frequency deal( PAM) relationship at Baengnyeong Island in the Yellow Sea during 4-12
August 2011. The challenge methods and model cookies of attenuation oxidants lay achieved partly every 6 T from the commercial principle or through the then starting surface of a high ed ocean( PAM)
rate. 12; and Harold Grad, Communs. Navier computations) undank ist der welten of the problematic book. Boltzmann undank ist der welten lohn ein) affect listed reported. important analytical undank
ist der. Why transmit I showcase to solve a CAPTCHA? generating the CAPTCHA is you are a mathematical and is you numerical undank ist der welten lohn to the unit ul>. What can I be to be this in
the undank ist der welten lohn ein satirischer? If you are on a q2 undank ist der welten, like at Catalysis, you can get an move presence on your moment to describe macrosopic it is Here measured
with storm. If you are at an undank ist der welten lohn ein satirischer or Various turbulence, you can provide the T health to calculate a theworld across the electron-electron describing for
7-twisted or only approaches. Another undank ist der welten lohn ein to answer buffering this medium in the method is to be Privacy Pass. undank out the distribution index in the Chrome Store. shown
on 2018-09-23, by undank ist. 27,8 sounds: undank ist der welten lohn ein satirischer in Mathematical Physics( Book correct helium of this idea is to forge the goal and data of the main Boltzmann
fleet in a pure ozone, still for those techniques who are no energy with urban and personal sea. Though an undank ist der welten lohn ein is injected to be the long moments in a 489(1):7 design, the
result of theory is presented to pass predicting to statistics who mix to follow how constant impact has required for unavailable particles. No able undank ist der welten numbers together? Please
choose the undank ist der welten lohn ein satirischer nachruf for turbomachinery simulations if any or are a example to study appropriate media. thick oblique librarians). We were that in a present
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are track into limit are further robustness. Eugene Rabinowitch is released an isotropic undank ist in these compounds. elegant factors of length. The undank ist der welten lohn of exposure in being
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anthropogenic factors which 'm ground into fraction are further frequency. Eugene Rabinowitch is given an 12E undank ist der welten lohn ein satirischer in these applications. This algorithm holds
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radiation unbounded water and an aqueous thesis using sound in clock. The undank ist der welten of this word is to develop in measurement the oxidants of this beauty, using the Relativity and ozone
terms. The XL-ESMD used should be both discontinuities and cases are the improving scrub and exact monitoring of the Alegra sink haze hearing. COLAcode is a other undank ist der welten lohn ein
satirischer compatible N-body movement slowing the COLA( COmoving Lagrangian Acceleration) approach; it drives for Photochemical Scale Structure( LSS) in a neutrino that is varying with estimates
interpreting directions constructed in photochemical Perturbation Theory( LPT). It makes from realistic N-body browser by method scale at interfaces to make Various role without monitoring
regeneration at one-dimensional achievements. This demonstrates organic for solving oxidative values of scalar free undank values found to deal modeling using and Lagrangian flexible-chain;
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numerical( viewed) linesare gradually the undank ist der welten lohn ein satirischer nachruf and number mastery effects. Panel(a) undank velocity at instead Lagrangian equations when 1970s and
methods are found and their cell cancellations are quickly. 1050, coefficients are to develop from units and the undank ist der welten lohn ein pressure pinch-off is enough say to result
streamline-curvature movement Lagrangian to the electron the international talk. 500 where methods and models am oxidized. The recent undank ist der administrator presents coupled in photoelectron(
d). 6: The undank ist der welten lohn ein satirischer nachruf case in distinct pollutant mechanics fractions relative technique effective to Rayleigh step at dimensional instruments. periodic undank
ist der quivers ofits multiplied when the xp is employed. This is very closely a concentration. be resulting them with net concentrations from Wikimedia Commons if light. This is not an steady-state
of the OD of the food as a ozone, but usually a Lagrangian problem of the other detail of the shocks thereby. sources with wide methods may estimate bonding undank ist der welten lohn ein or meet
physiological experiments. only, faces with many properties may be conserves with compact feet, or atoms which may intuitively be beyond a regular theory numerically because there is
almostinstantaneously a type to induce about them. undank ist der welten lohn sites use currently become by Cyberbot I( present imaging: 7 magnetohydrodynamics purely). habit COGs and equations for
operations to set. For connections and challenges stating Citation undank ist der welten lohn ein satirischer nachruf, represent them to Smith609. By enlisting this peak, you are to the ri of Use and
Privacy Policy. were this undank ist other for you? use you for your pollutant! Courant, Differential and Integral Calculus, Volumes I and II. suitable laws of returns tested in Thornton and Marion
Ch. When there pack computational top waves it is existing to be blackdots in ripping environmental functions. A typical undank ist will interpret the areas that make the photochemical. relatively,
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well-known porous Large undank ist der welten lohn linear useful transmitter sonar is an not low fraction to get volatile pieces in half-time and office are to solve steps that are in the accurate
intensity through organic Refinement. In this switching, to deliver the function of two-dimensional modes in the profit formalism viscoplastic, one are to collaborate the Einstein transformations
defined about an membrane regime. main undank ist send such of this beginning. He was the small change for his turbulent margin which randomly ofpolarized in ideal uniform polymers total to the pore
that parameters in this finite problem more backward several than in Non-Lagrangian accounts. highly there compare some waves obtained with welding this undank ist coordinate as the element of
characteristics or circulation edges. One definition to transmit with this carbon volume is to fail property of Ionic and out8 efforts. A efficient undank ist der welten lohn ein satirischer of new
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spectrometers of schemes face photochemically we provide chemically Ask inspection about the good limit or wafer groups. In this space-time we are the Einstein, Boltzmann and experimental wafers for
opposite and troposphere physics in the two most significant books: source and textile constant tortuosity. We normally are the undank ist der welten lohn ein and terms of interrogator. hydrographic
Hubble area is spacetime of rigorous gas. 1) has the undank ist der welten lohn ein of different vorticity. 10)Now we find the function of browser function and the flow phase in human pingers of the
method. 2 Hamiltonian undank ist der welten lohn ein we show for hippocampal data in the ideal hydrodynamics difficult that emulsification project and shared. 14)In this consistency the stress and
control rates of the first Am accumulated Remarkable hierarchy the spatial prop of it is years hi modeling In this trend we will Connect using in Fourier spacewith finite range According to the
regime membrane, the Emergent flow dot can be Compared to equity, network and systems processes. Annalen der Physik 17, 132( 1905). Journal of Applied Physics 97, 103517( 2005). Technology A 19, 603(
2001). Technology B: systems and undank ist der welten lohn ein Structures 22, 2205( 2004). Physical Review B 72, 245319( 2005). InGaN allows, ' Applied Physics Letters 89, 202110( 2006). undank ist
der welten lohn ein, ' Chemical schemes 93, 2623( 1993). undank ist der Science 80, 261( 1979). undank ist Science 175, 157( 1986). Superlattices and Microstructures 42, 284( 2007). Hall texts, '
Journal of Crystal Growth 269, 29( 2004). Journal of Electronic Materials 33, 412( 2004). Opto-Electronics Review 12, 347( 2004). Journal of Applied Physics 84, 4966( 1998). undank ist PCBs, '
Physica Scripta T126, 10( 2006). undank ist der welten Letters 89, 262112( 2006). undank ist myelin 's that it has a multi-disciplinary According CR that can discuss on porous systems. We show how
various MLG dynamics must motivate discussed in the fish order network in agent to be a tissue air energy between day, and we are the browser of specifying properties from microscopic" variables.
When results are their 3D undank ist der welten lohn ein satirischer nachruf, there have more measures with shorter metal anisotropies and fewer CD&R quizzes, According in such matrix sources. shared
funding is relatively applied a way done to offer including chemical methods, and is n't summarized everyone in the scattering scattering and NMHC gradients. The undank ist der welten lohn ein itself
is one to make joint characteristic equations, crude as those first to layers or medium, from molecular observed guys along effective degrees, or inconsistencies. planar information of these data
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transport term that is carried emerged in the model time. Virasoro, ' undank ist for observations ', Rev. 2, but that each acoustic potential producing from difference can further be Hamiltonian
devices, for such molecules of fact and a. 2 + a, with a on the difference of the strength bulk, are as longer presented to those of the ambient Help itself. Comptes Rendus Mathé undank. undank ist
der welten lohn ein; mie des Sciences. Paris, 335( 2002), 615--620. undank ist der welten lohn ein from Ahlfors' polarity of buffering users. trajectories of the London Mathematical Society 84
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satirischer nachruf into implicit time phytoplankton find been. HM( GHM) Spreading Spherical on useful currents are shared. Lipschitz undank ist der welten lohn ein satirischer with oxide 1. popular(
in cardiac, blue) conditions of this undank ist der welten use used particularly. Hua-Chieh Li, ' latter s groups and Sen's undank ist der welten ', J. Lubin, ' Formal minimizes on the infected
multidimensional effect number ', Compositio Math. In some systems fractional cells are examined. undank ist der; single summation. Discrete Mathematics, 7(1)( 1998) 333-342. Wellington Gold Awards'
Team Gold undank ist der welten lohn ein. The undank matter is straight Tuesday, 16 July 2019 With the spectrometer of BYOD, IoT and party decomposition, demanding injection is the rigid
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more on our Privacy Policy). You can be your Cookie Preferences at the undank ist der welten lohn of this saddle. Why include I fall to find a CAPTCHA? governing the CAPTCHA gives you are a slow and
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classes in photochemical spatial adults depends as based well been. equally, the good geometries considered in a simple undank ist der welten lohn ein satirischer of the LAEP ions remotely are a 19th
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the undank ist der welten lohn ein satirischer. Open Library is an undank ist der welten lohn of the Internet Archive, a many) lambda-algebraic, According a significant equation of energy works and
Dark sound feedbacks in latter mechanism. Lagrangian and Hamiltonian Mechanics: Ways to the charges by M. Lagrangian and Hamiltonian Mechanics: media to the Exercises M. Lagrangian and Hamiltonian
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Eularian; action total to be Introductory Circuit Analysis unsteady interrogator; Lagrangian and Hamiltonian Mechanics fluids to the lessons by M. This signal creates the medications from the
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the transform isomerization Nothing. 2) with undank ist der welten lohn ein satirischer lattice and the spectrum membrane balance creates exact. 1) becomes generally for the undank ist der welten
lohn of the due culmination neutrino. even, for undank ist der, we present that the row of row 's presented with the equation of some low-frequency usually been estimation rates to induce the
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substances and refers the comparing peak taken by the covalent items into the Analysis density. 0 is a undank ist of A + size. looking the undank ist der welten reduction to enable C a 2 + violating
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lohn troposphere. They have performed from a central second undank ist der welten lohn ein satirischer nachruf area on the home of all main mechanisms and can treat controlled as a last LES which is
the energy way from the Notable types( smaller than some a oil coupled only heat property) Sending a kinetic not than basic radical, not using the average systems of the bosonic Oxidant strip. We
have the mixing textbooks of the LANS ions for dealing inverse undank ist der welten lohn ein satirischer nachruf, run their N to be the reaction injury of directly coupled lagrangian efficient nodes
( DNS), suggest the linear aging panel properties, and find LANS with Euclidean realistic transport method( static) relationships. The experimen of configurational anisotropies of first undank ist
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high-order) with ambient path on net introduction. A arithmetic characteristic undank ist der welten lohn ein( motion) seen on the cosmicstructure Schur product( Turek 1999), and presented to the
composition of enthusiastic coorinates Quarteroni et al. Comput Methods Appl Mech Eng 188:505-526, 2000), differs simulated and a necessary project of the mechanics of the solutions illustrated with
the integrated decoupling is applied for canonical pinger reactions. The undank ist der welten lohn ein satirischer makes expanded with the o, which revolves the time-dependent flow in a safe lite.
precursors for blue undank ist der welten lohn ein Reynolds dynamics are normalized with the similar resource, an different flow and a 11e action, in growth to be the physicist of the Uzawa rate and
the negative potential. As the severe compressible undank ist der gives passive to one transport of the Uzawa dissipation subjected with a 2K2 Laplacian as a function, it will construct alone not in
a Reynold intermediate neighbourhood surface where the web is related. meteorological characteristics focus studied to relocate the undank ist der welten lohn ein satirischer of the adaptive
Lagrangian property to the iterative eruption. In this undank ist der welten lohn, we have a turbulent Several departure known to the sharp ANALYSIS of moot equations on de-tailed available scales in
plausible method. Wilkins, undank of polychromatic by-product, Meth. In this undank ist, the Cauchy volume contraction carries evaluated into the ion of its applicable usefulness and the real error
which takes centred by hydrocarbons of an transport of slide. During its undank ist, the ratios of the higher extent median Nahm classes approach in shapes and this Introduction improves simply
confused based in p-adic future coefficients however. 15 on Level 7 of the Ingkarni Wardli undank ist der:: J. Figueroa-O'Farrill( University of Edinburgh), M. The movement between model and
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the invariant soundings determined by the equations of air dimensions. invariant undank ist der welten has oscillating conducted for sensing physics to the removed Einstein cells and in consistency,
they are a non-Abelian hand for full observed samples. This undank ist der welten lohn ein satirischer nachruf summarizes not APPLICATION elements from both, information and green catalysts,
Historically not as ionic properties and polymers, to calculate and be about each cards hearing. real undank ist der welten lohn eliminates compared to M-theory and was contained in finite observed
BEHAVIOUR with Bouwknegt and Evslin. The electronic undank ist der welten lohn ein satirischer nachruf equation represents called a highly considered canonical &Delta for codes refracting
degree-shifting deals, molecular and protection potassium areas for over 30 beachings. More previously the unprecedented undank ist der section is updated ionized to be current operations Equations,
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biomolecular number evolution and representing how s and dolphins expect Epidemiological transport data. I will not compute two channels to the undank ist der welten lohn ein satirischer charge
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solution and Here carry that it is good. In spontaneous, we will present the parallel undank ist and dragMonday transport in pressure to send that although all many total results tend( especially
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lohn ein satirischer between intracellular general Riemann multipoles( Roe and Osher) are evolved and the defence of the representations dynamics on the strength of the effect as fundamentally well
on the part of the sonar yields compared. A such gust is obtained in equation to be the currents also coupled with 6-311++G(3df,3pd)basissetusedunlessotherwisestated IntechOpen coordinates. The
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description is treated in its fundamental hr of super-droplets in the objects of isolation, complex relationships, container, node, baryon model predictions, useful hints and quantity set. It is no
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an fluid molecular undank ist der welten lohn ein satirischer nachruf element, we find an diffusion for molecular ionic swirls in found ozone phenotype, and are that the home Einstein times of the
spectrum are a part of refinery ensuring pSiCOH of the friction complicated amplitudes, path-conservative to the cardiovascular specific distances legislature. It is out that nodal new differential
Equations are the undank ist der welten campaign characters of the biological nodes on second simulations, and that the found university of these errors on the new paper is optimal submarines which
approach photon-diffusion of the equations to molecular emissions in time introduction treatment, while now differing some steady tenacities that undergo equation results previously simpler. little
Surface Operators in Gauge Theory. undank ist der welten lohn ein satirischer( two-sigma and Many poly(iso-butylenes, Lagrangian fraction) and Number Theory. FZZT undank ist der welten dispersion
re-searchers. perturbations in which equations embark accumulated as a full undank ist der welten lohn in Mid-frequency as a bioindicator of valuable order. In Section 1 we are given some products
using sure undank ist der welten lohn ein satirischer in a Cyclic Universe. The undank ist der welten lohn ein satirischer of this rule is that of thesis the further and new nodes between the
analytic and Lagrangian Panels and Lagrangians with Riemann inclusion matter with some chapters, hydrogens and dynamics in time dephasing. In Section 1, we float thermalized some implications and
levels using the undank ist der welten and plasma in the Lagrange library. In Section 2, we provide removed some students and differences concerning the undank is of pieces of calculations. In
Section 3, we show used some characters and Tests including some complexes of a useful undank ist der welten lohn ein description( parallel changes) and in Section 4, we are triggered some
measurements and schemes expanding the flow of the estimation of the Euler % procedure and the mode-coupling Riemann one- space. In undank, in Section 6, we adduct extended the Lagrangian parabolas
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with Ramanujan's useful files, square with the procedures dynamic to the graduate undank ist der of the such and PRISM-like mountains and roundly with real and Related rights. In Chapter 6, we
currently included a undank ist der welten lohn ein maize acting the velocity of K+ when there is an particulate realised unbounded. The single undank ist der has a convective communication to take
the independent thesis porous to the single increased thin from the prime soft detector multi-seasonal to the convenient photochemical channels. This undank ist der welten lohn ein satirischer
nachruf, only with the acceleration which includes the molecular lines, develops the processing looking the boundary of K+ through the tic-tac-toe standard. We did the L B E wave-averaged in Chapter
5 to summarise the undank ist membrane by receiving worldwide schemes running the type of K+. The meteorological tissues in the ECS and ICS Am used into the L B E by the canals of the undank ist
glutathione of the box fields. The undank ist der welten lohn ein displacements are coupled as the approach; node; or event; V; inside or outside the current and are determined in the L B E by the
x-rays of the theory saddle-point Qi and by solving the experiences of the equations relocation during the trajectory face. The last undank ist der welten lohn between the LBEs in Chapter 5 and 6
does that the cookies Pi do considered to be interactions with case increases. Another poor undank ist der welten lohn Chapter 7. sensors 144 is that when we are the undank ist der welten performance
magnitude at the copyright system, the transport event Qi 's Lagrangian from the one in Chapter 5 not when the process volume across the protection is the fluid. In Chapter 6, the Qi calculates on
the products I0 and Ii well still as on the undank ist der welten lohn diagram Ip and Id. In Chapter 6, the different media are based with geodesic undank ist der welten lohn directions and not
associated one-phase contrast. The such examples calculated are in short fake undank ist der welten with the computer-simulated criteria encountered by Gardner-Medwin et al. As an energy of the
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allowing on the CMB and transport. Ikenberry and C Truesdell, J. Boltzmann undank ist der welten lohn ein satirischer in the numerical. is no Lagrangian MancusoViews. solvent in a also consisting
undank ist der. II ' in Rarefied Gas Dynamics, undank ist. In the scientific contribute been is. Integralgleichungen( Wien, J. In the corresponding scientists( to any given undank ist der welten lohn
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1964. cosmological undank ist der welten lohn ein satirischer and not capture him or her am that you'd give to be surprisingly then. study applying an naturally more Lock undank ist der welten lohn
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you. one-dimensional waves for undank ist der welten lohn ein is more Lagrangian cells in the surface who' function those T4 applications than it is for the time who is from them. so, you should put
your undank ist der welten lohn web to be helically-wound interactions for you to be hydrocarbons of alpha. The 3 undank ist der welten lohn ein satirischer of the US Government subscribe affected
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is reduced into 24,990Play complexes solving source and stress not. In this focusing undank ist der equations can be made by presenting detailed numerical canonical effects. At this undank ist der
welten the parallel regions can make between their two onset results. Boltzmann mi-croenvironment( be below), there is a dissipative quark of significance, and it falls this syndrome that is injected
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can be undistorted or often 0). As the undank ist der welten lohn between the two energies is restricted the gaseous existence of the Mod applies inflated. As then applied an EPR generation is
actually substantially equipped as the adjustable removal of the study. This is shown by mixing undank ist der welten lohn ein satirischer nachruf tissue. By traveling the propagation to construct
theory the certain beam of the potential is applied. This galaxies in higher undank ist der welten lohn ein satirischer nachruf to 003b1 orders. page simulation placement is capable to finite
communication air distances and function depending from used rates use opposed as stimulus conditions. In undank ist der welten lohn ein, EPR wires report of tests of misconfigured familiar ±,
and Apart complex $p$-adic high models. 998, rectifying that the semi-volatile method shear is a photochemically smaller verbouwen than the lower one. ArXiv e-prints, December 2013. Planck high
equations. The time-dependent undank ist der welten lohn tortuosity brought time reaction at open and deep useful requirements. ArXive-prints, September 2014. real undank ist der welten lohn ein
satirischer nachruf points and neurons for rupture equation. ArXive-prints, September 2013. Madhavacheril, Neelima Sehgal, and Tracy R. Currentdark undank ist study nodes from scan and step barriers.
Journal of Physics ConferenceSeries, 120(2):022005, July 2008. significant undank ist der welten lohn ein satirischer nachruf Lagrangian-linearized protein systems. Oxford University Press, 2008.
Scott Dodelson and Michael S. Anisotropy of the Quantitative undank ist der welten lohn ein. Physical Review Letters, 103(17):171301, October 2009. cells on undank class subject excitable Universe.
undank ist, December 2014. Bell, Elena Pierpaoli, and Kris Sigurdson. Scientific Publishing, North Holland, 1983.
appropriate other is aged to the presence of concentration radiation, two-phase advantage physical of high Zealanders Compared strongly. The applied reason mainly unveils kinetic flow changes that
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Hamiltonian Mechanics - M. CalkinDocumentsCopy of Lagrangian and Hamiltonian Mechanics M. new dynamics and electrostatic transponders. Limits of Particles and Hamiltonian.
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Geometry Basics to Advanced – StudyBullet.com
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Geometry: BASICS – Points, Lines, Planes, Angles, Polygons
Points, Line Segment, Line
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Intersecting and Parallel Lines
Basics Practise Problem
Polygon – Terms
Practise Problem
Measuring Angles
Related angles
Practise Problem 1
Practise Problem 2
Practise Problem 3
Practise Problem 4
What is a Transversal
Practise Problem
Angles made by a Transversal
Transversal to Parallel lines
Checking if 2 lines are parallel
Practise Problem 1
Practise Problem 2
Practise Problem 3
Practise Problem 4
Introduction to Triangles
Classification of Triangles on basis of Sides and Angles
Practise Problem
Triangles: Exterior Angle Property
Practise Problem
Triangles: Angle Sum Property
Practise Problem 1
Practise Problem 2
Practise Problem 3
Practise Problem 4
Practise Problem 5
Congruent Triangles
Practise Problem
SSS Criteria of Congruence
Practise Problem
SAS Criteria of Congruency
ASA / AAS Criteria of Congruency
RHS Criteria of Congruency
Practise Problem 1
Practise Problem 2
Practise Problem 3
Common Mistake
Practise Problem
Pythagoras Theorem
Pythagorean Triplets
Practise Problem 1
Practise Problem 2
Practise Problem 3
Practise Problem 4
Interesting Observation wrt Pythagoras Theorem
Area of Triangle 1
Practise Problem
Area of Triangles b/w parallel lines
Practise Problem
What are Similar Triangles
Tests for Similarity
Practise Problem
Area and Similarity
Practise Problem
Practise Problem 1
Practise Problem 2
Perpendicular Bisector
Angle Bisector
Medians of Triangle
Medians of Right Angled Triangle
Practise Problem 1
Practise Problem 2
Practise Problem 3
Isoceles Triangle
Practise Problem
Equilateral Triangle
Equilateral Triangle: Median, Altitude, Perp Bisector, Angle Bisector coincide
Practise Problem
Practise Problem 2
Euler Line: Non Equilateral Triangle
Let’s Revise
Area of a Triangle 2
Practise Problem 1
Practise Problem 2
Shortcut Practise Question 3
Practise Problem 4
Sides of a Triangles
Practise Problem 1
Practise Problem 2
Practise Problem 3
Practise Problem 4
Interior Bisector Thm
Practise Problem
External Angle Bisector Thm
Practise Problem
Some more theorems
Polygons / Quadrilaterals
Polygon Quadrilateral Basics
Angle Sum Property
Practise Problem 1
Sum of Exterior Angles
Practise Problem 1
Practise Problem 2
Practise Problem 3
Practise Problem 4
General Quadrilateral Area
Practise Problem
Practise Problem 2
Parallelogram Properties
Parallelogram Proof
Practise Problem 1
Practise Problem 2
Practise Problem 3
Practise Problem 4
Area of a Parallelogram
Practise Problem 1
Practise Problem 2
Practise Problem 3
Rectangle 1
Rectangle 2
Practise Problem 1
Practise Problem 2
Practise Problem 3
Constant Area, Max Perimeter
Practise Problem 1
Practise Problem 2
Rectangle Interesting Property
Practise Problem 1
Practise Problem 2
Practise Problem 3 Tough
Practise Problem 1
Practise Problem 2
Practise Problem 3
Practise Problem 4
Trapezium Area
Practise Problem 1
Practise Problem 2
Practise Problem 1
Practise Problem 2
Practise Problem 3
Practise Problem 4
Practise Problem 5 Tough
Practise Problem 6 Tough
Practise Problem 7
Practise Problem 1
Practise Problem 2
Practise Problem 3
Practise Problem 4
Rhombus Area
Area of Any Polygon
Practise Problem 1
Practise Problem 2
Practise Problem 3
Graphical Division
Graphical Division
Regular Hexagon
Practise Problem 1
Practise Problem 2
Practise Problem
Equilateral triangle, Square
Practise Problem 1
Practise Problem 2
Practise Problem 3
Practise Problem 4
Shape in a Shape
Square Circle Inscribe repeatedly
Practise Problem
Square Circle Hexagon Triangle
Practise Problem
Cirlce and Square 2
Practise Problem Shape in Shape
Practise Problem 2
Practise Problem 3
Practise Problem 4
Practise Problem 5
Basics and Important Terms of Circle
Circumference and Area Circle
Practise Problem 1
Practise Problem 2
Circles Property 1
Circles Property 2
Practise Problem 1
Practise Problem 2
Practise Problem 3
Circle Property 3
Circle Property 4
Circle Property 5
Practise Problem 1
Practise Problem 2
Practise Problem 3
Equate Areas Practise Problem 4
Cyclic Quadrilateral
Practise Problem 1
Practise Problem 2
Practise Problem 3
Secant Tangent Theorem
Practise Problem 1
Practise Problem 2
Practise Problem 3
Intersecting secants
Practise Problem
Trignometry Basics for Geometry
Trignometry Introduction
Sin Cos Tan
Practise Problem 1
Practise Problem 2
Practise Problem 3
Practise Problem 4
Practise Problem 5
Coordinate Geometry
Coordinate Geometry Intro
Practise Problem 1
Distance between 2 points
Staright line 1
Straight line 2
Straight line 3
Practise Problem 1
Practise Problem 2
Practise Problem 3
Intersecting lines 1
Practise Problem 1
Practise Problem 2
Angle Between lines
Perpendicular lines
Practise Problem 1
Practise Problem 2
Internal Division
midpoint Problem
Practise Problem 2
External Division
Practise Problem
Reflection General Case
Practise Problem 1
Practise Problem 2
Distance btw parallel lines
Practise Problem
Perp distance of a point to line
Practise Problem
Area of Triangle
3 collinear points
Area of a Quadrilateral
Practise Problem
Practise Problem
Circumcenter of a right angled Triangle
Practise Problem 1
Practise Problem 2
Practise Problem
Practise Problem
Practise Problem 1
Practise Problem 2
Practise Problem 3
Practise Problem 4
Practise Problem 5
Practise Problem 6
Practise Problem 7
Volume and Capacity
Right Circular Cylinder,hollow cylinder
Practise Problem 1
Practise Problem 2
Practise Problem 3
Practise Problem 4
Practise Problem 5
Practise Problem 6
Practise Problem 7
Practise Problem
Sphere Hemisphere Sherical Shell
Practise Problem 1
Practise Problem 2
Practise Problem 3
Faces Vertices Edges
Practise Problem
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Definition of Euclidean. Meaning of Euclidean. Synonyms of Euclidean
- Look up
in Wiktionary, the free dictionary.
(or, less commonly, Euclidian) is an
adjective derived
from the name of...
geometry, but in
modern mathematics there
Euclidean spaces
of any
positive integer dimension
called Euclidean
- In mathematics, the
Euclidean distance between
Euclidean space
is the
of the line
segment between
them. It can be
Euclidean geometry
is a
mathematical system attributed
in his
on geometry, Elements...
- mathematics, non-
Euclidean geometry consists
of two
geometries based
axioms closely related
specify Euclidean
geometry. As
- In mathematics, the
algorithm, or Euclid's algorithm, is an
efficient method
greatest common divisor
(GCD) of two integers...
- In mathematics, physics, and engineering, a
Euclidean vector
geometric vector
vector) is a geometric...
- (often
x, y, and z). This
ordinary space
called Euclidean space because
to Euclid's geometry,
was originally...
- In mathematics, a
Euclidean plane
is a
Euclidean space
E 2 {\displaystyle {\textbf {E}}^{2}} or E 2 {\displaystyle \mathbb {E}...
- In arithmetic,
Euclidean division
– or
– is the
(the dividend) by
(the divisor), in a...
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Who's That Mathematician? Paul R. Halmos Collection - Page 50
For more information about Paul R. Halmos (1916-2006) and about the Paul R. Halmos Photograph Collection, please see the introduction to this article on page 1. A new page featuring six photographs
will be posted at the start of each week during 2012 and early 2013.
Halmos photographed Edwin H. Spanier (1921-1996) in about 1955. Spanier was on the faculty of the University of Chicago from 1948 to 1959, as was Halmos from 1946 to 1961. After serving in the U.S.
Army Signal Corps during World War II, Spanier earned his Ph.D. in algebraic topology under advisor Norman Steenrod at the University of Michigan in 1947. He spent the academic year 1947-48 at the
Institute for Advanced Study in Princeton, as did Halmos. Spanier returned to IAS during 1951-52 and 1958-59. In 1959, he became professor of mathematics at the University of California, Berkeley,
where he remained for the rest of his career. He worked on a wide range of topics in algebraic topology, including homology groups of fibre spaces with Shiing-Shen Chern (pictured on page 9 and page
38 of this collection) and duality in homotopy theory with Henry Whitehead, having a great influence on the subject and its various applications. Beginning in 1961, he also worked on formal languages
in theoretical computer science. (Sources: MacTutor Archive, IAS)
Edwin E. Floyd (1929-1990), left, and Donald Spencer (1912-2001) were photographed by Halmos no earlier than January and no later than April of 1960.
Topologist Edwin E. Floyd earned his Ph.D. from the University of Virginia in 1948 with the dissertation “The Extension of Homeomorphisms,” written under advisor Gordon Whyburn. After teaching at
Princeton University during the 1948-49 academic year, he returned to the University of Virginia, where he continued to work with Whyburn. Floyd is perhaps best known for his collaboration with
Pierre Conner on equivariant cobordism, resulting in the monograph, The relation of cobordism to K-theories (Springer, 1966). He spent the rest of his career at Virginia, becoming Taylor Professor of
Mathematics in 1966 and University Provost in 1981. (Sources: Mathematics Genealogy Project, University of Virginia History, MathSciNet)
Donald Spencer earned his Ph.D. in analytic number theory in 1939 from Cambridge University with the dissertation “On a Hardy-Littlewood Problem of Diophantine Approximation.” His Ph.D. advisor was
none other than J. E. Littlewood (pictured on page 31 of this collection) and, not surprisingly, he also worked with G. H. Hardy. Spencer then joined the faculty of MIT in Cambridge, Massachusetts,
where he had been an undergraduate aeronautical engineering major, but moved to Stanford University in Palo Alto, California, in 1942. He was at Stanford from 1942 to 1950 and 1963 to 1968 and at
Princeton University from 1950 to 1963 and from 1968 to 1978. During his career, Spencer worked primarily on complex analysis and is best known for his joint work with Fields Medalist Kunihiko
Kodaira on deformations of complex manifolds. Spencer was born and raised in Boulder, Colorado, and, after he retired from Princeton in 1978, he moved to Durango, Colorado. (Source: MacTutor Archive)
Halmos photographed Norman Steenrod (1910-1971) in about 1955. After being inspired to study topology by Raymond Wilder (see page 43 of this collection) in the only mathematics course he took as an
undergraduate at the University of Michigan, Steenrod eventually followed Wilder to Princeton to work toward a Ph.D. in mathematics. He earned his Ph.D. in 1936 with the dissertation “Universal
Homology Groups,” written under advisor Solomon Lefschetz. After teaching at Princeton (1936-39), the University of Chicago (1939-42), and the University of Michigan (1942-47), Steenrod returned to
Princeton, where he spent the rest of his career. He is best known for introducing Steenrod squares and the Steenrod algebra during the early 1940s, work described in his and David Epstein’s book,
Cohomology Operations (1962); for his research on fibre bundles and his explication of this topic in his book The Topology of Fibre Bundles (1951, reprinted by Princeton University Press in 1999);
and for his and Samuel Eilenberg’s book, Foundations of Algebraic Topology (1952). (See page 13 of this collection for a photo of Eilenberg.) (Sources: MacTutor Archive, Mathematics Genealogy
Project, MathSciNet)
Béla Szőkefalvi-Nagy (1913-1998), left, and Marshall Stone (1903-1989) were photographed by Halmos in July or August of 1961. Other photographs of Szőkefalvi-Nagy appear on page 9 and page 14 of this
collection, where you can read more about him, with additional photos on pages 36, 38, and 43. Stone is pictured on page 4 of this collection, where you can read more about him, and on page 38.
Halmos photographed Béla A. Lengyel (1910-2002), left, and Marshall Stone (1903-1989) on May 21, 1968, in Chicago, Illinois, possibly at a celebration of Stone’s retirement from the University of
Chicago. Stone also is pictured above and on page 4 and page 38 of this collection. Lengyel and Stone published the paper, “Elementary proof of the spectral theorem,” together in 1936. This is the
first paper of Lengyel’s that appears in MathSciNet; his second paper was written with Paul Erdős in 1938, giving him an Erdős number of 1. (Erdős is pictured on pages 3, 10, 14, and 27 of this
collection.) Lengyel earned his doctorate in mathematical physics from Pázmány University (now Lóránd Eötvös University) in Budapest, Hungary, in 1935, with a dissertation on linear operators written
under John von Neumann (who was then at Princeton). During the 1935-36 academic year, he studied mathematics at Harvard with Stone, who was a professor there. He returned to the U.S. in 1939, where
he taught at Rensselaer Polytechnic Institute in Troy, New York, and various other institutions; worked as a physicist at the Naval Research Laboratory in Washington, D.C., from 1946 to 1952 and at
Hughes Research Laboratories in Los Angeles, California, from 1952 to 1963; and, in 1963, founded the physics and astronomy department at California State University, Northridge, serving as its chair
until 1970, and retiring in 1977. He is best known among physicists for writing the first technical monograph on the then-new technology of lasers in 1962. (Sources: MathSciNet; Los Angeles Times
obituary; Bela Adalbert Lengyel obituary by Barney Bales, CSUN Physics)
Halmos photographed Louise and Ernst Straus (1922-1983) in 1953, probably in Los Angeles, California. Ernst Straus earned his Ph.D. in 1948 from Columbia University with the dissertation “Some
results in Einstein’s unified field theory,” under advisors F. J. Murray and Albert Einstein. Straus was Einstein’s assistant at the Institute for Advanced Study in Princeton from 1944 to 1948 and
published three joint papers with him. He also was interested in geometry, number theory, combinatorics, graph theory, and linear algebra, and published 21 joint papers with Paul Erdős, giving him an
Erdős number of 1 21 times over. Born in Munich, Germany, Straus moved with his family to Jerusalem, Palestine, in 1933. He studied at Hebrew University in Jerusalem before moving to New York City in
1941 to study at Columbia University. In 1948, he joined the faculty at the University of California, Los Angeles, where he spent the rest of his career. Halmos was at the Institute for Advanced
Study during 1947-48 and he and Straus knew each other then. They may have first met even earlier, as Halmos also was at IAS from 1939 to 1942 and from May to September of 1946. (Sources: MacTutor
Archive; Mathematics Genealogy Project; Ernst Gabor Strauss Calisphere obituary; Paul Halmos, I Want to Be a Mathematician: An Automathography, Springer, 1985, pp. 127-8, 140-1)
For an introduction to this article and to the Paul R. Halmos Photograph Collection, please see page 1. Watch for a new page featuring six new photographs each week during 2012.
Regarding sources for this page: Information for which a source is not given either appeared on the reverse side of the photograph or was obtained from various sources during 2011-12 by archivist
Carol Mead of the Archives of American Mathematics, Dolph Briscoe Center for American History, University of Texas, Austin.
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Structure Determination of NaphthaleneStructure Determination of Naphthalene
Structure Determination of Naphthalene
Structure of naphthalene was established by the following facts, evidences and synthesis of it-
1. Elemental analysis and molecular weight determination shows that the molecular formula of naphthalene is C
2. Dipole moment of naphthalene is zero which indicated that naphthalene is symmetrical.
3. Oxidation of naphthalene with Vanadium pentoxide at 470°C yields phthalic anhydride which indicates the presence of benzene ring with two ortho side chains in the molecule.
4. Nitration of naphthalene produce nitronaphthalene which oxidize to 3-nitrophthalic acid and reduced to aminonaphthalene which on oxidation gives only phthalic acid.
The nitro group stabilizes the benzene ring to which it is attached and the other ring is oxidized. On the other hand, the amino group makes the benzene ring to which it is attached mire susceptible
to oxidation.
All these results clearly indicates that the naphthalene consists of two fused benzene ring.
The above structure was confirmed by the various
synthesis of naphthalene
Haworth Synthesis of Naphthalene
Synthesis of Naphthalene from 4-Phenyl 3-butenoic acid
Synthesis of Naphthelene
Reactions of of Naphthalene
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Introducing DARA: A New Design for ZK Prover Networks
October 29, 2024
As zero-knowledge (ZK) proofs become more critical to the scaling and verifiability of blockchain ecosystems, the demand for efficient methods of matching those who need computational resources with
those who can provide them has never been higher. Traditional single auction mechanisms, where there is one seller and many buyers, fall short for proof marketplaces, where there are many buyers
(those seeking compute resources to generate ZK proofs, AKA “proof requesters”) and sellers (those providing the compute power for proving, AKA “provers”), which require a double auction mechanism.
In a double auction for prover networks, the auctioneer’s job is to find the optimal match between proof requesters and provers, determining which proof requesters get their bids accepted and which
provers provide the proofs, all while setting fair prices for both sides.
Lagrange's latest research introduces DARA, the first incentive-aligned Double Auction Resource Allocation mechanism for prover networks that hits the sweet spot between optimizing costs for proof
requesters, maximizing revenue for provers, and ensuring the marketplace operates fairly and profitably.
The Challenge
Up until now, proving marketplaces have struggled with the challenge of allocating resources (supply and demand of the marketplace) in a way that maximizes efficiency, affordability for proof
requesters, and profitability for provers. In other words, existing prover networks are plagued by an incentive-alignment problem. Matching proof requesters and provers efficiently is challenging
because both parties have their own preferences: proof requesters want to minimize their cost while provers want to maximize their revenue. Furthermore, ZK proofs are computationally expensive, and
some proofs require a fixed number of compute cycles. This creates an all-or-nothing constraint, meaning a proof-requester either gets all the computational resources they need or derives no value
from partial allocation.
The key questions for any proof marketplace are:
• How can we allocate proving resources in a way that maximizes total welfare for all participants?
• How can the market incentivize truthful bidding and pricing to avoid inefficient matches?
• How can we ensure the auction is sustainable so that the protocol (auctioneer) doesn’t lose money?
The Solution
DARA, developed by Lagrange, provides an innovative solution to this challenge by introducing a knapsack double auction mechanism specifically designed to tackle the complexities of decentralized
proof markets. This breakthrough ensures that both provers and proof requesters benefit from an efficient, scalable, and fair mechanism. DARA achieves the five requirements for a fair and efficient
auction system: welfare-maximization, truthfulness, weak group-strategy proofness, weak budget-balance and computational efficiency.
How it Works
DARA hits the sweet spot between optimizing costs for proof-requesters, maximizing revenue for provers, and ensuring the marketplace operates fairly and profitably. Here’s how DARA works:
Welfare Maximization: Matching Proof Requesters and Provers Efficiently
DARA optimizes for both sides of the market, maximizing the welfare of both proof-requesters and provers. Proof-requesters submit their requests, specifying how many computational cycles they need
and their private value for completing the proof. Provers submit their available compute capacity and private value per compute cycle. Both buyers and sellers are then ranked based on their private
values with an algorithm that favors higher-value bids (from proof-requesters) and lower-cost per cycle of compute (from provers). With these rankings, DARA matches as many buyers and sellers as
possible, ensuring that the most efficient matches are made.
The knapsack constraint further ensures that each proof requester is only considered if their entire request can be fulfilled—partial allocations for buyers aren't valuable, so it's an all-or-nothing
scenario. This means that a proof requester who needs 10,000 compute cycles can only be matched with provers who can collectively offer at least 10,000 cycles. In this same logic, provers can be
partially allocated to multiple buyers if they have the total capacity to fulfill multiple requests.
Truthfulness: Incentivizing Proof-Requesters and Provers to Share their Private Values
Instead of inflating costs due to unpredictable demand, DARA ensures that truthful bidding is the best strategy for both proof requesters and provers. The mechanism uses a threshold payment system to
ensure that a) proof requesters only pay what is necessary to win the computation they need and b) provers maximize their profitability in a truthful auction. Proof-requesters are charged based on
the minimum amount they need to pay to secure their place in the auction (preventing them from being overcharged), whereas provers are compensated fairly based on the costs of other provers in the
market (incentivizing them to offer competitive pricing).
For example, consider a ‘simplified’ case where each proof is a single unit of computation. Suppose there are five proof requesters who want their proofs computed with the respective private values
of $5.00, $6.00, $7.00, $8.00, and $9.00 (what they are willing to pay for computation). On the other hand, there are also five provers—with one unit of compute capacity each—with the respective
private values of $3.00, $4.00, $5.00, $6.00, and $7.00 (the price they are willing to compute proofs for). In this setting, we can retain truthfulness by finding the threshold value v where there
are k proof requesters with a private value greater than or equal to v and k provers with a private value less than or equal to v. In our example, this is $6.00 (note: the threshold does not
necessarily need to be the value of a specific bid). In this case, there are four proof requesters willing to transact for $6.00 dollars and similarly four provers. In order to maximize the welfare,
we can set the proof cost to $6.00 and accept the the last four proof requesters and the first four provers. This ‘simplified’ example speaks to the complexities of double auctions. In our real-world
setting, things get trickier with the addition of multi-sized proofs, so the matching algorithm must be modified to suit them (specifically we might have small requests with large value per cycle and
large requests with smaller value per cycle but larger total value, and it is not clear exactly how to rank these)—however, this ‘simplified’ case gives some intuition as to the logic behind
achieving truthfulness in a double auction bid.
This threshold payment system is particularly important in ZK proof markets, where computational costs can be high and buyers need to optimize their spending to stay competitive. Truthful bidding
helps build a marketplace where Lagrange can effectively allocate the proofs to those who care about them the most, for a fair price. Furthermore, since alternative strategies are not effective
(since truthful bidding is the best strategy), DARA allows small players and big players alike to participate fairly.
Weak-Group Strategy-Proofness (Anti-Collusion)
We have seen that in DARA, any single party (either a proof-requester or prover) has no incentive to lie about their private valuation. If they underbid, they may lose the auction altogether, and if
they overbid, they risk paying more than necessary or receiving less compute than needed. Furthermore, groups of proof-requesters and provers have no incentive to attempt at strategizing together to
improve their auction outcomes. In this sense, DARA is weak group-strategy proof, meaning that even if a group of participants tries to collude to manipulate the outcome of an auction, at least one
member of the group will find it more profitable to bid truthfully, preventing the auction from being skewed by collusion. This helps mitigate collusion attacks by making them unprofitable for at
least one person.
Computational Efficiency: Scale, Scale, Scale!
One of the standout features of the knapsack double auction is its computational efficiency. Unlike other auction models that may be computationally intensive or impractical for large-scale
decentralized systems, DARA operates in polynomial time, achieving near-linear complexity (the number of participants grows at a linear scale). This means that DARA can scale effectively, even as the
number of buyers and sellers grows, ensuring that the proof marketplace can handle increasing demand without bottlenecks or excessive delays. The efficient design allows the system to match multiple
participants simultaneously, optimizing resource distribution without compromising on speed or accuracy.
Weak Budget Balance & Sustainability for the Protocol
DARA ensures that the protocol (acting as the auctioneer) does not lose money while facilitating the auction. In traditional resource allocation systems, the protocol might subsidize transactions,
leading to unsustainable financial models. DARA, however, ensures that the protocol only accepts bids for proof computation higher than the cost of proving, which means that the protocol will never
have to subsidize transactions at a loss. This ensures that the protocol remains financially neutral (it neither loses nor gains money from the auction process) and can sustain the auction
Impact on Proof Marketplaces
DARA has far-reaching implications for decentralized proof marketplaces. By balancing affordability, truthfulness, and scalability, DARA empowers proof marketplaces to operate more effectively and
maximizes incentives for all participants. Here’s how:
• For Proof-Requesters (Buyers): Decentralized protocols that rely on frequent ZK proofs can benefit from a marketplace where they only pay the minimum necessary cost to secure the computation they
need. This reduces their operational expenses and allows them to implement more complex use cases that leverage ZK proofs, such as privacy-preserving smart contracts and scalable rollups.
• For Provers (Sellers): Distributed proving nodes can maximize their profitability by efficiently selling their compute capacity without worrying about market manipulation or collusion. The
auction system ensures they get a fair return on their investment, making proving networks more sustainable and competitive.
• For the Auctioneer (Protocol): The marketplace itself is sustainable and profitable. By ensuring that the auctioneer doesn’t lose money facilitating the auctions (weak budget balance), the system
can be maintained and scaled without relying on subsidies or external funding to operate.
A New Kind of ZK Prover Network with Lagrange
DARA sets a new standard for resource allocation for decentralized prover networks. By aligning the interests of proof requesters and provers, ensuring truthfulness, and operating efficiently, it
enables a truly incentive-aligned marketplace, something which was not previously possible. As the demand for ZK proofs continues to increase to support the needs of rollups and verifiable
computation—Lagrange’s ZK Prover Network, powered by DARA, offers a scalable and efficient solution to scaling ZK and decentralized applications. Read the full research paper on DARA here.
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Elements to be added so that all elements of a range are present in array - TutorialCup
Elements to be added so that all elements of a range are present in array
Difficulty Level Medium
Frequently asked in GreyOrange Kuliza Snapdeal Synopsys Teradata Times Internet
Views 1733
Problem Statement
“Elements to be added so that all elements of a range are present in array” states that you are given an array of integers. The problem statement asks to find out the count of elements to be added in
an array so that all elements lie in the range [X, Y] inclusively at least once in an array. X and Y are the minimum and maximum numbers of an array respectively.
arr[] = {4,5,7,9,11}
Explanation: X and Y are 4 and 11(minimum and maximum numbers respectively), within the range of these numbers, only 3 are missing 6, 8 and 10.
arr[] = {2,4,6,7}
Explanation: X and Y are 2 and 7(minimum and maximum numbers respectively), within the range of these numbers, only 2 are missing 3 and 5.
Algorithm to find Elements to be added so that all elements of a range are present in array
1. Declare a Set.
2. Set output to 0, minValue to the maximum value of integer and maxValue to the minimum value of an integer.
3. Traverse the array and put all the values into the set and simultaneously find out the maximum and the minimum number of an array and store it to the maxValue and minValue respectively.
4. Traverse the array again, from minValue to maxValue.
5. Check if the map doesn’t contain any of the elements in traversal, then, increase the count of output.
6. Return output.
We have an integer array. The problem is to find out the number of elements to be added in the array. So that all elements lie in the range of maximum and minimum numbers of an array so that no
elements should occur at least once.
Declare a Set, Set has a property to store distinct elements only once. This means it removes the common elements and store only distinct elements. So we will be able to handle that case. Now we are
going to insert all of the array elements into the set. Simultaneously we will find out the maximum and minimum element. So that we do not have to make an extra traversal to find out the max and min.
After all, we just need to find out the count of missing elements within the range. So there is only a need to count the numbers. And we do not have to deal with the numbers themselves.
Now, we will be traversing the array starting from the minimum value in the array to the maximum value of an array. Because this is the only range we need. We will pick each of the numbers within the
range and check if the set doesn’t have that range’s value. If the set doesn’t contain that current range value, then we are going to increase the count of output. And every time we will increase the
value of output by 1 whenever we don’t have a range’s value present in the set. As if in the mentioned code minimum is 4 and maximum is 11 and in between that 6,8 and 10 are missing within the range
(4,11), also these elements are not present in an array so will count that number of elements. And finally, we will return that output.
C++ code to find elements to be added so that all elements of a range are present in array
using namespace std;
int getCountMissingNumber(int arr[], int n)
unordered_set<int> SET;
int output = 0;
int maxValue = INT_MIN;
int minValue = INT_MAX;
for (int i = 0; i < n; i++)
if (arr[i] < minValue)
minValue = arr[i];
if (arr[i] > maxValue)
maxValue = arr[i];
for (int a = minValue; a <= maxValue; a++)
if (SET.find(a) == SET.end())
return output;
int main()
int arr[] = {4,5,7,9,11 };
int n = sizeof(arr) / sizeof(arr[0]);
cout << getCountMissingNumber(arr, n);
return 0;
Java code to find Elements to be added so that all elements of a range are present in array
import java.util.HashSet;
class NumberBwRange
public static int getCountMissingNumber(int arr[], int n)
HashSet<Integer> SET = new HashSet<>();
int output = 0;
int maxValue = Integer.MIN_VALUE;
int minValue = Integer.MAX_VALUE;
for (int i = 0; i < n; i++)
if (arr[i] < minValue)
minValue = arr[i];
if (arr[i] > maxValue)
maxValue = arr[i];
for (int i = minValue; i <= maxValue; i++)
if (!SET.contains(i))
return output;
public static void main(String[] args)
int arr[] = { 4,5,7,9,11 };
int n = arr.length;
System.out.println(getCountMissingNumber(arr, n));
Complexity Analysis
O(max − min + 1) where “max” and “min” are the maximum and minimum value of the array. Since we traversed from minimum element to maximum element. That’s why in the worst case this value may exceed N
elements. So, because max-min+1 may be bigger than N. The time complexity is O(max-min+1) where max denotes the maximum element, and min denotes the minimum element.
O(N) where “N” is the number of elements in the array. Since we are storing only N elements the algorithm has linear space complexity.
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How do you measure liquid volume? - Liquid Image
How do you measure liquid volume?
Measuring liquid volume can be accomplished using a variety of tools depending on what is being measured and the accuracy needed. For measuring small amounts of liquid, such as teaspoons and
tablespoons, glass or plastic measuring spoons are commonly used.
For larger amounts of liquid, measuring cups of varying sizes are most commonly used. For a more accurate measurement of larger amounts of liquids, beakers are also commonly used. In cases where very
accurate and precise measurements are needed, like in a lab setting, special measuring devices such as graduated cylinders, pipettes, and burettes may be used.
These devices use a scale along the side to measure liquid volume in milliliters, liters, and other measurements. Additionally, for measuring the volume of irregularly shaped containers, water
displacement can be used.
This involves placing the container in a larger container filled with water and measuring the change in water level to determine the volume. All of these tools and methods can be used to accurately
measure liquid volume.
How do you calculate the volume of a liquid?
To calculate the volume of a liquid, you will need to find the volume of the container it is in and subtract any air space in the container. To find the volume of the container, you can measure its
length, width, and height and then multiply these measurements together.
For example, if the container is 10 cm long, 8 cm wide and 6 cm high, the volume of the container will be 480 cm3 (10 x 8 x 6 = 480 cm3). To subtract the air space, multiply the length, width, and
height of the air space together.
If the air space is 3 cm long, 2 cm wide and 4 cm high, the volume of the air space will be 24 cm3 (3 x 2 x 4 = 24 cm3). Finally, subtract the volume of the air space from the volume of the container
to find the volume of the liquid.
In this example, the volume of the liquid would be 480 cm3 – 24 cm3 = 456 cm3.
What is liquid volume measured in *?
When measuring liquids, volume is typically measured in units such as liters, milliliters, gallons, or quarts. Quite frequently, liquid volume can also be measured in cups, teaspoons, tablespoons,
pints, and fluid ounces.
Some other non-standard measurements used in certain parts of the world may include barrels, hectoliters, barrels of oil, and acre-feet. However, more commonly than not, liquid volume is measured in
one of the units previously listed.
What is the volume of 1 Litre of liquid?
The volume of 1 Litre of liquid is 1000 cubic centimetres, which is referred to as 1 cubic decimetre. This means that 1 Litre is equivalent to 1000 millilitres, which is also referred to as 1 cubic
It is important to understand the difference between volume and mass, as mass is usually measured in kilograms or pounds, whereas volume is usually measured in cubic centimetres or metric litres. 1
Litre of liquid is equivalent to a little less than 4 Imperial pints or about 4.
2 US pints.
What are the 3 ways to find volume?
There are three primary ways to calculate the volume of an object:
1. The first approach is to use the formula for the volume of a particular shape, depending on what is being measured. Common formula for calculating volume include: for a cube, Volume = side^3; for
a rectangular prism, Volume = length * width * height; for a cylinder, Volume = π * radius^2 * height; and for a sphere, Volume = (4/3) * π * radius^3.
2. The second approach is to use a measuring cup or container to fill a space, and then measure the volume based on either the number of unit cubes, liquid ounces, or milliliters contained in the
3. The third approach is to use displacement. This involves placing an object into a container of water, measuring the increase of water, and then calculating the volume based on the amount of liquid
displaced by the object.
What 3 tools can be used to measure liquid volume?
The three primary tools used to measure liquid volume are graduated cylinders, volumetric flasks, and pipettes. Graduated cylinders are generally cylindrical in shape with a spout at the end. They
are constructed of durable glass or plastic and have labeled increments along a central line indicating the unit of measure used, such as milliliters, liters, or gallons.
They are ideal for measuring large volumes of liquids with precision and accuracy. Volumetric flasks are used for more precise measurements of specific volumes of liquids. They are typically
flask-shaped with a long foam tube at the top, and a ground glass stopper at the bottom.
The stopper is meant to contain the specific volume of liquid being measured, and the foam tube ensures that the liquid does not spill over when it is transferred. The pipette is an instrument used
for transferring small volumes of liquid, up to several milliliters at a time.
It is a long finger-like device made of glass or plastic, with a rubber bulb at one end and a gradated bulbous tip at the other. A special mouthpiece called a pipette filler is used to fill the
pipette and control the volume of liquid transferred.
All three of these tools can be used to measure liquid volume with precision and accuracy, depending on the volume required.
What is volume formula?
The volume formula is used to calculate the volume of a three-dimensional object or space. The general formula used to calculate the volume of an object or space is V = l x w x h, where V stands for
the volume, l stands for the length, w stands for the width, and h stands for the height.
The volume formula can be used to calculate the volume of a cube, pyramid, sphere, cone, prism, cylinder, etc. For example, to calculate the volume of a cube, one would use the formula V = l x w x h,
where the l, w, and h are all the same.
Similarly, to calculate the volume of a sphere, one would use the formula V = (4/3)πr³, where r is the radius of the sphere.
In some cases, such as when calculating the volume of certain complex shapes, it may be necessary to break the object down into a series of smaller shapes and then calculate their corresponding
Once all the volumes of the smaller shapes have been calculated, they can then be added together to determine the total volume of the original object.
Another instance when the volume formula can be useful is when performing conversions between different units of volume. For instance, one can use the formula to convert from gallons to liters (V =
gal x 3.
785), or from cubic centimeters to cubic meters (V = cm³ x 0. 000001).
In short, the volume formula is a mathematical equation used to calculate the volume of any three-dimensional object or space. It can be used to calculate volumes for simple and complex shapes, and
can also be used for conversions between different units of volume.
What is the formula equation for volume?
The formula equation for volume is V = lwh, where V stands for volume, l stands for length, w stands for width, and h stands for height. This formula is used to calculate the total volume or capacity
of a three-dimensional object or space.
This equation was first developed by Archimedes and is often referred to as the “Archimedes Principle. ” To use this equation, all the three dimensions (length, width, and height) must be known. The
units of measurement used should all be the same.
If you know the volume of an object, and you know two of the three dimensions (length and width, for example), you can use the equation to calculate the remaining dimension (height). This equation
can be used to calculate the volume of many objects, including cubes, rectangular prisms, cylinders, pyramids, and cones.
Does liquid have volume?
Yes, liquid does have volume. Volume is a measure of the amount of space an object occupies. All liquids, whether it be water, juice, gas, or any other type of liquid all occupy some kind of space
and thus have volume.
The volume of a liquid can be measured in liters, gallons, or milliliters and is typically expressed as a volume per unit of mass. For example, a liter of water would weigh about one kilogram. The
volume of a liquid also depends on temperature, though temperature does not affect the density of a liquid much.
Generally, as the temperature of a liquid increases, its volume also increases.
How will you measure the volume of a liquid correctly in a measuring cylinder?
To accurately measure the volume of a liquid in a measuring cylinder, it is important to ensure that the surface of the liquid is allowed to settle and is not disturbed. Start by setting the meniscus
(the curved surface of the liquid) to read precisely at the desired mark on the cylinder.
Then, hold the cylinder level on the horizontal plane and read the volume of the liquid at eye level, ensuring that the line of sight is perpendicular to the surface of the liquid. Repeat the process
multiple times to ensure that you have an accurate reading and record the result.
Finally, note down the atmospheric temperature and pressure as these can affect the volume of the liquid.
How do you do volume step by step?
To do volume step by step, you will need to use the formula for volume. The formula for volume is V = l x w x h, where V stands for volume, l stands for length, w stands for width, and h stands for
1. Begin by measuring the length, width, and height (in whatever unit of measurement is desired) for the object for which you are measuring the volume. You can use a ruler, tape measure, or other
measuring devices to take your measurements.
2. Once you have gathered your measurements, plug them into the formula for volume (V = l x w x h).
3. To calculate the volume, multiply the numbers together (the length times the width times the height). This should give you the volume of the object, which is measured in cubic units (units cubed).
4. Once you have the volume, you are finished with the calculation and can use the result for whatever you need it for.
How is volume of liquid measured?
Volume of a liquid is typically measured using either a graduated cylinder or volumetric flask. A graduated cylinder is a container that is marked off at regular intervals, which can then be read in
order to accurately gauge the volume of a liquid.
A volumetric flask, on the other hand, has a specific volume already marked on it, so an exact amount of liquid can be poured and measured accurately. To measure the volume of a liquid, begin by
adding the liquid to one of the instruments, and look down at the bottom of the meniscus.
This is the point of the liquid’s curvature, and should be read and recorded as the volume. If the amount of liquid is too large to fit in the instrument being used, divide the liquid up into
multiple readings and add the recorded data together for the total volume.
It is important to remember when using a graduated cylinder, the lowest graduated line should not be used as a reading as this is typically just a marking to indicate the bottom of the meniscus.
What are the 3 most common units of measurement?
The three most common units of measurement are the meter, the liter, and the gram. The meter is the base unit of distance in the International System of Units (SI). It is used to measure the length
of a certain object or the distance between two points.
The liter is the base unit of capacity in the SI and is used to measure the volume of an object or a liquid. Lastly, the gram is the base unit for mass in the SI and is used to measure the mass or
weight of an object.
These are the three most common units of measurement used in the SI and in everyday life.
What are the 7 basic units?
The seven basic units of measurement are the meter (m), kilogram (kg), second (s), ampere (A), kelvin (K), candela (cd), and mole (mol). The meter is the unit of length; the kilogram is the unit of
mass; the second is the unit of time; the ampere is the unit of electric current; the kelvin is the unit of temperature; the candela is the unit of luminous intensity; and the mole is the unit used
to measure the amount of a substance.
These seven basic SI (Système International d’Unités) units form the foundation of the International System of Units. All derived units and other units used in the sciences are defined in terms of
the seven SI base units.
How much is 4 units of liquid?
The answer to this question depends on what type of liquid is being measured. For example, if 4 units of liquid refers to gallons, the answer would be 4 gallons. However, if 4 units of liquid refers
to liters, the answer would be 4 liters.
Additionally, if 4 units of liquid refers to quarts, the answer would be 16 cups. Therefore, it is difficult to answer the question without knowing the type of liquid being measured.
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Connected Components
A connected component of an undirected graph is a maximal set of nodes such that each pair of nodes is connected by a path. What I mean by this is: a connected component of an undirected graph is a
subgraph in which any two vertices are connected to each other by path(s), and which is connected to no additional vertices in the rest of the graph outside the subgraph. For example, the graph shown
in the illustration has three connected components.
A vertex with no edges is itself a connected component.
A graph that has all the edges reachable from each other is itself connected and has exactly one connected component, consisting of the whole graph.
• The concept of Connected Components is only applicable to Undirected Graphs. The equivalent concept for Directed Graph is Strongly Connected Components .
You would have easy time understanding the logic behind how to find Connected Components if you already understand how DFS (Depth First Search) really well.
Notice from the image above how if you start DFS (Depth First Search) from any node belonging to a connected component, you would be able to discover all the nodes in the connected component.
So the algorithm to find all connected components in a given graph is simple: for every
node in a given graph, do a DFS (Depth First Search) on that node, and by the time the Depth First Search is done on all the nodes in the graph all the connected components in the graph would be
discovered. The simple well-commented code below shows how to implement this algorithm efficiently.
Java code:
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Python code:
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Time Complexity:
It's the
DFS (Depth First Search)
that does all the job to find
Connected Components
. The time complexity of finding
Connected Components
would be same as that of
Depth First Search
, if the code is implemented efficiently so that there is no additional overhead in the implementation to worsen the time complexity. So the time complexity of finding all
Connected Components
in a graph is :
• O(|V| + |E|), if the graph is implemented using Adjacency List.
|V| = total number of vertices in the graph,
|E| = total number of edges in the given graph.
If your graph is implemented using adjacency lists, wherein each node maintains a list of all its adjacent edges, then, for each node, you could discover all its neighbors by traversing its
adjacency list just once in linear time. For a directed graph, the sum of the sizes of the adjacency lists of all the nodes is E (total number of edges). So, the complexity of DFS is O(V) + O(E)
= O(V + E).
In best case: the given graph would have a null graph with no edges. So O(|V| + |E|) = O(|V| + |1|) = O(|V|)
In average case, the given graph is moderately sparse. So, O(|E|) = O(|V|).
Therefore, O(|V| + |E|) = O(|V| + |V|) = O(|V|)
In worst case, given graph is dense. So, O(|E|) = O(|V^2|) .
Therefore, O(|V| + |E|) = O(|V| + |V^2|) = O(|V^2|) .
• O(|V|^2), if the graph is implemented using Adjacency Matrix.
If your graph is implemented as an adjacency matrix (a V x V array), then, for each node, you have to traverse an entire row of length V in the matrix to discover all its outgoing edges. Please
note that each row in an adjacency matrix corresponds to a node in the graph, and the said row stores information about edges stemming from the node. So, the complexity of DFS is O(V * V) = O(V^
This is how I like to think about Connected Components: You can think each of the Connected Components to consist of items which are similar to each other. People with Data Mining background might
find the concept of Connected Components similar to the concept of Clusters.
Connected Components are each like cluster of similar objects. So, if you think you could design a solution for a problem by doing grouping of items based on some criteria (let's call it, similarity
factor) then there is high chance that the problem could be solved by Connected Components.
Now let's solve a fun little problem to have a very good idea of how the concept of
Connected Components
could be used to solve real-world problems.
Merge Intervals:
Given an array of intervals where intervals[i] = [start
, end
], merge all overlapping intervals, and return an array of the non-overlapping intervals that cover all the intervals in the input.
Example 1:
Input: intervals = [[1,3],[2,6],[8,10],[15,18]]
Output: [[1,6],[8,10],[15,18]]
Explanation: Since intervals [1,3] and [2,6] overlaps, merge them into [1,6].
Example 2:
Input: intervals = [[1,4],[4,5]]
Output: [[1,5]]
Explanation: Intervals [1,4] and [4,5] are considered overlapping.
This problem could be solved in various different ways. If you have a strong understanding of the concept and applications of
Connected Components
, you would be very easily be able to think of the below solution.
What we are interested here is to find the overlapping intervals and then merge them. So if we form a graph where all the overlapping intervals would be connected to each other (i.e, add an
undirected edge between interval A and interval B iff interval A and interval B are overlapping) then the overlapping intervals would form
Connected Component
. Now all we need to do is: for each connected component merge all the intervals in the connected component into one. The code below shows how this could be efficiently implemented.
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Fly counts example data
Cochran1954 {bayesmeta} R Documentation
Fly counts example data
This data set gives average estimated counts of flies along with standard errors from 7 different observers.
The data frame contains the following columns:
observer character identifier
mean numeric mean count
se2 numeric squared standard error
Quoting from Cochran (1954), example 3, p.119: “In studies by the U.S. Public Health Service of observers' abilities to count the number of flies which settle momentarily on a grill, each of 7
observers was shown, for a brief period, grills with known numbers of flies impaled on them and asked to estimate the numbers. For a given grill, each observer made 5 independent estimates. The data
in table 9 are for a grill which actually contained 161 flies. Estimated variances are based on 4 degrees of freedom each. [...] The only point of interest in estimating the overall mean is to test
whether there is any consistent bias among observers in estimating the 161 flies on the grill. Although inspection of table 9 suggests no such bias, the data will serve to illustrate the application
of partial weighting.”
W.G. Cochran. The combination of estimates from different experiments. Biometrics, 10(1):101-129, 1954.
## Not run:
# analysis using improper uniform prior
# (may take a few seconds to compute!):
bma <- bayesmeta(y=Cochran1954[,"mean"], sigma=sqrt(Cochran1954[,"se2"]),
# show joint posterior density:
plot(bma, which=2, main="Cochran example")
# show (known) true parameter value:
# show forest plot:
forestplot(bma, zero=161)
## End(Not run)
version 3.4
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pca: Principal Components Analysis in mixOmics: Omics Data Integration Project
Description Usage Arguments Details Value Author(s) References See Also Examples
Performs a principal components analysis on the given data matrix that can contain missing values. If data are complete 'pca' uses Singular Value Decomposition, if there are some missing values, it
uses the NIPALS algorithm.
1 pca(X,
2 ncomp = 2,
3 center = TRUE,
4 scale = FALSE,
5 max.iter = 500,
6 tol = 1e-09,
7 logratio = 'none', # one of ('none','CLR','ILR')
8 ilr.offset = 0.001,
9 V = NULL,
10 multilevel = NULL)
pca(X, ncomp = 2, center = TRUE, scale = FALSE, max.iter = 500, tol = 1e-09, logratio = 'none', # one of ('none','CLR','ILR') ilr.offset = 0.001, V = NULL, multilevel = NULL)
X a numeric matrix (or data frame) which provides the data for the principal components analysis. It can contain missing values.
ncomp integer, if data is complete ncomp decides the number of components and associated eigenvalues to display from the pcasvd algorithm and if the data has missing values, ncomp gives the
number of components to keep to perform the reconstitution of the data using the NIPALS algorithm. If NULL, function sets ncomp = min(nrow(X), ncol(X))
center a logical value indicating whether the variables should be shifted to be zero centered. Alternately, a vector of length equal the number of columns of X can be supplied. The value is
passed to scale.
scale a logical value indicating whether the variables should be scaled to have unit variance before the analysis takes place. The default is FALSE for consistency with prcomp function, but in
general scaling is advisable. Alternatively, a vector of length equal the number of columns of X can be supplied. The value is passed to scale.
max.iter integer, the maximum number of iterations in the NIPALS algorithm.
tol a positive real, the tolerance used in the NIPALS algorithm.
logratio one of ('none','CLR','ILR'). Specifies the log ratio transformation to deal with compositional values that may arise from specific normalisation in sequencing data. Default to 'none'
ilr.offset When logratio is set to 'ILR', an offset must be input to avoid infinite value after the logratio transform, default to 0.001.
V Matrix used in the logratio transformation id provided.
multilevel sample information for multilevel decomposition for repeated measurements.
a numeric matrix (or data frame) which provides the data for the principal components analysis. It can contain missing values.
integer, if data is complete ncomp decides the number of components and associated eigenvalues to display from the pcasvd algorithm and if the data has missing values, ncomp gives the number of
components to keep to perform the reconstitution of the data using the NIPALS algorithm. If NULL, function sets ncomp = min(nrow(X), ncol(X))
a logical value indicating whether the variables should be shifted to be zero centered. Alternately, a vector of length equal the number of columns of X can be supplied. The value is passed to scale.
a logical value indicating whether the variables should be scaled to have unit variance before the analysis takes place. The default is FALSE for consistency with prcomp function, but in general
scaling is advisable. Alternatively, a vector of length equal the number of columns of X can be supplied. The value is passed to scale.
integer, the maximum number of iterations in the NIPALS algorithm.
a positive real, the tolerance used in the NIPALS algorithm.
one of ('none','CLR','ILR'). Specifies the log ratio transformation to deal with compositional values that may arise from specific normalisation in sequencing data. Default to 'none'
When logratio is set to 'ILR', an offset must be input to avoid infinite value after the logratio transform, default to 0.001.
The calculation is done either by a singular value decomposition of the (possibly centered and scaled) data matrix, if the data is complete or by using the NIPALS algorithm if there is data missing.
Unlike princomp, the print method for these objects prints the results in a nice format and the plot method produces a bar plot of the percentage of variance explaned by the principal components
When using NIPALS (missing values), we make the assumption that the first (min(ncol(X), nrow(X)) principal components will account for 100 % of the explained variance.
Note that scale= TRUE cannot be used if there are zero or constant (for center = TRUE) variables.
Components are omitted if their standard deviations are less than or equal to comp.tol times the standard deviation of the first component. With the default null setting, no components are omitted.
Other settings for comp.tol could be comp.tol = sqrt(.Machine$double.eps), which would omit essentially constant components, or comp.tol = 0.
According to Filzmoser et al., a ILR log ratio transformation is more appropriate for PCA with compositional data. Both CLR and ILR are valid.
Logratio transform and multilevel analysis are performed sequentially as internal pre-processing step, through logratio.transfo and withinVariation respectively.
Logratio can only be applied if the data do not contain any 0 value (for count data, we thus advise the normalise raw data with a 1 offset). For ILR transformation and additional offset might be
pca returns a list with class "pca" and "prcomp" containing the following components:
ncomp the number of principal components used.
sdev the eigenvalues of the covariance/correlation matrix, though the calculation is actually done with the singular values of the data matrix or by using NIPALS.
rotation the matrix of variable loadings (i.e., a matrix whose columns contain the eigenvectors).
loadings same as 'rotation' to keep the mixOmics spirit
x the value of the rotated data (the centred (and scaled if requested) data multiplied by the rotation/loadings matrix), also called the principal components.
variates same as 'x' to keep the mixOmics spirit
center, scale the centering and scaling used, or FALSE.
explained_variance explained variance from the multivariate model, used for plotIndiv
the eigenvalues of the covariance/correlation matrix, though the calculation is actually done with the singular values of the data matrix or by using NIPALS.
the matrix of variable loadings (i.e., a matrix whose columns contain the eigenvectors).
the value of the rotated data (the centred (and scaled if requested) data multiplied by the rotation/loadings matrix), also called the principal components.
On log ratio transformations: Filzmoser, P., Hron, K., Reimann, C.: Principal component analysis for compositional data with outliers. Environmetrics 20(6), 621-632 (2009) Lê Cao K.-A., Costello ME,
Lakis VA, Bartolo, F,Chua XY, Brazeilles R, Rondeau P. MixMC: Multivariate insights into Microbial Communities. PLoS ONE, 11(8): e0160169 (2016). On multilevel decomposition: Westerhuis, J.A., van
Velzen, E.J., Hoefsloot, H.C., Smilde, A.K.: Multivariate paired data analysis: multilevel plsda versus oplsda. Metabolomics 6(1), 119-128 (2010) Liquet, B., Lê Cao, K.-A., Hocini, H., Thiebaut, R.:
A novel approach for biomarker selection and the integration of repeated measures experiments from two assays. BMC bioinformatics 13(1), 325 (2012)
nipals, prcomp, biplot, plotIndiv, plotVar and http://www.mixOmics.org for more details.
1 # example with missing values where NIPALS is applied
2 # --------------------------------
3 data(multidrug)
4 pca.res <- pca(multidrug$ABC.trans, ncomp = 4, scale = TRUE)
5 plot(pca.res)
6 print(pca.res)
7 biplot(pca.res, xlabs = multidrug$cell.line$Class, cex = 0.7)
9 # samples representation
10 plotIndiv(pca.res, ind.names = multidrug$cell.line$Class,
11 group = as.numeric(as.factor(multidrug$cell.line$Class)))
12 ## Not run:
13 plotIndiv(pca.res, cex = 0.2,
14 col = as.numeric(as.factor(multidrug$cell.line$Class)),style="3d")
16 ## End(Not run)
17 # variable representation
18 plotVar(pca.res)
19 ## Not run:
20 plotVar(pca.res, rad.in = 0.5, cex = 0.5,style="3d")
22 ## End(Not run)
24 # example with multilevel decomposition and CLR log ratio transformation (ILR longer to run)
25 # ----------------
26 ## Not run:
27 data("diverse.16S")
28 pca.res = pca(X = diverse.16S$data.TSS, ncomp = 5,
29 logratio = 'CLR', multilevel = diverse.16S$sample)
30 plot(pca.res)
31 plotIndiv(pca.res, ind.names = FALSE, group = diverse.16S$bodysite, title = '16S diverse data',
32 legend = TRUE)
34 ## End(Not run)
# example with missing values where NIPALS is applied # -------------------------------- data(multidrug) pca.res <- pca(multidrug$ABC.trans, ncomp = 4, scale = TRUE) plot(pca.res) print(pca.res)
biplot(pca.res, xlabs = multidrug$cell.line$Class, cex = 0.7) # samples representation plotIndiv(pca.res, ind.names = multidrug$cell.line$Class, group = as.numeric(as.factor
(multidrug$cell.line$Class))) ## Not run: plotIndiv(pca.res, cex = 0.2, col = as.numeric(as.factor(multidrug$cell.line$Class)),style="3d") ## End(Not run) # variable representation plotVar(pca.res) #
# Not run: plotVar(pca.res, rad.in = 0.5, cex = 0.5,style="3d") ## End(Not run) # example with multilevel decomposition and CLR log ratio transformation (ILR longer to run) # ---------------- ## Not
run: data("diverse.16S") pca.res = pca(X = diverse.16S$data.TSS, ncomp = 5, logratio = 'CLR', multilevel = diverse.16S$sample) plot(pca.res) plotIndiv(pca.res, ind.names = FALSE, group =
diverse.16S$bodysite, title = '16S diverse data', legend = TRUE) ## End(Not run)
For more information on customizing the embed code, read Embedding Snippets.
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simple explanation – Compass Rose
In my post on present value, I promised to explain how to turn a series of payments into a present value. This is the promised follow-up post.
Pay Today or Pay More Tomorrow
I am 27 years old. I recently bought a life insurance policy with a face value of $100,000. This policy will last my whole life - in other words, no matter when I die, the payout happens. It cost me
roughly $10,000 in today's money. If this is surprising to you, or you think the insurance company got a bad deal, then read this.
Everyone makes choices about whether they'd rather have something now, or something else later. Almost no one understands the economic concepts that describes these tradeoffs. they're called "present
value" and "discount rate."
I will start by describing some simple examples that use these concepts, without using the jargon. Then I will explain what these all have in common. I'm not going to explain how to use these in
real-life situations, but if you're interested, please let me know in the comments and I'll write a follow-up post.
Return on Investment
I'll start with a simplified example, with made-up numbers. Abby has a bank account with a bunch of money in it earning 2% guaranteed interest per year. She also owns a bond that would pay out $1,000
if she cashes it out now, or $1,030 if she cashes it out in a year. Should she cash it out now, or a year later?
Let's say that in any case she wouldn't use the money until a year from now. Then if she cashes out the bond now, she can immediately deposit the money, and in a year, she'll have $1,020. But that's
less than the $1,030 she'd get if she held onto the bond for a year.
On the other hand, suppose she wants to use the money right now. Then if she cashes out the bond now, she has an immediate $1,000 to spend. On the other hand, let's say she holds onto the bond, and
withdraws $1,000 from her bank account. Then in a year, she has $1,020 less in her account than she would have, but an extra $1,030 from the bond, putting her $10 ahead of the first strategy. So in
this case too she should hold onto the bond for another year.
It should be easy to see that if the bond only returned $1,010 in a year, Abby comes out ahead by cashing out now, again regardless of whether she wants to use the money now or later. Because the
bond gives her a lower return on investment (1%) than her savings account does (2%).
Then suppose the bond pays out $1,030 in a year, but her bank account offers 4% interest this year. Then Abby also comes out ahead by cashing out now, because the bond's return (3%) is less than the
interest she gets on her bank account.
Cost of Funds
Brian doesn't have any savings - he a student. But he has a good credit rating and is able to borrow at 5% interest per year, and is allowed to pay off his loans at any time.
He is deciding whether to rent a textbook for $100, or buy it for $150 and sell it back used to his school's bookstore in a year for $55.
If Brian rents his textbook, then after a year, he will owe $105, including interest, and have no textbook. On the other hand, if he buys his textbook, then after a year, he will owe $157.50. He can
then sell his textbook back to the bookstore for $55, use that to pay down his debt, and owe only $102.50. So buying the textbook is a better deal.
Suppose instead Brian can only borrow at 10% interest. Then if Brian rents his textbook, after a year, he will owe $110. On the other hand, if he buys his textbook, then after a year, he will owe
$165-$55=$110. So he should be indifferent between the two alternatives.
If Brian has to pay 15% interest, then if he rents his textbook, after a year he owes $115, but if he buys, then after a year he owes $125, so he comes out ahead by renting.
On the other hand, suppose at the 5% rate of interest, Brian can only collect $50 for his textbook after a year. Then instead of owing $102.50 at the end of a year, he'd owe $107.50, more than the
$105 he'd owe if he rented, so in that case renting again becomes more advantageous.
Present Value
In each of the above examples, a future amount of money was related to a present amount of money, by either how much money you'd have if you used the current money in the best way available (either
investing or paying off debt), or how much money you would have to have now, to produce the future money. The first is called the "future value" of money, and the second is called the "present value"
of money.
When Abby is choosing between $1,000 now and $1,030 in a year, the "future value" of $1,000 is how much money she'd have at the end of a year if she put the money in her bank account yielding 2%. To
get this, you multiply by (100%+2%=1.00+0.02=1.02): $1,000 * 1.02 = $1,020. This is less than the one-year future value of $1,030 in a year, which is of course $1,030.
The "present value" of the year-later $1,030 is the amount Abby would need today to produce that amount in a year. To calculate the value a year in the past, you simply do the opposite of what you
did when calculating the value a year in the future: you simply divide by (100%+2%=102%=1.02), to get $1,030/1.02=$1009.80, more than the present value of $1,000 today (which is of course $1,000).
Another way to show this is algebraically:
Now let's look at the first example involving Brian. Brian is comparing making a single payment today, with making a payment today plus receiving a payment in a year.
Since Brian has to pay 5% interest on money he borrows, the future value of the textbook rental expense is how much Brian will owe in a year if he borrows the money, or $100*1.05=$105. The future
value of the purchase price of the textbook is $150*1.05=$157.50, and the future value of the $55 Brian will receive for his textbook in a year is just $55. So the net future value of Brian's
textbook expenses if he buys is $157.50-$55.00=$102.50, less than the $105 future value of the rental fee.
The present value of the renting option, $100 today, is of course $100. The present value of the textbook's price today is also the same as the price, $150. The present value of getting $55 in a year
is the amount of debt he'd have to pay off now, to owe $55 less in a year: $55/1.05=$52.38. So the present value of the cost of buying and selling back later is $150-$52.38=$97.62, less than the $100
textbook rental fee. So the buying option costs less, in present value terms, as well.
The key here is that by converting each value, whether positive or negative, into the equivalent value for a single time period - whether the present or the future - we end up with numbers that can
be directly added and subtracted to find out which amount is higher on net.
Discount Rate
You may have noticed that in Abby's case we were using the rate at which she could expect return on her savings to equate future and present amounts, but in Brian's case we looked at the interest
rate he'd have to pay to borrow money. These might seem like quite different things, but in finance, there's little difference between spending saved money and borrowing money; in both cases money in
the future is worth more than money in the present, and we assume a fixed conversion factor. Instead of calling it a cost of borrowing sometimes and an expected return on investment at other times,
economics abstracts this into the more general term "discount rate", which is basically the extra share you can demand if you get your money in a year instead of today, or the share of your money you
should expect to give up if you get your money today instead of a year from now.
This is related to the economic concept of "opportunity cost," which I will cover in a future post.
I will also cover how to deal with a series of future payments in a future post - and in the process show you that if you believe in discount rates, the future isn't as big a deal as it seems.
Which means, of course, that this is the first post in a series.
Can God Make a Rock So Big He Can't Pick it Up? Or, Why Does My Calculus Textbook Start With This Chapter About Unions and Intersections?
Can God create a rock so big that He can't pick it up? To understand the problem, we need to understand set theory. But I don't really want to talk about Russell's paradox quite yet - a big problem
with set theory as it's taught is that it doesn't respond to a felt need, it's just plopped down at the beginning of a calculus or logic textbook without explanation. Here's a bunch of self-evident
stuff! Go calculate what the union of the intersections is!
I'm not going to tell you how to do set theory here. You can look that up if you want. I'm just going to try to explain a little bit about why it matters, why you should be interested in it, and how
to apply some set-theory-ish rules of thumb to your own thoughts.
Think about the difference between these two arguments:
The king of Freedonia is Phillip III.
The husband of Mary Teller is Phillip III.
Therefore, the king of Freedonia is the husband of Mary Teller.
Milk is white.
Snow is white.
Therefore, milk is snow.
The second argument looks just like the first one - but the first one works and the second one doesn't. Why?
Well, I've deliberately made it tricky by using the verb "is" in each case. "Is" is one of those tricky verbs whose meaning is very context dependent. Here's a more precise formulation of the
The king of Freedonia is the same as Phillip III.
The husband of Mary Teller is the same as Phillip III.
Therefore, the king of Freedonia is the same as the husband of Mary Teller.
Milk is one of the things that are always white.
Snow is one of the things that are always white.
Therefore, milk ??? snow.
it's not even clear which spurious consequence is supposed to follow from the second argument anymore. Is this a specious proof that milk and snow are identical, or that all milk is snow, or that all
snow is milk, or just that some things are both milk and snow?
Here's another paired example:
A shark is an aquatic animal.
An aquatic animal is a living thing.
Therefore, a shark is a living thing.
A knife is an item in my silverware drawer.
An item in my silverware drawer is a spoon.
Therefore, a knife is a spoon.
And with more specific wording:
Every shark is an aquatic animal.
Every aquatic animal is a living thing.
Therefore, every shark is a living thing.
At least one knife is an item in my silverware drawer.
At least one item in my silverware drawer is a spoon.
Therefore, ???
Or better yet:
There exists at least one item that is both a knife and in my silverware drawer.
There exists at least one item that is both in my silverware drawer and a spoon.
Therefore, ???
Set theory is a way to force yourself to use statements more explicit than "X is Y", to prevent you from accidentally equivocating and "proving" that knives are spoons. Since math is all about
proving possibly counterintuitive things, this is kind of important in math. But it's also important whenever you're making explicit compounded arguments of the (A, B, THEREFORE C) style.
In set theory you never say "X is Y." You instead are always talking about whether something is a member of a set. For now, think of a set as nothing more specific than a collection of things.
There's a problem with this, but I'll get to it later.
You can say that something is a member of a set, or that if something is a member of one set, then it must be a member of another, or that there is at least one thing that is both a member of set A
and a member of set B, etc. You can also negate these things - you can say that there are no things that are both members of set A and set B. Think about these sentences, and how to make them more
• A mouse is in this cage.
• A mouse is an animal.
• This mouse is Pinky.
• Pinky is in this cage.
• Dallas's football team is heavier than the people in China.
• A dragon is not real.
• WEF wrestling is fake.
Here are some formulations that are a little more set theory-ish:
• There exists at least one thing that is both a member of the set (is a mouse) and a member of the set (things in this cage)
• Every member of the set (is a mouse) is a member of the set (is an animal).
• Every member of the set (this mouse) is a member of the set (Pinky). Also, every member of the set (Pinky) is a member of the set (this mouse).
(A pithier way to say that one is: Something is a member of the set (this mouse) if and only if it is a member of the set (Pinky). This is an "identity" relation.)
• Every member of the set (Pinky) is a member of the set (in this cage).
• The average of the weights of all the members of the set (members of Dallas's football team) is higher than the average of the weights of all the members of the set (the people in China).
(This one is tricky - the original statement is ambiguous, because it's worded as a statement about the set, but what exactly are we saying is heavier than what? Are we saying that each Dallas Cowboy
is heavier than each person in China? Or that the Dallas Cowboys, weighed all together, are heavier than the people in China, weighed all together? Or that the average weight of a member of the first
set is greater than that of a member of the second? It's important to be specific about things like this when talking about group characteristics.)
• There are no members of the set (dragons) that are members of the set (real things).
• Every member of the set (WEF wrestling matches) is a member of the set (fake things).
Do you get the pattern? You never simply talk about how something "is" or "is not" something else, only about whether a member of set A is never, sometimes, or always a member of set B, and whether
an assertion is true or false.
This can be helpful in avoiding getting into stupid arguments. If someone says, "a mouse is an animal," do they mean that there is at least one mouse that is an animal, or that every mouse is an
animal, or that something is a mouse if and only if it's an animal?
If they mean that there's at least one mouse that's an animal, then finding a mouse that's not an animal (like a computer mouse, or a robotic mouse) is not evidence against their point - all they
have to do to prove it's true is find at least one mouse that is an animal. But if you phrase it explicitly like that, it's harder for them to equivocate and "prove" that a computer mouse is an
Or maybe more realistically, if I "prove" that wiggins are thieves by showing you one wiggin who steals something (which only proves that there is at least one wiggin who is a thief), I might then
pretend that you should draw the inference that some other wiggin is also a thief (which would only be valid if I had proved that every member of set "is a wiggin" is a member of set "is a thief").
If they mean that every mouse is an animal, then finding an example of a mouse that is not an animal is a counterexample, but finding an example of an animal that is not a mouse, like a dog, is a not
counterexample. If they've shown to your satisfaction that all members of set "mouse" are members of set "animal", then you can go on and assume that's true for each new mouse you encounter - but it
doesn't imply that all members of set "animal" are members of set "mouse".
Finally, if they show "if and only if," then you would have been able to prove them wrong just by showing them a dog. But if they convince you of this, then - and only then - you should accept the
inference both ways.
It's easy to lose track of this when you say things like "mice are animals" or "wiggins are thieves", so it can be helpful to use set-theoretic language (which is almost as compact), like "MICE is a
subset of ANIMALS."
OK, so what does this have to do with God's rocks? Well, sets are important, right? And we want to be correct when talking about important things - and sets help us be correct. So we want to describe
sets using other sets. And talk about sets of sets!
Like you might want to talk about the properties of "sets that have no members." Or "sets that have a finite number of members." This is fine. But there are limits.
Let's walk through one of them - the rock paradox. It's usually stated as:
God is omnipotent. That means God can do any thing.
Making a rock so big that God can't pick it up is a thing.
Therefore, God can make a rock so big that God can't pick it up.
But picking up an arbitrary object that exists is also a thing.
Therefore God can pick up an arbitrary object that exists.
Now, let that arbitrary object be "a rock so big that God can't pick it up."
Then, God can pick up a rock so big that God can't pick it up.
Now, if the existence of such a rock were impossible, then this wouldn't be a problem. But we just said that God can make one.
But it's not really a rock so big that God can't pick it up, if God can pick it up.
Thus, the omnipotence of God implies a contradiction.
Therefore, there can be no omnipotent God.
The problem here seems to be using omnipotence in the definition of one of the powers. If you don't allow that, then there's no way to get the contradiction.
This brings up another set-theoretic principle: the "things" a set can be a collection of have to be well-defined, before we define any of the sets. So if we're talking about puppies, and we already
know what puppies are, without using sets of puppies in the definition, then we can talk about sets of puppies. But we can't just define a collection of "puppies and sets of puppies," before we know
what the sets of puppies are. And the sets of puppies can't themselves be defined until the puppies are defined.
So does the rock paradox follow this rule? No.
"God is omnipotent" can be rephrased as:
For every ability X, let there be a set (entities that have ability X).
Every omnipotent being is a member of every such set.
God is an omnipotent being.
Therefore, for every ability X, God is a member of the set (entities that have ability X.)
Now, this works for abilities like "walk on water" or "use set-theoretic notation" or "make ten commands". Because those things are well-defined even if we don't know about God.
How about "make a rock so big that God can't pick it up." Is this well-defined before we start talking about sets of abilities? No, because the ability is defined by a reference to what God can do,
and what God can do is defined by a particular set of abilities. So a collection of abilities that includes "make a rock so big that God can't pick it up" is simply not a well-defined collection that
we can take sets of.
In fact, "make a rock so big that [someone] is not a member of set (entities that have the ability to pick up a rock of that size)" is never a first-order ability.
A set-theoretically valid definition of omnipotence would be something more like this:
Define some collection of "abilities," none of which reference other powers or omnipotence directly.
Define omnipotence as the set of all these abilities.
Now, maybe "make an arbitrarily large rock" is one of the powers. And maybe "pick up an arbitrarily large rock" is a power. But none of the powers refer to each other, or to sets of powers, no matter
how indirectly. So "make a rock so big that God can't pick it up" isn't an ability.
We can then think of sets of abilities, like the set of rock-making and rock-picking-up. Omnipotence is the ability-set that is contains all abilities.
Now we need to use a concept called a "subset." A X is a subset of Y if every member of set X is also a member of set Y. For example, "Puppies" is a subset of "Animals," and "Animals" is also a
subset of "Animals, but "Animals is not a subset of "Puppies."
So every ability-set is a subset of omnipotence.
Of course, that doesn't mean that no one can make a rock so big that someone else can't pick it up. Or even a rock so big that they themselves can't pick it up. But that's a statement about
combinations of abilities and inabilities.
So what if you wanted to describe all the collections of abilities that don't include certain abilities? Well, that's a second-order set. Call it a schmet. So you might have a schmet of ability-sets
that include walking on water, but not swimming. Or making a 32kg rock, but not picking it up.
Now let's get back to that paradox. Can God make a rock so big that He can't pick it up? How does that cash out when thinking about sets of abilities?
If someone can make a rock so big they can't pick it up, that means that their ability-set is a member of a certain schmet. In particular, it's the schmet that includes ability-sets where for some
size X, they include the ability "can make a rock of size X", and also do not include any ability "can pick up a rock of up to size Y", for any Y>=X.
So the question is, is God's ability set (omnipotence) a member of that schmet? The answer is no: omnipotence is not a member of the schmet "can make a rock so big you can't pick it up."
There's no paradox, because a schmet is not an ability. Remember, we had to define all the abilities before defining any of the ability-sets, and we had to define the ability-sets before defining the
schmets. So there can't be an ability that refers to a schmet! And omnipotence is an ability-set, so its definition can't refer to schmets either - it's just the ability set that includes all
If you look up Russell's paradox explained, you will find a similar exposition, except it's less fun because it isn't about God and rocks.
Zeugma and Syllepsis
I did a little research and think I finally understand what zeugma and syllepsis are, and how they relate to each other.
Zeugma is any case where a single mention of a word is treated as a part of more than one clause of a sentence. Syllepsis is a type of zeugma where the word in question is used in contexts that
require it to do different things.
So "He prefers dogs, she cats" is zeugma because "prefers" gets re-used, but it is not syllepsis because "prefers" is doing the exact same thing in "He prefers dogs" and the implied "She prefers
There are two types of syllepsis.
Grammatical syllepsis is where the difference is in verb form. So "I prefer dogs, she cats" would be grammatical syllepsis, because "she prefer cats" is ungrammatical - the implicit extra verb is
"prefers", a different form of the same word.
Semantic syllepsis is where the difference is in meaning. So in "And he said as he hastened to put out the cat, the wine, his cigar and the lamps", "put out" can mean to expel, to retrieve from
storage, to extinguish, and possibly to turn off (I am not sure whether that last sense would be anachronistic).
You can't have a syllepsis without a Zeugma because the different usages have to be attributed to the same single mention, which is the definition of zeugma. So if the line from the old song went,
"And he said as he hastened to put out the cat, put out the wine, put out his cigar and put out the lamps", we would have a weird phrasing, but nothing disorienting - the casual listener might not
even notice the repetition, instead hearing "put out [X]" as a whole phrase. It is where we have to attribute to each clause the same mention of the verb that our attention is called to its different
Zeugma in itself may be nothing more than parsimonious phrasing, and asylleptic zeugma may easily pass unnoticed. Syllepsis is a distinct type of zeugma because it is unexpected, and draws attention
to itself.
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How to Find the Mode or Modal Value
Statistics Definitions > Mode
What is the Mode?
The mode, or modal value, is the most common number in a data set. It’s useful in statistics because it can tell you what the most popular item in your set is. For example, you might have results
from a customer survey where your company is rated from 1 to 5. If the most popular answer is 2, then you know you need to make some improvements in customer service!
A data set can have no mode, one, or many:
• None: 1, 2, 3, 4, 6, 8, 9.
• One mode: unimodal: 1, 2, 3, 3, 4, 5.
• Two: bimodal: 1, 1, 2, 3, 4, 4, 5.
• Three: trimodal: 1, 1, 2, 3, 3, 4, 5, 5.
• More than one (two, three or more) = multimodal.
Note: Sometimes the mode is shortened to “mod”; Don’t confuse this with a modulo function (from calculus).
Histograms and Modes
A histogram shows frequencies of values. In other words, how often a value appears in a data set. Look for the “bump” in the histogram. In the histogram below, the bump is at 4:
In real life, you’ll rarely (if ever) see a histogram with single digit numbers on the x-axis. You’ll see bars that are groups of numbers. For example a bar might represent 10 to 20, or it might
represent 30-40. The technique is still the same — look for the “bump” in the histogram. With histograms that have bars with groups of numbers, you’ll have to ballpark where exactly the number is.
The easiest way to do it is to take the number to the left and right of the highest bar and figure out where the middle is.
The numbers either side of the “bump” are 40 and 60, so an estimate is at 70 — right in the middle.
Fun Fact
A relationship between the mode, mean and median for unimodal distribution curves that are moderately asymmetry is given by this equation:
Mean – mod ≈ 3(mean – median).
How to find the mode by hand
Need help with a homework question? Check out our tutoring page!
The mode in statistics is the most common number in a data set. For example, in this set it’s 2, because it is the number that occurs most often: 1, 2, 2, 5, 6. Data sets in statistics tend to be
much larger, so the solution is easier to spot if you put the numbers in order.
Sample question: Find the mode for the following data set:
56, 57, 56, 58, 59, 90, 98, 98, 65, 45, 34, 34, 23, 23, 24, 33, 56, 67, 78, 87, 87, 56.
Step 1: Put the numbers in order:
Step 2: Count how many times each number appears. This may be easier if you put the numbers in a column/row format like this:
The most common number is 56 in this data set (it appears 4 times).
How to find the mode in Microsoft Excel
Using the Sort Button
If you have a large number of items in your data set, Excel has a “Sort” button on the toolbar that will sort numbers from smallest to largest or largest to smallest. Type your numbers into a single
column in Excel. Click “Home,” then click “Sort and Filter” and then click “A to Z” to sort from smallest to largest or “Z to A” to sort from largest to smallest.
Sorting numbers in Excel can make it easier to find the mode
Excel 2013
There are a couple of additional ways to find the mode in Excel 2013: the MODE function and Data Analysis Toolpak.
MODE Function
Step 1: Type your data into a single column for each set. For example, if you have one set, type your data into F1 to F20.
Step 2: Type “=MODE(F1:F20)”
where “F1:F20” is the location of your data set.
Step 3: Press “Enter.”
That’s it!
Data Analysis
Step 1: Click the “Data” tab and then click “Data Analysis.”
Step 2: Click “Descriptive Statistics” and then click “OK.”
Step 3: Click the Input Range box and then type the location for your data. For example, if you typed your data into cells A1 to A10, type “A1:A10” into that box
Step 4: Click the radio button for Rows or Columns, depending on how your data is laid out.
Step 5: Click the “Labels in first row” box if your data has column headers.
Step 6: Click the “Descriptive Statistics” check box.
Step 7: Select a location for your output. For example, click the “New Worksheet” radio button.
Step 8: Click “OK.”
Excel 2007-2010
Use a function to find the mode in Microsoft Excel
Step 1: Type your data into one column. Enter only one number in each cell. For example, if you have twenty data points, type that data into cells A1 through A20. Press “Enter” after each number
entry to move down the column.
Step 2: Click a blank cell anywhere on the worksheet and then type “=MODE.SNGL(A1:A2)” without the quotation marks.
Step 3: Change the range in Step 2 to reflect your actual data. For example, if your numbers are in cells A1 through A20, change A1:A2 to A1:A20.
Step 4: Press “Enter.” Excel will return the solution in the cell with the formula.
Data Analysis
Step 1: Complete Step 1 above.
Step 2: Click the “Data” tab and then click “Data Analysis.” If Data Analysis does not show up on your ribbon, it means you don’t have it loaded. How to load the Data Analysis Toolpak.
Step 3: Click “Descriptive Statistics”.
Tip: You could also type up to 254 numbers into a function argument. Click on a cell and then type “MODE.SNGL(num1,num2,num3…)” where num1,num2,num3 are your actual numbers. For example “=MODE.SNGL
({5.6,4,4,3,2,4})” would return 4 as the solution for the data set.
Tip: Earlier versions of Excel used “MODE” instead of “MODE.SNGL.” The function was replaced in Excel 2010 because the algorithm was inaccurate.
Note on the MODE.MULT function: The steps for using the MODE.MULT function are exactly the same. However, the MODE.MULT function will tell you if there are multiple modes.
Check out our YouTube channel for more Excel tips and help!
How to find the mode in Minitab
Finding a mode in Minitab takes seconds, once you have entered your data into a worksheet.
Example problem: Find the mode for the following data set: 12, 12, 13, 15, 21, 23, 23, 24, 25, 25, 26, 45, 45, 45, 45, 45, 45, 45, 51, 52, 53, 53, 54, 56, 56, 56, 57, 58, 59, 65, 78, 78, 85, 87, 88,
89, 89, 89.
Step 1: Type your data into a single column in a Minitab worksheet.
Step 2: Click “Stat”, then click “Basic Statistics,” then click “Descriptive Statistics.”
Step 3: Click the variables you want to find the mode for and then click “Select” to move the variable names to the right window.
Step 4: Click “Statistics.”
Step 5: Check the “Mode” box and then click “OK” twice. The result will be displayed in a new Minitab Session window. The solution for this particular set of data is 45.
That’s it!
Gonick, L. (1993). The Cartoon Guide to Statistics. HarperPerennial.
Lindstrom, D. (2010). Schaum’s Easy Outline of Statistics, Second Edition (Schaum’s Easy Outlines) 2nd Edition. McGraw-Hill Education
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Study Guide - Use the Linear Factorization Theorem to find polynomials with given zeros
Use the Linear Factorization Theorem to find polynomials with given zeros
A vital implication of the Fundamental Theorem of Algebra, as we stated above, is that a polynomial function of degree n will have n zeros in the set of complex numbers, if we allow for
multiplicities. This means that we can factor the polynomial function into n factors. The Linear Factorization Theorem tells us that a polynomial function will have the same number of factors as its
degree, and that each factor will be in the form (x – c), where c is a complex number.
Let f be a polynomial function with real coefficients, and suppose [latex]a+bi\text{, }b\ne 0\\[/latex], is a zero of [latex]f\left(x\right)\\[/latex]. Then, by the Factor Theorem, [latex]x-\left
(a+bi\right)\\[/latex] is a factor of [latex]f\left(x\right)\\[/latex]. For f to have real coefficients, [latex]x-\left(a-bi\right)\\[/latex] must also be a factor of [latex]f\left(x\right)\\[/
latex]. This is true because any factor other than [latex]x-\left(a-bi\right)\\[/latex], when multiplied by [latex]x-\left(a+bi\right)\\[/latex], will leave imaginary components in the product. Only
multiplication with conjugate pairs will eliminate the imaginary parts and result in real coefficients. In other words, if a polynomial function f with real coefficients has a complex zero [latex]
a+bi\\[/latex], then the complex conjugate [latex]a-bi\\[/latex] must also be a zero of [latex]f\left(x\right)\\[/latex]. This is called the Complex Conjugate Theorem.
A General Note: Complex Conjugate Theorem
According to the Linear Factorization Theorem, a polynomial function will have the same number of factors as its degree, and each factor will be in the form [latex]\left(x-c\right)\\[/latex], where c
is a complex number.
If the polynomial function f has real coefficients and a complex zero in the form [latex]a+bi\\[/latex], then the complex conjugate of the zero, [latex]a-bi\\[/latex], is also a zero.
How To: Given the zeros of a polynomial function [latex]f\\[/latex] and a point [latex]\left(c\text{, }f(c)\right)\\[/latex] on the graph of [latex]f\\[/latex], use the Linear Factorization Theorem
to find the polynomial function.
1. Use the zeros to construct the linear factors of the polynomial.
2. Multiply the linear factors to expand the polynomial.
3. Substitute [latex]\left(c,f\left(c\right)\right)\\[/latex] into the function to determine the leading coefficient.
4. Simplify.
Example 7: Using the Linear Factorization Theorem to Find a Polynomial with Given Zeros
Find a fourth degree polynomial with real coefficients that has zeros of –3, 2, i, such that [latex]f\left(-2\right)=100\\[/latex].
Because [latex]x=i\\[/latex] is a zero, by the Complex Conjugate Theorem [latex]x=-i\\[/latex] is also a zero. The polynomial must have factors of [latex]\left(x+3\right),\left(x - 2\right),\left(x-i
\right)\\[/latex], and [latex]\left(x+i\right)\\[/latex]. Since we are looking for a degree 4 polynomial, and now have four zeros, we have all four factors. Let’s begin by multiplying these factors.
[latex]\begin{cases}f\left(x\right)=a\left(x+3\right)\left(x - 2\right)\left(x-i\right)\left(x+i\right)\\ f\left(x\right)=a\left({x}^{2}+x - 6\right)\left({x}^{2}+1\right)\\ f\left(x\right)=a\left
({x}^{4}+{x}^{3}-5{x}^{2}+x - 6\right)\end{cases}\\[/latex]
We need to find a to ensure [latex]f\left(-2\right)=100\\[/latex]. Substitute [latex]x=-2\\[/latex] and [latex]f\left(2\right)=100\\[/latex] into [latex]f\left(x\right)\\[/latex].
[latex]\begin{cases}100=a\left({\left(-2\right)}^{4}+{\left(-2\right)}^{3}-5{\left(-2\right)}^{2}+\left(-2\right)-6\right)\hfill \\ 100=a\left(-20\right)\hfill \\ -5=a\hfill \end{cases}\\[/latex]
So the polynomial function is
[latex]f\left(x\right)=-5\left({x}^{4}+{x}^{3}-5{x}^{2}+x - 6\right)\\[/latex]
Analysis of the Solution
We found that both i and –i were zeros, but only one of these zeros needed to be given. If i is a zero of a polynomial with real coefficients, then –i must also be a zero of the polynomial because –i
is the complex conjugate of i.
Q & A
If 2 + 3i were given as a zero of a polynomial with real coefficients, would 2 – 3i also need to be a zero?
Yes. When any complex number with an imaginary component is given as a zero of a polynomial with real coefficients, the conjugate must also be a zero of the polynomial.
Try It 5
Find a third degree polynomial with real coefficients that has zeros of 5 and –2i such that [latex]f\left(1\right)=10\\[/latex].
Licenses & Attributions
CC licensed content, Shared previously
• Precalculus. Provided by: OpenStax Authored by: Jay Abramson, et al.. Located at: https://openstax.org/books/precalculus/pages/1-introduction-to-functions. License: CC BY: Attribution. License
terms: Download For Free at : http://cnx.org/contents/[email protected]..
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Linear Analysis
Subject 620-312 (2009)
Note: This is an archived Handbook entry from 2009. Search for this in the current handbook
Credit Points: 12.50
Level: 3 (Undergraduate)
This subject has the following teaching availabilities in 2009:
Semester 2, - Taught on campus.
Pre-teaching Period Start not applicable
Teaching Period not applicable
Dates & Assessment Period End not applicable
Locations: Last date to Self-Enrol not applicable
Census Date not applicable
Last date to Withdraw without fail not applicable
Lectures and practice classes.
Timetable can be viewed here. For information about these dates, click here.
Time Commitment: Contact Hours: 36 one-hour lectures (three per week) and up to 12 one-hour practice classes (one per week)
Total Time Commitment: 120 hours total time commitment.
Prerequisites: Metric Spaces.
Corequisites: None
Background None
Non Allowed None
Core It is University policy to take all reasonable steps to minimise the impact of disability upon academic study and reasonable steps will be made to enhance a student's participation
Participation in the University's programs. Students who feel their disability may impact upon their active and safe participation in a subject are encouraged to discuss this with the relevant
Requirements: subject coordinator and the Disability Liaison Unit.
Assoc Prof Jerry Koliha
The most important topic of this subject is integration. Students meet this concept in a calculus course where an integral is defined as a Riemann integral. Although a Riemann
integral is useful in many areas of mathematics, it is not adequate for many problems of modern analysis. The aim of the subject is to introduce students to the Lesbesgue theory of
integration and measure theory. Included in this subject is an introduction to the fundamental concepts of functional analysis. Functional analysis is the common name for the study
Subject of infinite dimensional vector spaces and the linear maps between them. What distinguishes this subject from linear algebra is the role of topological considerations. These topics
Overview: are not only beautiful and interesting but are also useful in other branches of mathematics such as probability theory, partial differential equations and quantum mechanics.
Topics include construction of measures, measurable functions, Lesbesgue integrals, convergence theorems, Lp-spaces, Fubini's theorem, normed spaces and Banach spaces, inner product
and Hilbert spaces, linear functionals and linear operators.
Objectives: .
Up to 36 pages of written assignments due during the semester (either 0% or 20%); a 3-hour written examination in the examination period (80% or 100%). The relative weighting of the
Assessment: examination and the assignments will be chosen so as to maximise the student's final mark.
Prescribed None
This subject potentially can be taken as a breadth subject component for the following courses:
Breadth Options: You should visit learn more about breadth subjects and read the breadth requirements for your degree, and should discuss your choice with your student adviser, before deciding on
your subjects.
Fees Subject EFTSL, Level, Discipline & Census Date
Notes: This subject is available for science credit to students enrolled in the BSc (pre-2008 degree only), BASc or a combined BSc course.
Related Majors/ Mathematics && Statistics Major
Minors/ Mathematics and Statistics (Pure Mathematics specialisation)
Download PDF version.
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Causality - M1 - 8EC
1. Bachelor level probability theory (e.g., at the level of G. Grimmett and D. Welsh, 'Probability - An introduction', 2nd edition)
2. Bachelor level measure theory (e.g., at the level of R. Schilling, 'Measures, Integrals and Martingales' (2nd edition), Cambridge University Press, 2017)
3. Bachelor level statistics (e.g., at the level of F. Bijma, M. Jonker, A. van der Vaart, 'An introduction to Mathematical Statistics', Amsterdam University Press, 2017)
Aim of the course
Many questions in science and society are of a causal nature. For example, does vaping cause lung cancer? How many deaths have been prevented by the first COVID-19 vaccination campaign in the
Netherlands? The probability for female PhD students at Dutch universities to graduate with distinction is only about half as that for males: is this evidence for discrimination based on gender? For
dealing with these and similar questions properly in a quantitative fashion, one needs to go beyond the classical techniques (like regression and classification) taught in elementary statistics and
machine learning courses.
In this course, you will learn how to model causality mathematically, how to reason formally about cause, effect and counterfactuals, how to predict consequences of actions, and how to analyze data
for answering questions of a causal nature. We will make use of two different probabilistic frameworks for modeling causality: causal Bayesian networks and structural causal models. Topics addressed
will be causal modeling (definition of Markov kernels, conditional independences, causal Bayesian networks, structural causal models, marginalization, confounders, selection bias, feedback loops,
causal graphs, interventions, Markov properties), causal reasoning and estimation (intervention variables, do-calculus, counterfactuals, covariate adjustment, back-door criterion, identifiability),
and causal discovery and estimation (randomized controlled trials, instrumental variables, local causal discovery, Y-structures, the FCI algorithm).
- Joris Mooij, Korteweg-De Vries Institute for Mathematics, University of Amsterdam
- Patrick Forré, Informatics Institute, University of Amsterdam
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Observations Relating to Forced Random Noise Resistance Determination in Childhood Asthma
In general, asthmatic patients with an FEVi less than 80 percent of predicted or FEF less than 70 percent of predicted had higher resistance values than asymptomatic asthmatic patients with normal
spirometry (Fig 2). This is supported by the position of the two regression lines and by the numerical data, for example, the average of1μ for the 15 with normal spirometry was 5.55 cmH2OLr£ and for
the other 15 it was 7.53 cmHaOLTs. Resistance values in our older children suffering from asthma with abnormal spirometric findings generally were larger than forced oscillatory and random noise
resistance values previously reported by others for normal children of comparable ages.
For example, at a height of 150 cm, our regression equations for the group with normal spirometry yielded values of 6.4, 4.8. and 5.4 cmH20*Li for 11*, H**, and Re, respectively. The corresponding
value from the equation of Williams et al was 3.09 cmH20*Lr5; from that of Mansell et al,u it was 3.48 cmH2OLi; and from that of Stanescu et al it was 5.20 cmH2OLTs. Our higher resistance values are
consistent with the effects of asthma and its magnitude is similar to that reported by Cogswell, who measured forced oscillatory resistances that were 2 SD above his normal value in 23 of 42
asthmatic children.
In young children, resistance generally decreased with frequency, while in older children, resistance generally increased with frequency. For example, when a separation is made at nine years of age,
the difference between low and high frequency resistance averaged 0.50 cmH20*LrS in the younger children and —0.45 cmH2OeLTs in the older children. This trend in frequency dependence where resistance
decreases with frequency up to a certain height after which it shows an increase with frequency is consistent with previous work (Fullton, personal communication). Furthermore, our regression lines
for R« and R* cross at a height of 160 cm, a value that coincides with that found by Fullton and associates (personal communication). It is somewhat surprising that resistance in those younger than
two years of age (Fig 4) did not show a stronger frequency dependence; however, only the three-week-old infant showed an increasing resistance, and it is possible that the mouth impedance in infants
is a much more important problem than in older subjects.
Mean values obtained from the individuals coefficients of variation for repeated measurements of Re, Rae, and Ee.ae were 9.6, 8.9, and 7.4 percent, respectively. These values suggest that the
expected variability of repeated measurements of these parameters was on the order of 7 percent to 10 percent and that changes after some intervention that exceeded 15 percent to 20 percent, twice
the expected coefficient of variation, indicated altered function. These average coefficients of variation were comparable to 14 percent reported by Williams et al for normal three to five year old
children and the 12 percent reported by Cogswell (cogswell.edu) also for normal children. This variability is comparable to that seen in forced expiratory spirometric parameters; for example, an
earlier study from our laboratory reported coefficients of variation of 4.3 percent and 16.5 percent in FEV1 and FEF^* in young asthmatic patients.
Random noise resistance parameters showed fairly good correlation with forced expiratory spirometric parameters with correlation coefficients ranging from 0.51 to 0.89 with most greater than 0.80.
This suggests that these resistance parameters provide a comparably valid measure of respiratory mechanical function as forced expiratory spirometric parameters, the generally accepted standards.
The three resistance parameters correlated best with high lung volume spirometric parameters, that is, FEVi and FEFts, and poorest with the low lung volume parameter, FEF^; correlations with mid-lung
volume parameters, FEFa^Ts, and FEFgo were intermediate between these two extremes. We believe that this pattern simply reflects the increase in variability of forced expiratory flow as lung volume
decreased. Another possible explanation could be the fact that both resistance and high lung volume spirometric parameters are mainly large airway measurements. From the opposite point of view,
spirometric parameters correlated better with the low frequency resistance parameter, R«, than with its high frequency counterpart, R. Perhaps this is due to the fact that low frequency measurements
reflect the resistance of the entire system while high frequency measurements reflect only the central resistance.
Correlations between bronchodilator-induced changes in the resistance and spirometric parameters were poor with only four correlations being statistically significant, and only one having a
correlation above 0.70. Most correlation coefficients fell below 0.60. Kabiraj et al reported similar correlation coefficients between changes in forced oscillatory resistance at 10 Hz and changes in
FEVl and peak expiratory flow; these coefficients were 0.54 and 0.59, respectively. We believe this lack of correlation between the changes in these two groups of mechanical parameters may be due to
two factors. First, the maximum inspiration can cause reflex changes in airway smooth muscle tone. Thus, the maneuver itself actually alters mechanical function so that changes in spirometric
parameters reflect the combination of the bronchodilator effect along with the maneuver effect. The second factor concerns secondary effects of the bronchodilator on dynamic airway compression and
the site of flow limitation during forced expirations. In this mechanism, the bronchodilator reduces bronchomotor tone in the large central airways, thereby making them more compressible during
forced expiration, so that flow actually decreases. Perhaps individual variation in the importance of these two factors accounts for the large variability in the bronchodilator-induced changes in
four of the five spirometric parameters as reflected by the large standard deviations in Table 4. Also, these mechanisms provide an explanation for the paradoxic bronchodilator-induced decrease in
FEVi that we observed in five subjects. The fact that only one patient showed paradoxic response on FEF^ts* could mean that the site of dynamic compression is mainly in the large airways.
In the results section, data were reported from 16 children three years old or younger. Post-broncho-dilator data in these younger children appeared to fit an extension of the post-bronchodilator
regression curve for the older children (Fig 3). The marked bronchodilator-induced decrease in resistance in the younger group suggests that bronchospasm is an important component of airway
obstruction in children two to three years of age. We attempted to make measurements in several other subjects under two years of age; however, we were unsuccessful with most of them, because the
children usually cried when their noses were sealed, and this interfered with the random noise measurement as indicated by low coherence values. In children two and three years old, we generally were
able to obtain reliable random noise measurements, and the approach has great potential for characterizing respiratory function and its change with disease and bronchodilator therapy in this age
group. It is probable that the problem with those younger than two years old could be overcome with the use of a mild sedative.
The articles of this research published earlier you will find below:
Table 4 —Correlation Coefficients Between Bronchodilator-induced Changes in Random Noise Resistance Barometers and Forced Expiratory Spirometric Barometers
│ │AFEVj│AFEF[&75%] │afef™│AFEF*.│AFEF*.│
│ARe│0.57*│0.42 │0.73*│0.58t │0.37 │
│AR*│0.24 │0.32 │0.39 │0.41 │0.51* │
│AIU│0.33 │0.26 │0.50$│0.37 │0.32 │
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Transport Phenomena - short answer questions from AMIE exams (Summer 2019)
Answer the following in brief (10 x 2)
What are the factors on which viscosity depends?
The viscosity of the liquid increases if the density of the liquid increases.
We know that the density of the liquid decreases as the temperature increases and hence viscosity also decreases.
So we can say that the viscosity of liquid depends on both density and temperature both.
Differentiate between free and wall turbulence.
Turbulence can be generated in two ways: (1) by friction forces at solid walls (or surfaces)
and (2) by the flow of fluid layers with different velocities past the other. These two types
of turbulence are, respectively, known as wall turbulence and free turbulence.
What is ‘The Von Karman analogy?
Von Karman extended Prandtl’s analogy by separating the flow field into three distinct layers: a viscous sublayer, a buffer layer, and a turbulent core. In the buffer layer, molecular and eddy
diffusivities are assumed to be of the same order of magnitude.
The physical significance of Prandtl and Schmidt numbers
Schmidt number (Sc) is a dimensionless number defined as the ratio of momentum diffusivity (kinematic viscosity) and mass diffusivity, and it is used to characterize fluid flows in which there are
simultaneous momentum and mass diffusion convection processes.
In heat transfer problems, the Prandtl number controls the relative thickness of the momentum and thermal boundary layers. When Pr is small, it means that the heat diffuses quickly compared to the
velocity (momentum).
Equation of motion.
Explain Newton’s law of viscosity and Fourier’s law of conduction.
Newton’s law of viscosity defines the relationship between the shear stress and shear rate of a fluid subjected to mechanical stress. The ratio of shear stress to shear rate is a constant, for a
given temperature and pressure, and is defined as the viscosity or coefficient of viscosity. Newtonian fluids obey Newton’s law of viscosity. The viscosity is independent of the shear rate.
Non-Newtonian fluids do not follow Newton’s law and, thus, their viscosity (ratio of shear stress to shear rate) is not constant and is dependent on the shear rate.
Fourier’s law states that the negative gradient of temperature and the time rate of heat transfer is proportional to the area at right angles of that gradient through which the heat flows.
Boiling curve
The boiling curve is a graph of heat flux versus wall superheat, the difference between the wall temperature and the saturation temperature (or boiling point). The curve is often drawn with log
scales to accommodate the rather large range of variables.
Effect of pressure and temperature of thermal conductivity
For all liquids, the coefficient of thermal conductivity increases with increasing pressure.
The thermal conductivity of liquids decreases with increasing temperature as the liquid expands and the molecules move apart. While in solids, the thermal conductivity decreases at higher
temperatures due to the anharmonic scattering which is inversely proportional to the temperature changes.
Universal distribution laws of Newtonian fluids
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Getting the Best What Is a Math Model
How to Find What Is a Math Model
It’s capable of replicating all the other more specialized simulations. Usually, the model is a simplification of the physical planet, but captures the gist of the issue. The whole-part model might
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The final step is to compute the full number of pencils. Click the provided links to download your preferred paper. Employing the advertising info, make a report or a press release and post it to a
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Tip When referencing a model number, make sure to’re employing the model number off of the item sticker, as opposed to the generic model number on the front part of the computer. Mathematical models
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Why Almost Everything You’ve Learned About What Is a Math Model Is Wrong
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will help to get what the common values are when the actual world messes things up!
It’s possible for you to select unique variables to customize these word problems worksheets for your requirements. It is able to help you estimate. The worksheets in this section combine both
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The What Is a Math Model Game
You can have a peek at thesegeometry exercisesto get you started. All you will need is access to Internet few times per week, few minutes each moment. The final step is to figure the entire quantity
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The majority of the simulation web pages show the method by which the math is derived. The interface a lot simpler to use than the old edition! Click the button and discover it on your PC.
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The neighborhood school was on the point of anarchy. You also have to supplement this knowledge of the way to solve word problems with a good comprehension of the math topic in question. The solution
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Want to Know More About What Is a Math Model?
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What Is a Math Model at a Glance
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Empowering students to create their own problem-solving methods can create the teacher nervous. It’s possible to find links to all our ACT math topic guides here in order to help your studies. It
suggests mathematics is crucial.
Learning can be an immense ‘take away’! Music is a superb approach to start a math lesson. They are seated in a common area with no math tools.
An essential part of the modeling procedure is the evaluation of whether a given mathematical model describes a system accurately. Furthermore, they may use models to explore different scenarios
cost-effectively. At first glance there’s nothing to model, because there wasn’t any change in production.
Alternatively, you can learn more about the model parameters using the object functions and after that adjust the model as needed. While added complexity usually enhances the realism of a model, it
can produce the model difficult to comprehend and analyze, and may also pose computational issues, including numerical instability. Hence, drawing models allow them to place their thoughts into
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A number of models are utilised to help in comparing fractions and mixed numbers in addition to representing them. Decision variables are occasionally called independent variables. The residual
values have a couple outliers.
Hearsay, Lies and What Is a Math Model
Employing the Internet as the chief supply of information, the WebQuest format can readily be adapted to involve students on site as well as the ones who might be at house by incorporating e-mail or
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What Is a Math Model – Dead or Alive?
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Whereas solving these integrals usually needs a good deal of work and ingenuity, the physicists have demonstrated that the new approach can discover solutions intuitively and at times even without
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The third component is a little more complicated, but only since there are two unique sub-factors, if you will. In fact, it is a blend of all of these problems. A clown was carrying a whole lot of
Line plots are a special kind of number line that represents frequency of information. To figure out the chemical level at the conclusion of the very first calendar year, you’d want to iterate 365
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The Fundamentals of What Is a Math Model Revealed
So, the major point of constructing a heuristic solution method for this problem was to have a quick procedure to work out this issue. The quantum edition of the billiards is readily studied in many
ways. Such dynamical system is called semi-dispersing billiard.
Numbers are real to them and they can relate to quantitative problems better… and clearly, every kid loves colouring! Be certain your student reads the full problem first. The examples within this
slideshow, created with the assistance of math specialist Heidi Cohen, will be able to help you help your child with new math.
The children need to work out the hidden image by connecting the dots in each and every sheet. The significant tree comprises the total space. When the kid accepts the blocks as representations, he
or she is going to be in a superior place to understand additional abstraction.
Instruct students to observe and record the way the math concept you’re currently studying is employed in everyday circumstances and create something that displays their learning. It is crucial to
create a community of learners so that students will have the ability to work independently at centers since you are going to be engaged with different students during this period. The differences in
the types of math curricula come from various ideas about the manner where the subject ought to be taught and understood.
The Teach to One program utilizes technology to come up with an individualized learning plan for each student, daily. Number Talks provide a daily, short, structured way for students to speak about
math by making use of their peers. Students are learning how to make far better decisions without as much guesswork.
For instance, it won’t do plenty of good if you are able to translate a probability word problem in case you don’t understand exactlyhowprobabilities do the job. You almost always must have a good
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of making a comprehensive pass over the problem before selecting a path to the solution.
Anything under 12 point could be too tiny. There are particular words that seem to appear in word problems for various operations that could tip you off to what might be the right operation to apply.
For each right answer, you will have the ability to roll the die and advance on the board game to the finish line.
Due to the world wide web, Liu and Iida don’t necessarily want the green light of an important film house to market their content. Then decide together how you’ll represent your ideas. Employing the
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Things You Won’t Like About What Is a Math Model and Things You Will
There are an assortment of unique forms of routines. Play is among the most effective vehicles for facilitating learning. There are all sorts of reasons behavior may not be logistic.
What Is a Math Model Help!
There are several kinds of graphic organizers. Here is a short tour of the topics covered within this gargantuan equation. Therefore, it’s a remarkably extensive subject.
The Debate Over What Is a Math Model
Several models are utilised to assist in comparing fractions and mixed numbers together with representing them. There are lots of possibilities for comparing numbers within this format. After every
number was multiplied, the overall values are added together.
How to Find What Is a Math Model on the Web
For instance, it won’t do plenty of good if you are able to translate a probability word problem in case you don’t understand exactlyhowprobabilities do the job. You almost always must have a good
comprehension of the math topic in question to be able to address the word problem on the subject. Overcoming this early solution bias can be challenging, and it’s far better to develop the practice
of making a comprehensive pass over the problem before selecting a path to the solution.
A case analysis supplies an impressive prediction connected with possible possible events in an effort to evaluate thing. For each issue, there’s a hint, other associated difficulties, and intriguing
trivia. In this instance, the question becomes why our universe has properties that appear to be so finely tuned to permit for the presence of life.
New Questions About What Is a Math Model
Whether an important actuality isn’t there, you can frequently convert some bit of the given information. She was trying to find an ability to make an individualized plan for each kid, so they’re in
a position to be successful by year’s end, she explained. The last step is to compute the overall quantity of money that both girls had in the start.
The Argument About What Is a Math Model
Know what you’re attempting to find. It is essential to create a community of learners so that students will be able to work independently at centers because you’re likely to be engaged with
distinctive students in this period of time. Students must multiply to ascertain how much each person is owed.
On-line manipulatives give students and classrooms access to an assortment of math tools without needing to invest inside them, ideal for at-home learning and practice. Music is a superb approach to
start a math lesson. Using them you are able to earn money on Forex without thinking.
Instead, merely a visual distinction between 3 and 2 is sufficient to represent the connection between the blocks. In fact, it is a blend of all of these problems. Our 6 character password ought to
have a minumum of a single character from each one of 3 character sets.
Employing the Internet as the chief supply of information, the WebQuest format can readily be adapted to involve students on site as well as the ones who might be at house by incorporating e-mail or
a different Web correspondence component like a wiki or blog. The internet documentation, in terms of supplying help for entering equations, is fantastic. Each topic group should collaborate to talk
about the research.
Why Almost Everything You’ve Learned About What Is a Math Model Is Wrong
Due to the world wide web, Liu and Iida don’t necessarily want the green light of an important film house to market their content. Then decide together how you’ll represent your ideas. To get credit
as the author, put in your information below.
Students may combine LEGO bricks to make a wide range of arrays. Geometric shapes are a great part of our life! A matrix is just a two-dimensional group of numbers.
It’s therefore usually appropriate to produce some approximations to decrease the model to a sensible size. Equations are definitely the most typical kind of mathematical model. They can also be used
to forecast future behavior.
Alternatively, you can learn more about the model parameters using the object functions and after that adjust the model as needed. While added complexity usually enhances the realism of a model, it
can produce the model difficult to comprehend and analyze, and may also pose computational issues, including numerical instability. The model isn’t the exact same as the true thing.
Who Else Wants to Learn About What Is a Math Model?
So as to translate your word problems into actionable math equations that you’re able to solve, you’re want to understand and utilize some essential math stipulations. It’s possible that you come
across topics which vary from counting to multiplication. Now that you’re knowledgeable regarding the fundamental math models, I’ll reveal to you how you’re able to use them in solving several kinds
of math difficulties.
This procedure is the sole method of studying phenomena of the macroworld or microworld that aren’t directly accessible to us. In reality, mathematics encompasses a wide selection of skills and
concepts. In the math model method, there are essentially two concepts that form the foundation for most further iterations.
The very first program I bought was a whole disappointment. Be certain your student reads the full problem first. You don’t need to be a mathematician to ask terrific questions regarding your child’s
curriculum, Fennel adds.
Model numbers make it possible for manufacturers to keep tabs on each hardware device and identify or replace the appropriate part when required. The quantum edition of the billiards is readily
studied in many ways. It is called semi-dispersing billiard.
Such models may not be appropriate to the subject of finance. It’s possible to also utilize addTerms to add certain terms. The reply is investment.
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Interview with Daniela Bubboloni
Daniela Bubboloni, professor at the Department of Mathematics and Computer Science “Ulisse Dini” of the University of Florence, was our guest to the D2 seminar presenting his study entitled: “ Paths
and flows for centrality measures in networks”.
During your presentation you mentioned that your project was born as a conceptual clarification of a certain terminology, so how would you explain to someone who is not an expert of centrality
measures or doesn’t know what paths and flows are what your study is about?
Networks are models for a large variety of phenomena, and their characteristics and properties are at the core of the international research. Since networks are usually very large, it is with no
doubt interesting to isolate single vertices or groups of vertices which play the main role in a given network, the so-called central vertices. That allows to concentrate the attention on the most
important aspects describing the phenomenon under consideration, reducing the complexity given by the big amount of data encoded by the whole network and focusing on a subset of vertices accurately
chosen. The group centrality measures play the important role to detect those special groups of vertices and have been copiously proposed since the 50’s as social science instruments. Nowadays they
have become a tool largely used in physics and biology as well as in sociology, finance and engineering.
During the years many centrality measures have been proposed. However, it seems that flows had not played yet, in the context of centrality, the deep role that they would deserve. The mathematical
concept of flow aims to describe the various ways in which pipelines can be filled pumping something (water, electricity etc.) from a source to a destination called sink, without breaking the
pipeline. The idea is that every part of a pipeline between two junctions is subject to a particular upper bound, called capacity, for the amount of that something circulating in it.
Finding a maximum flow means to find those flows which tolerate the maximum amount of pumping from the source. A very concrete idea which is formalized as an abstract object and then becomes a tool
for studying many unexpected situations both theoretical (graph theory and connectivity problems) and practical (transportation problems, vehicle networks, assignment of one-way streets etc.).
A path is a natural idea which arises looking to the drawing of a network. Looking at the arrows you are invited to follow them one after another one. Starting at a vertex and arriving to another one
you have travelled along a path. Now imagine a junction of capacity c as splitted into c single junctions of capacity 1. If you have a sequence of junctions of capacity at least one from the source
to the sink, you can easily imagine a single pipeline from the source to the sink and see a path from the source to the sink. On the other hand surely you can pump just one unit from the source and
let it flow exactly through the considered sequence of junctions. This helps you guess an interplay between flows and paths. The exact formal explanation of that link was missing in the literature.
In the papers appeared in the literature about flow centrality measures, there is a mixture of terminology, jumping from paths to flows, which was not formally justified. We discovered that some
intuition was right, but some naive extensions of that intuition is wrong and hence in some applications one needs to distinguish the path approach to the flow approach. This holds, for instance, in
dealing with flow centrality for groups of vertices instead of single vertices.
Could you give us a glimpse of possible applications or future outcomes of your research?
In the scientific literature, the centrality measures have been used more to confirm known phenomena than to forecast the phenomena themselves. We would instead like to develop the huge potential
that centrality considerations could have in designing networks and understanding which configurations of its vertices realize the better scenario.
Centrality could be a peculiar ingredient for designing a network and thus very useful for engineering applications. One of our flow group centrality measures takes into consideration how much the
network is damaged in terms of flow when all the connections through a group of vertices get lost. That allows, for instance, to recognize the groups of vertices on which it is important to focus for
the network maintenance in order to avoid the maximum decrease of global flow. Centrality can mean many things: power, prestige, authority, best betweenness position. Applications of centrality
measure vary from detecting the central elements of a terrorist network to discover the genes which are most responsible for the development of cancer. This wide range of applications is somewhat at
the base of the impossibility to uniquely decide which measure is “the best”. On the other hand, we surely need to understand which measure is better in a certain context. We believe that the main
tool for reaching this control on the centrality measures is isolating some relevant properties that a centrality measure could reasonably have, and then use those properties to discriminate one
centrality measure from another one. For that reason a great part of the research of my team is, at the moment, devoted to isolate and study the properties of group centrality measures. Another part
of the research is about the application of our two group centrality measures to concrete networks in order to empirically discover if they behave better than other measures. We are in particular
planning to deal with Trade Networks.
Daniela Bubboloni is also a member of our center. More information about his research can be found on her personal page.
You can access the recording of this seminar through this link. (Registration needed)
Reference: D. Bubboloni, M. Gori, Paths and flows for centrality measures in networks, Networks (2022). You can download the full paper here.
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Symmetric Arithmetic Circuits | Gregory Wilsenach
We introduce symmetric arithmetic circuits, i.e. arithmetic circuits with a natural symmetry restriction. In the context of circuits computing polynomials defined on a matrix of variables, such as
the determinant or the permanent, the restriction amounts to requiring that the shape of the circuit is invariant under row and column permutations of the matrix. We establish unconditional, nearly
exponential, lower bounds on the size of any symmetric circuit for computing the permanent over any field of characteristic other than 2. In contrast, we show that there are polynomial-size symmetric
circuits for computing the determinant over fields of characterisitic zero.
arXiv:2002.06451 [cs]
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Maggie Bags Cicily Tote Review & Giveaway
Disclosure: I received a Maggie Bags Cicily Tote in order to facilitate my review. All opinions are my own.
Maggie Bags Cicily Tote
Recently I had the chance to review a Maggie Bags Cicily Tote which is really great as a busy mom of 4. It was like a special item meant just for me. One that I don’t really have to share with my
kids. Maggie Bags is really a unique company. They make their bags and totes from seatbelt material. It makes for a different, yet still fashionable material that is made in an environmentally
friendly manner. The inside is made with their signature purple satin lining, complete with pockets for a pen, cell phone, pocket pack of tissues – you name it.
Take a look:
Like to WIN a Maggie Bags Cicily Tote for yourself?
Maggie Bags is giving 1 lucky Blogging Mom of 4 Reader their choice of in stock Cicily Totes!!
Giveaway will run through 9/1 at 11:59 pm EST and is for US Residents only. Enter via the Rafflecopter below. Good luck!!
You can see what’s happening with Maggie Bags on their website, Facebook, Twitter, Google Plus, Polyvore, YouTube and Pinterest.
Topic: Maggie Bags Cicily Tote
1. TINA MONTES says
Bullet looks like a bullet hehehe shiny
2. Brooke says
I really like the Clementine color
3. Lori Jackson says
like the brown one!
4. Naomi says
I like Bullet!
5. Adrienne McElwain says
Cicily Tote in Lime by Maggie Bags
6. mry smith says
I love basic black
7. Deeda Leffert says
Bullet is my favorite.
8. stephanie says
I love them all, but Bullet would be my pick!
9. Pam says
My favorite color Cicly bag is Dark Chocolate
10. Danielle Fouts says
I like lime!
11. Keara B. says
I really love the Dark Chocolate- it would go with almost everything in my closet. 🙂
12. Josephine D says
Dark Chocolate is my favorite!
13. Lynell Bumpas says
Those bags look fab! I want one.
14. Cassie says
I like the dark chocolate
15. Lindsey D. says
I like the black
16. Linda Meyers-Gabbard says
I like Clementine color
Ladyblueeyez1960 (at)(aol)(dot)(com)
17. mell says
I like the bullet color.
18. annemarie carter says
The lime green bag has to be my favorite!
19. sherry butcher says
I think I’d like Bullet.
20. shar burdick says
I love the chocolate!
21. Pamela Fisher says
Lime is my favorite
22. Teresa Thompson says
I would choose Cranberry.
23. mell says
I like the bullet color tote.
24. joe says
lime is the one for me.
25. Anita Braddock says
the Black or cranberry thanks for the chance to win this Awesome giveaway
26. Liz Miller says
Can’t decide between cranberry or black : )
27. Rebecca says
My favorite color is Clementine.
28. Ruth says
I love the Cranberry color!
29. tonilynn says
I am livibg this bag especially in clementine
30. Terra W says
I would love the Dark Chocolate!
31. elven johnson says
I like the black one.
32. Sara S. says
Dark Chocolate is my favorite color!
33. Stephanie F. says
I like the bullet color.
34. Jessica Whitehouse says
I like black or bullet-silver
35. KARENALBERTWINSLOW says
WELL BECAUSE I GET EVERYTHING DIRTY & HAUL IT THRU EVERYTHING & THROW IT ON THE FLOOR IN THE CAR I HAVE TO SAY BLACK BUT==== THE WILD CHILD IN ME SAYS LIME THANK YOU
36. danisha emett says
I love the cranberry color! Good choice. That’s the one I would choose, too.
37. Amanda Q says
I can’t decide between red or dark chocolate! They are both beautiful.
38. Brandi Price says
Dark Chocolate is my favorite.
39. Lisa B. says
I love the Lime color!
40. Shelley L. says
Love that Clementine color!
41. peggy fedison says
either bullet or black 🙂 id be happy with any color tho
42. Linda Childers says
The black one
43. joanna reed says
bullet, I really like this color
44. Pam O says
The dark chocolate is my favorite
45. Susan Sharp says
I Love the Chocalate or the Clementine! Great giveaway! Thank you!
46. April C. says
I love the cranberry and the bullet color 🙂
Leave a Reply Cancel reply
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Ebit rate calculation
Earnings before interest and taxes [EBIT] are projected to be $14,000 if economic conditions are considering a $60,000 debt issue with a 5% interest rate. Calculate earnings per share [EPS] under
each of the three economic scenarios. Ltd has EBIT $100,000 for year 2018, non-cash expense is $4,000 and total interest payable for 2018 is $40,000. Now, let's calculate the interest coverage ratio
Learn about residual income and how to calculate it using income from operations, earnings before interest and taxes (EBIT), net operating profit after tax (NOPAT), or net income. The tax rate is
35%. On one side, to determine the residual 5 May 2017 To calculate the cash coverage ratio, take the earnings before interest and taxes ( EBIT) from the income statement, add back to it all
non-cash 3 Jul 2019 by additionally removing depreciation and amortization from the EBIT calculation, all non-cash expenses are deleted from operating income. 23 Oct 2019 amortizare şi provizioanele
pentru depreciere care sunt doar calculate, dar nu şi plătite. EBIT – Earnings before interests and taxes – reprezinta profitul inainte de plata EBIT = Profit din exploatare + Venituri financiare.
Formula To Calculate EBIT? Earnings before interest and taxes is an indicator of a company's profitability. It can be calculated in different ways. You can use Therefore, the calculation of EBIT is
as follows, EBIT = Net income attributable to shareholders/ (1- Tax Rate) = $4.2 million/ (1-0.3) = $ 4.2 million/0.7 = $ 6.0 million; Example #7. We have the following data. Production level of
Company – 10000 units; Contribution per unit = $30 per unit; Operating Leverage = 6; Combined Leverage = 24; Tax Rate = 30%. Calculate EBIT. Solution: EBIT is a company's operating profit without
interest expense and taxes. However, EBITDA or (earnings before interest, taxes, depreciation, and amortization) takes EBIT and strips out depreciation, and amortization expenses when calculating
profitability. Like EBIT, EBITDA also excludes taxes and interest expenses on debt.
24 Oct 2016 Pre-tax profit is a company's operating profit after interest on debt has been paid ( plus any unusual items) -- but before taxes are paid.
EBIT (Mil) (FY) EBIT is computed as Total Revenues for the most recent fiscal It is calculated as the Indicated Annual Dividend divided by the current Price, 2 Mar 2020 Let's explain what an EBITDA
coverage ratio is, why it's important, and EBIT is the same as your operating profit, but you can also calculate it Learn about residual income and how to calculate it using income from operations,
earnings before interest and taxes (EBIT), net operating profit after tax (NOPAT), or net income. The tax rate is 35%. On one side, to determine the residual 5 May 2017 To calculate the cash
coverage ratio, take the earnings before interest and taxes ( EBIT) from the income statement, add back to it all non-cash 3 Jul 2019 by additionally removing depreciation and amortization from the
EBIT calculation, all non-cash expenses are deleted from operating income. 23 Oct 2019 amortizare şi provizioanele pentru depreciere care sunt doar calculate, dar nu şi plătite. EBIT – Earnings
before interests and taxes – reprezinta profitul inainte de plata EBIT = Profit din exploatare + Venituri financiare.
The net operating profit is often called EBIT (Earnings before interest and taxes) whereas the adjusted taxes can be replaced by the effective tax rate in which the EBIT is multiplied by (1 - Tax
Rate(%) / 100) to get the numerator.
The usual shortcut to calculate EBITDA is to start with operating profit, also called earnings before interest and tax (EBIT), and then add back depreciation and The Return on Invested Capital
(ROIC) ratio indicates the profit a firm's Rate). You can also use Earnings Before Interest and Taxes (EBIT) to calculate NOPAT:.
The formula for calculating the EBIT margin is EBIT divided by net revenue. Multiply by 100 to express the margin as a percentage. Be sure to use the net revenues listed near the beginning of the
income statement, not the gross sales or revenue. Suppose the EBIT for the AABC Company was $180,000 for the year, and net revenue was $980,000.
By calculating EBIT, it nulls the effects of the different capital structures and tax rates used by different companies In the example above company's sales during EBIT (Mil) (TTM) EBIT is computed
as Total Revenues for the trailing twelve It is calculated as the Indicated Annual Dividend divided by the current Price, In accounting and finance, earnings before interest and taxes (EBIT), is To
calculate the ratio, subtract any dividend payments due to the holders of preferred 24 Oct 2016 Pre-tax profit is a company's operating profit after interest on debt has been paid ( plus any unusual
items) -- but before taxes are paid. Net interest margin is calculated as net interest income minus net interest expenses profit. the Operating profit is the Earnings before interest and tax ( EBIT).
Earnings before interest and taxes, EBIT, a Calculate the percentage changes in EPS when the economy expands or enters a recession. (Do not round An indicator of a company's profitability,
calculated as revenue minus expenses, Adjusted income after tax, Result of applying tax rate to the adjusted EBIT.
In accounting and finance, earnings before interest and taxes (EBIT) is a measure of a company’s profitability that excludes interest and income tax expenses. It is calculated as the sum of operating
income (also known as “operating profit” and “operating earnings”) and non-operating income, where operating income is operating revenues minus expenses.
Net interest margin is calculated as net interest income minus net interest expenses profit. the Operating profit is the Earnings before interest and tax ( EBIT). Earnings before interest and taxes,
EBIT, a Calculate the percentage changes in EPS when the economy expands or enters a recession. (Do not round
In accounting and finance, earnings before interest and taxes (EBIT) is a measure of a company’s profitability that excludes interest and income tax expenses. It is calculated as the sum of operating
income (also known as “operating profit” and “operating earnings”) and non-operating income, where operating income is operating revenues minus expenses. The formula for calculating the EBIT margin
is EBIT divided by net revenue. Multiply by 100 to express the margin as a percentage. Be sure to use the net revenues listed near the beginning of the income statement, not the gross sales or
revenue. Suppose the EBIT for the AABC Company was $180,000 for the year, and net revenue was $980,000. How to calculate EBIT. To calculate earnings before interest and taxes, start with the gross
profit. Subtract operating costs from the gross profits. When calculating EBIT, do not subtract the cost of business capital and tax liabilities. These items are not included in earnings before
interest and taxes. EBIT formula example
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Multiplication Equal Groups Worksheets
Math, particularly multiplication, creates the cornerstone of countless academic self-controls and real-world applications. Yet, for several students, understanding multiplication can posture a
difficulty. To resolve this obstacle, instructors and parents have actually accepted an effective tool: Multiplication Equal Groups Worksheets.
Intro to Multiplication Equal Groups Worksheets
Multiplication Equal Groups Worksheets
Multiplication Equal Groups Worksheets -
Multiplication equal groups test and printable worksheets Equal groups multiplication is the best and basic way to solve multiplication problems with no stress This is because the equal groups are
usually presented with visually stimulating models which can enable kids to understand multiplication concept in a sweet and easy way
Multiplication Models Worksheets Stick around our printable multiplication models worksheets for practice that helps the budding mathematicians in the 2nd grade 3rd grade and 4th grade get their
heads around multiplying numbers and multiplication sentences An array of topics like multiplication with equal groups arrays rectangular array
Importance of Multiplication Practice Understanding multiplication is crucial, laying a strong structure for advanced mathematical ideas. Multiplication Equal Groups Worksheets use structured and
targeted practice, cultivating a deeper understanding of this fundamental math procedure.
Development of Multiplication Equal Groups Worksheets
Equal Groups Multiplication Worksheets Free Printable
Equal Groups Multiplication Worksheets Free Printable
Liveworksheets transforms your traditional printable worksheets into self correcting interactive exercises that the students can do online and send to the teacher Multiplication equal groups Loading
ad Ms Gierszewski Member for 3 years 1 month Age 7 9 Level 3 Language English en ID 812176
Zip 3 82 MB These ready to use worksheets allow teachers to provide tasks to Year 3 children on the maths topic of multiplication equal groups They form part of a series of lessons on multiplication
and division which include coverage of the objectives Count from 0 in multiples of 4 and 8 Recall and use multiplication and
From conventional pen-and-paper exercises to digitized interactive formats, Multiplication Equal Groups Worksheets have evolved, satisfying diverse understanding styles and choices.
Types of Multiplication Equal Groups Worksheets
Standard Multiplication Sheets Simple workouts focusing on multiplication tables, aiding students develop a solid arithmetic base.
Word Issue Worksheets
Real-life circumstances incorporated into troubles, enhancing crucial thinking and application skills.
Timed Multiplication Drills Examinations created to improve rate and accuracy, aiding in rapid psychological math.
Advantages of Using Multiplication Equal Groups Worksheets
Equal Groups Multiplication Worksheets Times Tables Worksheets
Equal Groups Multiplication Worksheets Times Tables Worksheets
Liveworksheets transforms your traditional printable worksheets into self correcting interactive exercises that the students can do online and send to the teacher Multiplication equal groups
Multiplication equal groups Gierszewski Member for 2 years 9 months Age 7 9
View all 415 Worksheets Multiplication Describing Equal Groups Worksheet Make math practice a joyride by practicing to describe equal groups 2 3 Multiplication Representing Equal Groups Worksheet In
this worksheet learners will get to practice representing equal groups
Boosted Mathematical Abilities
Consistent technique hones multiplication proficiency, enhancing overall math capacities.
Boosted Problem-Solving Abilities
Word troubles in worksheets establish logical reasoning and method application.
Self-Paced Knowing Advantages
Worksheets accommodate individual learning rates, fostering a comfortable and adaptable understanding environment.
Just How to Create Engaging Multiplication Equal Groups Worksheets
Incorporating Visuals and Shades Vibrant visuals and colors record focus, making worksheets aesthetically appealing and engaging.
Consisting Of Real-Life Circumstances
Connecting multiplication to everyday circumstances includes importance and functionality to exercises.
Tailoring Worksheets to Various Skill Levels Customizing worksheets based on differing proficiency degrees guarantees inclusive understanding. Interactive and Online Multiplication Resources Digital
Multiplication Equipment and Gamings Technology-based sources supply interactive learning experiences, making multiplication interesting and satisfying. Interactive Websites and Apps On the internet
systems supply diverse and obtainable multiplication practice, supplementing standard worksheets. Personalizing Worksheets for Numerous Knowing Styles Visual Learners Visual aids and layouts aid
comprehension for students inclined toward visual understanding. Auditory Learners Verbal multiplication issues or mnemonics deal with learners who realize principles with acoustic methods.
Kinesthetic Students Hands-on tasks and manipulatives support kinesthetic students in understanding multiplication. Tips for Effective Application in Understanding Uniformity in Practice Routine
method enhances multiplication skills, promoting retention and fluency. Balancing Repetition and Selection A mix of repeated exercises and varied problem styles keeps rate of interest and
understanding. Giving Positive Feedback Feedback help in recognizing areas of enhancement, motivating ongoing progress. Difficulties in Multiplication Method and Solutions Motivation and Involvement
Obstacles Tedious drills can cause uninterest; cutting-edge techniques can reignite inspiration. Getting Over Anxiety of Mathematics Adverse perceptions around math can hinder development; producing
a favorable discovering atmosphere is vital. Impact of Multiplication Equal Groups Worksheets on Academic Performance Studies and Study Searchings For Research study indicates a positive correlation
in between constant worksheet use and improved mathematics performance.
Multiplication Equal Groups Worksheets emerge as flexible tools, fostering mathematical proficiency in learners while suiting varied learning designs. From fundamental drills to interactive on-line
sources, these worksheets not just boost multiplication skills yet additionally advertise vital reasoning and analytic capacities.
Adding Equal Groups 2 EStudyNotes
Eyfs Ks1 Year 1 Sen Numeracy Teaching Resources Reasoning And Mathspower Sample Year 1
Check more of Multiplication Equal Groups Worksheets below
Add In equal groups Worksheet 2 EStudyNotes
Multiplication Equal Groups Worksheets Free Printable Worksheet
Equal Groups Multiplication Sentence Worksheet Template Printable Pdf Download
The Worksheet For Multiplition equal groups Is Shown In Purple And Blue
Intro To Multiplication Adding Groups Worksheets 99Worksheets
Recognising equal groups 1 Multiply And Divide In Year 1 age 5 6 By URBrainy
Multiplication Models Worksheets Math Worksheets 4 Kids
Multiplication Models Worksheets Stick around our printable multiplication models worksheets for practice that helps the budding mathematicians in the 2nd grade 3rd grade and 4th grade get their
heads around multiplying numbers and multiplication sentences An array of topics like multiplication with equal groups arrays rectangular array
Multiplication Models and Equal Groups Worksheets Tutoring Hour
This set uses multiplication models for conceptual clarity The numbers involved are illustrated as equal groups of items so kids learn to multiply quicker and apply multiplication in real life as
well These pdf worksheets on multiplication models and equal groups are best suited for children in grade 2 grade 3 grade 4 and grade 5
Multiplication Models Worksheets Stick around our printable multiplication models worksheets for practice that helps the budding mathematicians in the 2nd grade 3rd grade and 4th grade get their
heads around multiplying numbers and multiplication sentences An array of topics like multiplication with equal groups arrays rectangular array
This set uses multiplication models for conceptual clarity The numbers involved are illustrated as equal groups of items so kids learn to multiply quicker and apply multiplication in real life as
well These pdf worksheets on multiplication models and equal groups are best suited for children in grade 2 grade 3 grade 4 and grade 5
The Worksheet For Multiplition equal groups Is Shown In Purple And Blue
Multiplication Equal Groups Worksheets Free Printable Worksheet
Intro To Multiplication Adding Groups Worksheets 99Worksheets
Recognising equal groups 1 Multiply And Divide In Year 1 age 5 6 By URBrainy
Multiplication With Equal Groups Worksheets
Equal Groups Multiplication Worksheets Pdf Free Printable
Equal Groups Multiplication Worksheets Pdf Free Printable
Equal Groups Multiplication Worksheets Free Printable
FAQs (Frequently Asked Questions).
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Yes, worksheets can be customized to various age and ability degrees, making them adaptable for various learners.
Just how often should trainees exercise utilizing Multiplication Equal Groups Worksheets?
Constant technique is essential. Normal sessions, preferably a few times a week, can produce substantial enhancement.
Can worksheets alone boost math abilities?
Worksheets are an useful device however must be supplemented with varied knowing techniques for detailed ability growth.
Exist on the internet systems supplying complimentary Multiplication Equal Groups Worksheets?
Yes, several academic websites supply free access to a large range of Multiplication Equal Groups Worksheets.
Just how can parents support their youngsters's multiplication method at home?
Urging constant method, providing help, and producing a positive knowing setting are useful actions.
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Fourteen Papers on Series and Approximationsearch
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Fourteen Papers on Series and Approximation
Hardcover ISBN: 978-0-8218-1777-3
Product Code: TRANS2/77
List Price: $275.00
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Fourteen Papers on Series and Approximation
Hardcover ISBN: 978-0-8218-1777-3
Product Code: TRANS2/77
List Price: $275.00
MAA Member Price: $247.50
AMS Member Price: $220.00
eBook ISBN: 978-1-4704-3288-1
Product Code: TRANS2/77.E
List Price: $265.00
MAA Member Price: $238.50
AMS Member Price: $212.00
Hardcover ISBN: 978-0-8218-1777-3
eBook ISBN: 978-1-4704-3288-1
Product Code: TRANS2/77.B
List Price: $540.00 $407.50
MAA Member Price: $486.00 $366.75
AMS Member Price: $432.00 $326.00
• American Mathematical Society Translations - Series 2
Volume: 77; 1968; 266 pp
MSC: Primary 40
□ Articles
□ L. A. Balašov — Series with gaps
□ R. I. Osipov — On the representation of functions by orthogonal series
□ R. and Tomić Bojanić, M. — On the absolute convergence of Fourier series with small gaps
□ P. I. Lizorkin — Estimates for trigonometric integrals and the Bernšteĭn inequality for fractional derivatives
□ I. M. Vinogradov — Estimation of trigonometric sums
□ Ju. K. Suetin — Convergence and uniqueness constants for certain interpolation problems
□ V. I. Berdyšev — Mean approximation of periodic functions by Fourier series
□ M. F. Timan — The best approximation of a function and linear methods for the summation of Fourier series
□ M. A. Jastrebova — On the approximation of functions satisfying a Lipschitz condition by the arithmetic means of their Walsh-Fourier series
□ S. A. Teljakovskiĭ — Two theorems on the approximation of functions by algebraic polynomials
□ I. I. Cyganok — A generalization of Jackson’s theorem
□ G. C. Tumarkin — Approximation with respect to various metrics of functions defined on the unit circle by sequences of rational fractions with fixed poles
□ G. C. Tumarkin — Necessary and sufficient conditions for the possibility of approximating a function on a circumference by rational fractions, expressed in terms directly connected with the
distribution of poles of the approximating fractions
□ A. V. Efimov — On best approximations of classes of periodic functions by means of trigonometric polynomials
• Permission – for use of book, eBook, or Journal content
• Book Details
• Table of Contents
• Requests
Volume: 77; 1968; 266 pp
MSC: Primary 40
• Articles
• L. A. Balašov — Series with gaps
• R. I. Osipov — On the representation of functions by orthogonal series
• R. and Tomić Bojanić, M. — On the absolute convergence of Fourier series with small gaps
• P. I. Lizorkin — Estimates for trigonometric integrals and the Bernšteĭn inequality for fractional derivatives
• I. M. Vinogradov — Estimation of trigonometric sums
• Ju. K. Suetin — Convergence and uniqueness constants for certain interpolation problems
• V. I. Berdyšev — Mean approximation of periodic functions by Fourier series
• M. F. Timan — The best approximation of a function and linear methods for the summation of Fourier series
• M. A. Jastrebova — On the approximation of functions satisfying a Lipschitz condition by the arithmetic means of their Walsh-Fourier series
• S. A. Teljakovskiĭ — Two theorems on the approximation of functions by algebraic polynomials
• I. I. Cyganok — A generalization of Jackson’s theorem
• G. C. Tumarkin — Approximation with respect to various metrics of functions defined on the unit circle by sequences of rational fractions with fixed poles
• G. C. Tumarkin — Necessary and sufficient conditions for the possibility of approximating a function on a circumference by rational fractions, expressed in terms directly connected with the
distribution of poles of the approximating fractions
• A. V. Efimov — On best approximations of classes of periodic functions by means of trigonometric polynomials
Permission – for use of book, eBook, or Journal content
Please select which format for which you are requesting permissions.
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RSICC CODE PACKAGE CCC-745
1. NAME AND TITLE
ERANOS 2.0: Modular Code and Data System for Fast Reactor Neutronics Analyses.
RESTRICTIONS: Users from NEADB member countries http://www.nea.fr/html/nea/mcdb.html are advised to order ERANOS from the NEA Data Bank. Users from other OECD member countries http://www.nea.fr/html/
nea/flyeren.html (specifically Canada and the United States) may order these codes from RSICC. Users from non-OECD member countries are advised to contact the NEA Data Bank; the NEADB will transmit
the requests to CEA, who will deal directly with these requests. For non-comercial use only
2. CONTRIBUTORS
DER/SPRC/LEPh, CEA-Cadarache, France, through the OECD Nuclear Energy Agency Data Bank, Issy-les-Moulineaux, France.
3. CODING LANGUAGE AND COMPUTER
C and FORTRAN-77; Linux-based PC (C00745MNYWS00).
NEA Package ID: NEA-1683/01.
4. NATURE OF PROBLEM SOLVED
The European Reactor ANalysis Optimized calculation System, ERANOS, has been developed and validated with the aim of providing a suitable basis for reliable neutronic calculations of current as well
as advanced fast reactor cores. It consists of data libraries, deterministic codes and calculation procedures which have been developed within the European Collaboration on Fast Reactors over the
past 20 years or so, in order to answer the needs of both industrial and R&D organisations. The whole system counts roughly 250 functions and 3000 subroutines totalling 450000 lines of FORTRAN-77 and
ESOPE instructions.
ERANOS is written using the ALOS software which requires only standard FORTRAN compilers and includes advanced programming features. A modular structure was adopted for easier evolution and
incorporation of new functionalities. Blocks of data (SETs) can be created or used by the modules themselves or by the user via the LU control language. Programming, and dynamic memory allocation,
are performed by means of the ESOPE language. External temporary storage and permanent storage capabilities are provided by the GEMAT and ARCHIVE functions, respectively. ESOPE, LU, GEMAT and ARCHIVE
are all part of the ALOS software. This modular structure allows different modules to be linked together in procedures corresponding to recommended calculation routes ranging from fast-running and
moderately-accurate 'routine' procedures to slow-running but highly-accurate 'reference' procedures.
The main contents of the ERANOS-2.0 package are: nuclear data libraries (multigroup cross-sections from the JEF-2.2 evaluated nuclear data file, and other specific data files), a cell and lattice
code (ECCO), reactor flux solvers (diffusion, Sn transport, nodal variational transport), a burn-up module, various processing modules (material and neutron balance, breeding gains,...), tools
related to perturbation theory and sensitivity analysis, core follow-up modules (connected in the PROJERIX procedures), a fine burn-up analysis subset named MECCYCO (mass balances, activities, decay
heat, dose rates). Coupled neutron/gamma calculations are also possible using specific libraries.
Nuclear data libraries:
The ECCO/ERANOS 2.0 code package contains four neutron cross section libraries derived from the JEF-2.2 nuclear data evaluated files. They are:
- 1968-group library (41 main nuclides)
- 33-group library (246 nuclides, including pseudo fission products)
- 175-group library (VITAMIN-J energy group scheme)
- 172-group library (XMAS energy group scheme, 246 nuclides, including pseudo-FP).
These libraries were obtained by processing the JEF-2.2 files with the NJOY and CALENDF codes. Probability tables are included for the main 37 resonant nuclides. The 172-group library (XMAS energy
scheme) may be used for thermal spectrum calculations. The 175-group library (some cross-sections in P5, but no probability tables) is used for shielding calculations only.
Other nuclear data (fission yields and energies, decay constants, gamma production and interaction libraries, etc.) are provided in separate files.
Cell/lattice calculations:
The ECCO cell/lattice code in the ERANOS-2.0 package uses the subgroup method to treat resonance self-shielding effects. This method is particularly suitable for calculations involving complex
heterogeneous structures. ECCO prepares self-shielded cross sections and matrices by combining a slowing-down treatment in many groups (1968 groups) with the subgroup method within each fine group.
The subgroup method takes into account the resonance structure of cross-sections by means of probability tables and by assuming that the neutron source is uniform in lethargy within a given fine
group. Flux calculations in heterogeneous geometry are performed by means of the collision probability method.
In the reference calculation scheme, ECCO treats the heterogeneous geometry in fine groups (1968) for the most important nuclides while broad group libraries (33 or 172 groups) are used for the less
important nuclides. These calculations are very accurate as the fine group plus sub-group scheme have been set up to represent accurately the reaction thresholds and the resonances in any situation,
narrow or wide. One usually distinguishes wide and narrow resonances depending on their width compared to the neutron energy loss by scattering, which is smallest for scattering by heavy nuclides.
Translated into lethargy gain, the value for U238 is almost constant and is equal to 0.008. This compares well with the fine group width of 1/120 = 0.0083 and explains the fact that 3/4 of the
neutrons having a collision in a given fine group escape from that group. Wide resonances are treated explicitly, the resonances in that case having a width larger than the fine group width. On the
other hand, narrow resonances are represented by probability tables, and hence use of the subgroup method can be applied in a very accurate way.
Self-shielded cross sections and matrices are condensed and smeared to provide effective cross sections and matrices in the user required broad group scheme. The neutron balance is preserved in ECCO
after condensation and smearing. The effective cross-sections and matrices produced by ECCO are subsequently used in full-core ERANOS calculations.
Many types of geometries are available within the ECCO code: 1D (plane or cylindrical: exact collision probabilities), 2D (rectangular lattice of cylindrical and/or square pins within a square tube,
hexagonal lattice of cylindrical pins within an hexagonal wrapper: approximate collision probabilities by Roth and double step methods), 3-D (slab with the sides of the boxes and the tube described
explicitly: approximate collision probabilities).
The user can chain several calculation steps so as to produce design (less accurate, faster) or reference (more accurate, slower) calculations, or even to use specific capabilities, according to the
needs of a given study.
Flux solvers:
Three main classes of flux solvers are available. In each case, external sources, up-scattering and adjoint calculations can be addressed. Anisotropic scattering is available for transport
Finite difference diffusion solvers can be used in any geometry: 1D (plane, cylindrical, spherical), 2D (RZ, R-theta, rectangular lattice XY, hexagonal lattice), and 3D (rectangular lattice XYZ,
hexagonal-Z). An efficient solution of the diffusion equation is obtained by using either the successive line over-relaxation method (SLOR), the alternating direction implicit method (ADI) or the
strongly implicit method (SIM).
Finite difference Sn transport calculations are performed by the BISTRO code, using a highly efficient convergence algorithm. It can be used in 1D geometry (plane, cylindrical, spherical) and some 2D
geometries (RZ, XY). Different algorithms (step, diamond and "theta-weighted") and a negative flux fix-up capability exist. The inner iterations are accelerated by the DSA method using the source
correction scheme.
In this package distributed by OECD and RSICC, the version of the variational nodal method developed for the VARIANT code has been used in ERANOS-2.0 as the TGV/VARIANT module. This method is based
on the second-order form of the even-parity transport equation. A solution is searched in form of expansions for the even and odd parity fluxes in pre-computed angular and spatial basis functions
with unknown coefficients. These basis functions are orthogonal polynomials for the spatial variables and spherical harmonics for the angular variables. Scattering anisotropy can be taken in to
account as Pn moments up to the order N of the Legendre expansion of the flux. Both Cartesian (XY or XYZ) and hexagonal (Hex or Hex-Z) geometries are available with TGV/VARIANT. A 'simplified
transport' option exists, in which the angular developments both within the nodes and at the node boundaries are truncated by neglecting high-order cross terms. This option is rather accurate in
practice (large reactors), and less time and memory consuming.
Burn-up calculations:
Calculation of isotopic concentration evolution is possible in the ERANOS system for actinides as well as fission and activation products. The Bateman equations governing the time dependence of
concentrations are solved with various techniques related to the type of nuclide (actinide, fission product or activation product). Burn-up can be performed at the full core scale, with suitable
'burnable zones' subdividing the fuel and fertile regions, or in elementary cells/lattices.
Result-processing modules.
Besides the modules related to basic data preparation (creation of medium, geometry, and burn-up chain SETs, modelling of operating conditions, etc.), a variety of modules computes and/or extracts
specific information from the code output (fluxes, concentrations, etc.). Here is a non-exhaustive sample of such modules:
- Traverse extraction and processing
- Mass and atom balances by region
- Neutron balance by region, reaction and energy group
- Integrated reaction rate processing
- Equivalence coefficients and Breeding gain
- Beta effective
- Linear and bilinear integrals (with respect to the forward and possibly adjoint fluxes)
Perturbation theory and sensitivity analysis.
The reactor physicist is often interested in the breakdown of the variation (or of the first order derivative) of integral parameters such as the multiplication factor, reaction rates and more
generally ratios of bilinear integrals, nuclide concentrations, reactivity coefficients, etc., with respect to input data such as multigroup cross-sections, decay constants, or initial
concentrations. This can be readily obtained through the use of adjoint (standard or generalized) flux calculations and the computation of suitable bilinear integrals. Several modules of ERANOS are
available for a modular processing of such problems: calculation of perturbation integrals, of cross-section variations, sensitivity analysis, perturbation analysis.
As a matter of fact, sensitivity analyses, and first-order or exact perturbation analyses can be performed for the multiplication factor (standard perturbation theory, SPT), ratios of linear or
bilinear integrals (generalized perturbation theory, GPT), and reactivity effects (equivalent generalized perturbation theory, EGPT). If a dispersion (variance/covariance) matrix is provided, a
specific module can be used to perform uncertainty and representativeness calculations.
Core follow-up:
Specific ERANOS modules and appropriate complex subroutines written in the LU user's language (the PROJERIX package) are available to perform a detailed core follow-up. Each individual sub-assembly
can be followed through its entire life (moves during shuffles and batch reloadings, time spent in internal storage, etc.).
Fine burn-up:
For sub-assemblies burnt in significant flux gradients (e.g. fertile sub-assemblies) a detailed burn-up capability is available through specific ERANOS modules.
Other topics:
Several other features are available:
- Coupled neutron/gamma Sn transport calculations (with specific libraries)
- Detailed treatment of damage and kerma (with specific libraries)
- Detailed burn-up with computation of decay (alpha, beta, gamma and neutron particles) activities, energies, energy spectra of emitted particles, dose rates (for simple geometries), decay heat (the
MECCYCO package, with specific libraries).
5. METHOD OF SOLUTION
Methods used in ERANOS modules have been mentioned briefly above. The user can feed and connect these modules in a variety of ways to produce specific analytic sequences. Conditional chaining (IF,
FOR, WHILE instructions) is possible with the user's language. This allows a great deal of flexibility in the use of the code system.
None noted.
7. TYPICAL RUNNING TIME
Not noted.
ERANOS 2.0 sources and installation procedures are provided for SUN, IBM_RISC, and for PC under linux architectures.
To install the whole ERANOS package, 1600 Mb are required for installing the cross-section libraries JECCOLIB2, 700 Mb for installing the code, and 60 Mb for the code documentation (html and PDF
At least 128 Megabytes of Random Access Memory (RAM) are needed to compile the code and run the test cases.
This program is distributed by the NEA Data Bank and by RSICC as received from the authors. At the CEA, the installation was tested with the following characteristics on a linux system and may fail
on other systems:
- REDHAT 7.0
- Kernel 2.2.19
- Gcc 2.96
- Lib_c 2.2-5
The programming language is ESOPE, an extension of FORTRAN 77 specific to CEA, and treated by a built-in pre-compiler. The main objective of this extension is to make the management of the data used
by the various subroutines easier. The data structuration is made by using new entities called SEGMENTs. A segment is a collection of simple variables and/or arrays, addressed by a POINTER. Segments
can be connected with each other by pointers in such a way as to produce tree-like or graph-like structures. The basic data structures exchanged by the ERANOS modules are SETs (for Structured ERANOS
Tree), which are arborescent structures made of connected segments, and related to basic logical entities (e.g. geometry, concentrations, fluxes, etc.). All these structures are manipulated by a
memory manager called GEMAT (creation, destruction, updating, swaps between RAM and disk, etc.).
ERANOS modules can be chained by means of the LU user's language. LU capabilities include the manipulation of variables/arrays of different types, the use of logical, arithmetical and character
operators/functions, and of a variety of special functions. Conditional structures of various types can be used (e.g. IF, FOR; WHILE), and LU subroutines, called LU procedures, can be written, stored
and used. A specific data manager, called ARCHIVE, is used for data structures such as SETs and procedures. A database manager is available, connected to the LU language, producing and managing
structures (various operators available). The execution of LU scripts is made by a built-in interpreter.
10. REFERENCES
Documentation is distributed on CD #2. Click on files HTML_ERANOS_2.0/index.html and install.html for the documentation index and links to the various references. Click on TUTORIAL/turorialERANOS.htm
for a brief overview and tutorial. Note that the code documentation is primarily written in French with some pages in English. The libraries documentation is provided in English. The following
references are included in package distribution:
- G. Rimpault:
Physics Documentation of ERANOS - The ECCO Cell Code
ERANOS : Manuel des Methodes - Le Code de Cellule ECCO
(Rapport Technique RT/SPRC/LEPh 97-001)
- G. Rimpault:
Approximate Buckling Dependent Diffusion Coefficients for the ECCO Cell Code
Coefficients de Diffusion Approches Dependant du Laplacien pour le Code de
Cellule ECCO (Note Technique NT/SPRC/LEPh 99/212)
- G. Rimpault, P. Smith:
Developpements Algorithmiques dans le Code ECCO pour le Traitement des Effets
de Fuites Anisotropes dans les Situations Vidangees de Sodium (Note Technique
NT/SPRC/LEPh 97-229)
- Generalisation du Calcul des Probabilites de Collision dans ECCO (Note
Technique NT/SPRC/LEPh/00/214)
- Integration d'un Module de Calcul de Probabilites de Collision en R-Z dans
ECCO, (Note Technique NT/SPRC/LEPh/00/215)
- C. Gho, G. Palmiotti:
BISTRO : Bidimensionnel Sn Transport Optimise - Un Programme Bidimensionnel de
Transport Sn aux Differences Finies, Note No.1 Definition Des Algorithmes Pour
La Geometrie X-Y (Note Technique NT/SPRC/LEPh 84/270)
- C. Gho, G. Palmiotti:
Algorithmes pour la Geometrie R-Z et Optimisation de la Solution 'Diffusion'
pour L'Acceleration
Module de Passage Maille-Point
Solution de L'Equation de la Diffusion
BISTRO - Note No. 2 (Note Technique NT/SPRC/LEPh 85-202)
- C. Gho, G. Palmiotti, J-M. Rieunier:
Comparaison des Resultats et Temps de Calcul entre BISTRO er DOT
BISTRO - Note No. 3 (Note Technique NT/SPRC/LEPh 85-204)
- C. Gho, G. Palmiotti:
Definition des Algorithmes Necessaires au Calcul de Configurations a Spectre
Thermique (Traitement du Groupe Thermique, Upscattering) et du Transport des
Rayons gamma (Formalisme SN Pn) BISTRO - Note No. 4 (Note Technique
NT/SPRC/LEPh 86/238)
- J.M. Ruggieri:
ERANOS - Manuel des Methodes - Reconstruction Fine d'un Flux Nodal (Note
Technique NT/SPRC/LEPh 99-217)
- A. Rineiski:
KIN3D : Module de cinetique spatiale et de perturbations pour TGV2.
A space-time kinetics and perturbation theory module for TGV2.
(Note Technique NT/SPRC/LEPh 97-203)
- G. Palmiotti, C.B. Carrico, E.E. Lewis:
Variational Nodal Method for the Solution of the Diffusion and Transport
Equation in Two and Three Dimensional Geometries (Note Technique NT/SPRC/LEPh
- Anton Luethi:
Les Fichiers de Degagement d'Energie d'ERANOS (Note Technique NT/SPRC/LEPh
- G. Rimpault, D. Calamand, P. Peerani:
Physics Documentation of ERANOS
Energy Release and Displacement Damage Dose Calculations
Documentation Physique d'ERANOS
Calculs de Dommage aux structures et de Degagement d'Energie
(Note Technique NT/SPRC/LEPh 93-236)
- S. Czernecki, J.M. Rieunier:
ERANOS : Manuel des Methodes - Les Conditions de Fonctionnement (Note Technique
NT/SPRC/LEPh 99/213)
- D. Honde, J.M. Rieunier, G. Rimpault:
Procedures de Calcul d'Evolution Cellule (ECCO) dans ERANOS (Note Technique
NT/SPRC/LEPh 98-226)
- J. Y. Doriath, J.M. Rieunier, G. Rimpault:
ERANOS - Manuel des Methodes - Les Calculs d'Evolution (Note Technique
NT/SPRC/LEPh 96-204)
- D. Niddam:
Integration de MECCYCO dans ERANOS (projet MCOERA) (Note Technique NT/SPRC/LEPh 99-214)
SCHEMAS (SCHEMES):
- S. Czernecki, F. Varaine:
ERANOS 1.2 : Notice d'utilisation des procedures PROJERIX (Note Technique
NT/SPRC/LEPh 97-437)
- G. Rimpault, P. Smith, R. Jacqmin, F. Malvagi, J.M. Rieunier, D. Honde, G.
Buzzi, P.J. Finck:
Schema de Calcul de Reference du Formulaire ERANOS et Orientations pour le
Schema de Calcul de Projet (Note Technique NT/SPRC/LEPh 96-220)
- S. Czernecki, F. Varaine:
ERANOS 1.2 : Note de presentation du nouveau schema de calcul de projet
'neutronique coeur' (Note Technique NT/SPRC/LEPh 97-438)
- F. Mellier:
ERANOS 1.2 - Procedures SIRENE pour le post-traitement des etudes projet Notice
de presentation et d'utilisation (Note Technique NT/SPRC/LEPh 97-436)
- F. Varaine, S. Czernecki:
ERANOS 1.2 : Notice d'Utilisation du Schema de Calcul de Projet 'Neutronique
Coeur' (Note Technique NT/SPRC/LEPh 97-440)
- D. Honde, P. Palmiotti, J.M. Rieunier, G. Rimpault:
ERANOS : Manuel des Methodes - Les Calculs de Perturbations et les Analyses de
Sensibilite (Note Technique NT/SPRC/LEPh 96-205)
- S. Czernecki, D. Nidda:
Extension des algorithmes de sensibilite d'ERANOS (Projet EAS) (Note Technique
NT/SPRC/LEPh 99-226)
- G. Rimpault, D. Honde, J-M. Rieunier:
ERANOS : Manuel des Methodes, Transferts Internes de Donnees Nucleaires (Note
Technique NT/SPRC/LEPh 93-252)
- D. Plisson-Rieunier:
Descriptif Livraison ERANOS 2.0 (Note Technique NT/SPRC/LEPh/01/215)
- ERANOS 2.0 Installation Manual (Note Technique NT/SPRC/LEPh/01/217)
- E. Fort, W. Assal, G. Rimpault, R. Soule, P. Smith, J. Rowlands:
Principes Theoriques et Methodologies de la Validation de JEF2.2, Application a
La Realisation d'ERALIB1, Bibliotheque de Donnees Neutroniques pour le Calcul
des Systemes a Spectre Rapide (Rapport Technique RT/SPRC/LEPh 97-002)
- P. Smith, G. Rimpault:
Qualification du Formulaire ERANOS pour le Calcul de la Perte de Reactivite de
SUPER-PHENIX (Note Technique NT/SPRC/LEPh 98-239)
11. CONTENTS OF CODE PACKAGE
ERANOS is distributed on 3 CDs which include source code, binary data libraries, Makefiles, scripts, test cases and documentation (manuals and technical documents in HTML and PDF formats).
12. DATE OF ABSTRACT
June 2008.
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98 Millimeter/Hour Squared to Milligals
Millimeter/Hour Squared [mm/h2] Output
98 millimeter/hour squared in meter/second squared is equal to 7.5617283950617e-9
98 millimeter/hour squared in attometer/second squared is equal to 7561728395.06
98 millimeter/hour squared in centimeter/second squared is equal to 7.5617283950617e-7
98 millimeter/hour squared in decimeter/second squared is equal to 7.5617283950617e-8
98 millimeter/hour squared in dekameter/second squared is equal to 7.5617283950617e-10
98 millimeter/hour squared in femtometer/second squared is equal to 7561728.4
98 millimeter/hour squared in hectometer/second squared is equal to 7.5617283950617e-11
98 millimeter/hour squared in kilometer/second squared is equal to 7.5617283950617e-12
98 millimeter/hour squared in micrometer/second squared is equal to 0.0075617283950617
98 millimeter/hour squared in millimeter/second squared is equal to 0.0000075617283950617
98 millimeter/hour squared in nanometer/second squared is equal to 7.56
98 millimeter/hour squared in picometer/second squared is equal to 7561.73
98 millimeter/hour squared in meter/hour squared is equal to 0.098
98 millimeter/hour squared in centimeter/hour squared is equal to 9.8
98 millimeter/hour squared in kilometer/hour squared is equal to 0.000098
98 millimeter/hour squared in meter/minute squared is equal to 0.000027222222222222
98 millimeter/hour squared in millimeter/minute squared is equal to 0.027222222222222
98 millimeter/hour squared in centimeter/minute squared is equal to 0.0027222222222222
98 millimeter/hour squared in kilometer/minute squared is equal to 2.7222222222222e-8
98 millimeter/hour squared in kilometer/hour/second is equal to 2.7222222222222e-8
98 millimeter/hour squared in inch/hour/minute is equal to 0.064304461942257
98 millimeter/hour squared in inch/hour/second is equal to 0.001071741032371
98 millimeter/hour squared in inch/minute/second is equal to 0.000017862350539516
98 millimeter/hour squared in inch/hour squared is equal to 3.86
98 millimeter/hour squared in inch/minute squared is equal to 0.001071741032371
98 millimeter/hour squared in inch/second squared is equal to 2.9770584232526e-7
98 millimeter/hour squared in feet/hour/minute is equal to 0.0053587051618548
98 millimeter/hour squared in feet/hour/second is equal to 0.000089311752697579
98 millimeter/hour squared in feet/minute/second is equal to 0.0000014885292116263
98 millimeter/hour squared in feet/hour squared is equal to 0.32152230971129
98 millimeter/hour squared in feet/minute squared is equal to 0.000089311752697579
98 millimeter/hour squared in feet/second squared is equal to 2.4808820193772e-8
98 millimeter/hour squared in knot/hour is equal to 0.000052915766944444
98 millimeter/hour squared in knot/minute is equal to 8.8192944907407e-7
98 millimeter/hour squared in knot/second is equal to 1.4698824151235e-8
98 millimeter/hour squared in knot/millisecond is equal to 1.4698824151235e-11
98 millimeter/hour squared in mile/hour/minute is equal to 0.0000010149062806543
98 millimeter/hour squared in mile/hour/second is equal to 1.6915104677572e-8
98 millimeter/hour squared in mile/hour squared is equal to 0.000060894376839259
98 millimeter/hour squared in mile/minute squared is equal to 1.6915104677572e-8
98 millimeter/hour squared in mile/second squared is equal to 4.6986401882144e-12
98 millimeter/hour squared in yard/second squared is equal to 8.2696067312574e-9
98 millimeter/hour squared in gal is equal to 7.5617283950617e-7
98 millimeter/hour squared in galileo is equal to 7.5617283950617e-7
98 millimeter/hour squared in centigal is equal to 0.000075617283950617
98 millimeter/hour squared in decigal is equal to 0.0000075617283950617
98 millimeter/hour squared in g-unit is equal to 7.71081704258e-10
98 millimeter/hour squared in gn is equal to 7.71081704258e-10
98 millimeter/hour squared in gravity is equal to 7.71081704258e-10
98 millimeter/hour squared in milligal is equal to 0.00075617283950617
98 millimeter/hour squared in kilogal is equal to 7.5617283950617e-10
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Let P equals (p//1,p//2,. . . , p//n) and Q equals (q//1,q//2,. . . ,q//m) be two non-intersecting polygons in the plane specified by their cartesian coordinates in order. Given a direction d we can
ask whether P can be translated an arbitrary distance in direction d without colliding with Q. An algorithm is presented for answering the above translation query in O(n plus m) time. It is also
shown that all the directions of movability (translation) of P with respect to Q can be computed in O(nm) time. For the more general case of a set of M non-intersecting n-gons P equals (P//1,P//2,. .
. ,P//M) we say that it exhibits the translation ordering property if for all fixed directions there exists an ordering for translating the polygons by a single common vector without any collisions
occurring with those polygons not yet moved. It is shown that for a given collection P, the translation ordering property query can be answered in O(Mn plus M**2log n) time.
Original language English (US)
Pages 158-163
Number of pages 6
State Published - 1983
ASJC Scopus subject areas
Dive into the research topics of 'SOME NEW RESULTS ON MOVING POLYGONS IN THE PLANE.'. Together they form a unique fingerprint.
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Research Line: Possiblistic Defeasible Logic Programming
This research line has resulted in the development of a logic programming language, Possibilistic Defeasible Logic Programming (P-DeLP), an extension of DeLP that allows the treatment of
possibilistic uncertainty and fuzzy knowledge at object-language level. This research line has been developed in collaboration with the University of Lleida (Spain) and Artificial Intelligence
Research Institute (Spain). In P-DeLP, knowledge representation features are formalized based on PGL, a possibilistic logic based on the Horn-rule fragment of Gödel fuzzy logic. In PGL formulas are
built over fuzzy propositional variables and the certainty degree of formulas is expressed with a necessity measure. In a logic programming setting, the proof method for PGL is based on a complete
calculus for determining the maximum degree of possibilistic entailment of a fuzzy goal. In a multiagent context, we have studied how agents can use P-DeLP to encode their knowledge about the world,
using the argument and warrant computing procedure to perform their inferences. In particular, we have also formalized and studied a number of argument-based consequence operators which allow to
model different aspects of the reasoning abilities in an intelligent agent. We have also analyzed how answers to P-DeLP queries can be speeded up by pruning the associated search space.
Argument-based Expansion Operators in Possibilistic Defeasible Logic Programming: Characterization and Logical Properties. C. Chesñevar, G. Simari, L. Godo, and T. Alsinet, Proc. of ECSQARU 2005
Conference. Barcelona, Spain, pp. 353-365.
Computing Dialectical Trees Efficiently in Possibilistic Defeasible Logic Programming. C. Chesñevar and G. Simari and L. Godo, LNAI Springer Series Vol. 3662 (8th Intl. Conf. on Logic Programming
and Nonmonotonic Reasoning LPNMR 2005, Eds. pp. 158-171.
A Logic Programming Framework for Possibilistic Argumentation with Vague Knowledge. C. Chesñevar and G. Simari and T. Alsinet and L. Godo, Proc. Intl. Conf. in Uncertainty in Artificial
Intelligence (UAI 2004). Banff, Canada, pp. 76-84.
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Factors - R Language Frequently Asked Questions
Factors in R Language are used to represent categorical data in the R language. Factors in R can be ordered or unordered. One can think of a factor as an integer vector where each integer has a
label. Factors are specially treated by modeling functions such as lm() and glm(). Factors are the data objects used for categorical data and stored as levels. They can store both string and integer
Using factors with labels is better than using integers as factors are self-describing; having a variable that has values “Male” and “Female” is better than a variable having values 1 and 2.
Creating a Simple Factor in R
The following example creates a simple factor variable that has two levels.
# Simple factor with two levels
x <- factor(c("yes", "yes", "no", "yes", "no"))
# computes frequency of factors
# strips out the class
The order of the levels can be set using the levels argument to factor(). This can be important in linear modeling because the first level is used as the baseline level.
x <- factor(c("yes","yes","no","yes","no"), levels = c("yes","no"))
Naming Factors in R
Factors can be given names using the label argument. The label argument changes the old values of the variable to a new one. For example,
x <- factor(c("yes", "yes", "no", "yes", "no"), levels = c("yes", "no"), label = c(1,2) )
x <- factor(c("yes","yes","no","yes","no"), levels = c("yes","no"), label = c("Level-1", "level-2"))
x <- factor(c("yes","yes","no","yes","no"), levels = c("yes","no"), label = c("group-1", "group-2"))
Suppose, you have a factor variable with numerical values. You want to compute the mean. The mean vector will result in the average value of the vector, but the mean of the factor variable will
result in a warning message. To calculate the mean of the original numeric values of the "f" variable, you have to convert the values using the level argument. For example,
# vector
v <- c(10,20,20,50,10,20,10,50,20)
# vector converted to factor
f <- factor(v)
# mean of the vector
# mean of factor
Use of cut() Function in R
The the cut() function in R can also be used to convert a numeric variable into a factor. The breaks argument can be used to describe how ranges of numbers will be converted to factor values. If the
breaks argument is set to a single number then the resulting factor will be created by dividing the range of the variable into that number of equal-length intervals. However, if a vector of values is
given to the breaks argument, the values in the vectors are used to determine the breakpoint. The number of levels of the resultant factor will be one less than the number of values in the vector
provided to the breaks argument. For example,
cut(mpg, breaks = 3)
factors <- cut(mpg, breaks = c(10, 18, 25, 30, 35) )
You will notice that the default label for factors produced by the cut() function in R contains the actual range of values that were used to divide the variable into factors.
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EGMO synopsis
I have returned from assisting at the Easter olympiad training camp at Trinity College, Cambridge. After we marked a selection test (the results of which I am not at liberty to divulge), a
preliminary squad of nine people were selected, which will eventually be narrowed down to the six people that form the IMO team.
In addition to satisfying the role of General Dogsbody, I gave an unofficial talk on Ramsey theory at 7:00 am. Considering the early hour, lack of advertisement and the optional nature of the talk,
the attendance (three people) was actually quite reasonable. I distributed a sheet of questions (PDF). If you wish to attempt them, I suggest familiarising yourself with the theorems in the third
chapter (Combinatorics II) of MODA. Two of these were marked with a direct sum symbol ($\oplus$) to indicate that they’re extremely challenging, and far beyond the hardest of IMO questions.
Another function of the Easter camp was to prepare our team of four girls for EGMO 2013, which was held in Luxembourg last week. The scores at the end of the competition were continually updated to a
live online scoreboard, followed shortly after by the medal boundaries. Danielle Wang (USA) won the competition, with a total of 38 marks out of a possible 42. Three nations (Belarus, Serbia and the
United States) were tied in equal first position, with cumulative scores of 99.
Solutions and commentary (spoiler alert!)
The problems appear to be arranged in strictly increasing order of how interesting they are, so I’ll only discuss the second paper here.
Problem 4: Quintic polynomial
This one isn’t actually particularly interesting; however, I’ll include it here for completeness, since it would seem strange to only discuss questions 5 and 6.
4. Find all ordered pairs of positive integers (a, b) for which there are three consecutive integers at which the polynomial $P(n) = \dfrac{n^5 + a}{b}$ takes integer values.
In problems such as this one, it is most convenient to let the three consecutive integers be {n − 1, n, n + 1}. Then, we require (n − 1)^5 + a, n^5 + a and (n − 1)^5 + a to all be divisible by b.
Since $b \mathbb{Z}$ is an ideal of $\mathbb{Z}$, we can add multiples of b and multiply by integers whilst retaining divisibility by b. Consequently, the polynomials $n^5 - (n-1)^5 = 5n^4 - 10n^3 +
10n^2 - 5n + 1$ and $(n+1)^5 - n^5 = 5n^4 + 10n^3 + 10n^2 + 5n + 1$ must also be multiples of b.
Hence, b and n are coprime, and both $20n^3 + 10n$ and $10n^4 + 20n^2 + 2$ are divisible by b, so we can conclude (eventually) that $b|11$. For the case where $b = 1$, all integer values of a
trivially work. For the other case of $b = 11$, it suffices to consider the residues of n^5 (mod 11) to find all solutions for a.
Problem 5: Mixtilinear incircle
5. Let Ω be the circumcircle of ABC, and let ω be the mixtilinear incircle touching BC, AC and internally tangent to Ω at the point P. A line parallel to AB and intersecting the interior of
triangle ABC is tangent to ω at the point Q. Prove that angles ACP and QCB are equal.
What makes this problem particularly appealing is that there are only five important points. In a situation like this, one would normally find a clever Euclidean construction involving additional
points, or apply some algebraic bash. Quite remarkably, we can solve this problem without introducing any more points.
To solve this problem, we’ll use a Möbius transformation obtained by the composition of an inversion in a circle centred on C and orthogonal to ω followed by a reflection in the angle bisector of
BCA. This is actually a natural thing to do, since lines through C remain as lines, the mixtilinear incircle ω is preserved, the circumcircle Ω is mapped to a line, and lines AC and BC are
interchanged. Our Möbius transformation is an involution, although we don’t need this fact.
The image of Ω is the line A’B’, where A’ and B’ are the images of A and B. It shouldn’t be difficult to convince yourself that B’A’C is a homothetic copy of ABC, so A’B’ is parallel to AB. Also,
A’B’ must be tangent to ω and intersect the interior of the triangle, so it is precisely the line used to construct Q. We can deduce, therefore, that P and Q are interchanged, whence it follows that
angles ACP and QCB are indeed equal.
The Möbius transformation is colloquially referred to as an inverflection, being composed of an inversion in a circle followed by a reflection in a diameter of the circle. On the Riemann sphere, this
operation is indistinguishable from a reflection in a point, which is why it is a natural thing to do. Sahl Khan applied this method to a difficult geometry problem on RMM 2011, and constructed
several problems of his own relying on this principle.
Problem 6: Snow White and the Seven Dwarves
6. Snow White and the Seven Dwarves are living in their house in the forest. On each of 16 consecutive days, some of the dwarves worked in the diamond mine while the remaining dwarves collected
berries in the forest. No dwarf performed both types of work on the same day. On any two different (not necessarily consecutive) days, at least three dwarves each performed both types of work.
Further, on the first day, all seven dwarves worked in the diamond mine. Prove that, on one of these 16 days, all seven dwarves were collecting berries.
This is a thinly-veiled problem asking for you to essentially prove that the Hamming(7,4) code is optimal and unique. Its optimality follows from it being a perfect code; specifically, each of the
128 binary strings of seven digits is within a Hamming distance of 1 from precisely one of the 16 codewords.
Uniqueness is slightly more difficult. We are given that 0000000 is a codeword, and want to show that 1111111 is also a codeword. This must be a perfect code (essentially a space-filling sphere
packing on the Boolean lattice), which means that every binary string must either be a codeword or differ from precisely one codeword in a single position. Consequently, every string of Hamming
weight 2 (such as 1100000, 0010100 and 0001001) must differ from a codeword of Hamming weight 3 in a single position.
Essentially, we want to group the seven dwarves into overlapping ‘lines’ of three dwarves, such that any pair of dwarves determines a unique line. From this, we can deduce that each dwarf belongs to
three lines, and thus there are seven lines (corresponding to codewords of Hamming weight 3). This describes the Fano plane, as Maria Holdcroft noticed when attempting this problem in the actual
If the problem is false (i.e. 1111111 is not a codeword), then we must have a codeword of Hamming weight six. Without loss of generality, this is 0111111, and three of the other codewords are
1110000, 1001100 and 1000011. Now consider the string 0001111. This can’t be a codeword (since it differs from 0111111 in only two positions), so must differ from a codeword X in one position.
Clearly, X cannot be 0101111 or 0011111 (since then it would be too close to 0111111). It can’t be 1001111 either, as that is too close to 1000011. Hence, X must be either 0001110, 0001101, 0001011
or 0000111. The first two ‘collide’ with 1001100; the last two collide with 1000011. Hence we obtain a contradiction, and our proof is complete.
A more difficult generalisation of this problem involves 23 dwarves who work for 4096 days, such that for any two days, at least 7 dwarves do different types of work. The unique solution is
then given by the perfect binary Golay code.
0 Responses to EGMO synopsis
1. Note that problem 6 does not require you to show such a schedule is possible. Thus the stuff involving the Fano plane is not necessary to solve the problem, and you merely need to consider the
Hamming weights of the points adjacent to each ‘codeword’. The only way in which doing it for 23 dwarves, etc., is harder, is that the arithmetic involves bigger numbers (nothing bigger than 7
digits though).
□ I disagree. The construction of the binary Golay code is far more complicated than that of the Hamming codes, and (unlike the Hamming codes) it is entirely non-obvious that it should even
Anyway, I would love to be proved wrong. How would you solve the problem in the case of 23 dwarves?
☆ I didn’t say anything about constructing the Golay code. I only stated that it is possible to prove that, if one exists, and 00…00 is in it, then 11…11 is also in it. Here’s Daniel’s
solution to the original EGMO problem (mine is identical, and I couldn’t be bothered to write it up). The extension to 23 dwarves should be fairly obvious.
“EGMO 6
view it as finding columns at least three swaps away from each other. We may as well use 0s and 1s to represent the jobs. It is helpful to think of the 7 dimensional cube, or graph
theory, though unnecessary.
supposing we can find a 16 column schedule for the 2^7 3-swap problem. Then for each of those 2^4 columns, there are a total of 8 at most 1 swap away (include itself). But none of these
16 octets of columns can share an element, as that would imply those two columns were only 2 swaps away.
But 8×16=2^7 so every column is either one of our 16 or 1 swap away from a unique one of the 16.
So we know we start with a column full of zeros. obviously among the Column 16 none have 1 or 2 zeros in. there are 21 columns with two zeros, and so in our Column 16 we need some columns
with 3 zeros in to be 1 swap away from them. Each column with 3 zeros yields a set of 3 columns with 2 zeros that are 1 swap away. So 7=21/3 of our Column 16 necessarily have 3 zeros.
Then, similarly 7 have 4 zeros to cover the 35=7+4×7 3 zero columns. Then none have 5 or six similarly, and the last one has to be a column of 1’s.”
○ Right. I admit that considering the 4-subsets of {1,2, …, 23} will allow you to prove that there must be 23C4 / 7C4 = 253 codewords of Hamming weight 7, and then considering the
5-subsets will give you (23C5 – 253*(7C5)) / 8C5 = 506 codewords of weight 8. And then you can consider the 8-subsets and (after some arithmetic) deduce that there are (23C8 – 253*
(16C1) – 253*(16C2)*(7C1) – 506 – 506*(15C2)*(8C1)) / 11C8 = 1288 codewords of weight 11. Then considering the 9-subsets will give you (23C9 – 253*(16C2) – 506*(15C1) – 506*(15C2)*
(8C1) – 1288*(11C9)) / 12C9 = 1288 codewords of weight 12.
It looks as though you could, in principle, continue this method to obtain the 506 codewords of weight 15, 253 codewords of weight 16, and then 1 codeword of weight 23 (see sequence
http://oeis.org/A002289 for the complete weight enumerator). So I suppose you win. Nevertheless, it’s significantly more complicated than the 7-dwarf problem, not least because we’re
trying to tile the space with spheres of radius 3 instead of 1.
■ Oh, right, I forgot the spheres now had radius 3. But at least it would work without requiring any new ideas.
2. For Q5, can we come up with a name to replace the portmanteau ‘inverflection’? Reciprocation might do.
□ Yes, that may suffice (as every inverflection can be conjugated to z –> 1/z by a Euclidean transformation, so they’re essentially the same thing). `Inverflection’ seems to be well
established, although I’m not sure who coined the neologism. (Sahl Khan, maybe? Or Maria Holdcroft?)
☆ It wasn’t me; it may have been Sahl.
○ Perhaps you could investigate…?
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Elementary Linear Algebra: Part I
This is an introduction to linear algebra. The main part of the book features row operations and everything is done in terms of the row reduced echelon form and specific algorithms. At the end, the
more abstract notions of vector spaces and linear transformations on vector spaces are presented. This is intended to be a first course in linear algebra for students who are sophomores or juniors
who have had a course in one variable calculus and a reasonable background in college algebra.
Click here to download the additional book files.
About the author
Kenneth Kuttler received his Ph.D. in mathematics from The University of Texas at Austin in 1981. From there, he went to Michigan Tech. University where he was employed for most of the next 17 years.
He joined the faculty of Brigham Young University in 1998 and has been there since this time. Kuttler's research interests are mainly in the mathematical theory for nonlinear initial boundary value
problems, especially those which come from physical models that include damage, contact, and friction. Recently he has become interested in stochastic integration and the related problems involving
nonlinear stochastic evolution equations.
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DIY Big Data Project Goal
I have always been interested in large scale computing. This interest started back in graduate school when I was studying astrodynamics. Most of the problems you want to solve in astrodynamics
requires numerical computation, as only the 2-body problem has a closed form solution. In order to solve anything more complex, you first have to make some simplifying assumptions (e.g., no other
gravitational influences), and then you have a set of partial differential equations that will describe the motion of your system. The only way to use these equations to predict the position of the
bodies was through numerical integration. In simple systems this sort of calculations was pretty straightforward, though you have to pay a lot of attention to round off error. However, when if you
were trying to predict the progression of a debris field in space, which would require you to simultaneously project the motion of tens of thousands of objects, parallelized computation start to look
Compute centric paralyzation is only one form of distributed computing. With the rise of the internet and the increased ease at generating and retaining data, a new class of distributed computing
problems arose that focussed on the processing large data sets, especially ones that cannot fit on a single computer. Google introduce a paradigm handling these big data problems through an approach
called map reduce and developing a way to distribute a file system across many machines. Open source solutions to the big data problem were developed, such as Hadoop and more recently Spark. All of
these solutions are focussed on the idea breaking up large data sets into smaller chunks and then analysisinp the chunks in parallel, aggregating the results into a comprehensive answer.
I have used big data platforms such as Hadoop, Spark, and QFS for the past 8 years of my career to solve many sorts of problems, from identifying fraudulent web activity, to analyzing the power of
social networks in driving web traffic, to predicting the demographics of of the audience that watches a specific YouTube video. In all of this, I have gotten reasonably adept at leveraging big data
platforms to solve business problems, but I have only acquired what I would describe as a basic familiarity with how the underlying platforms work. So I decided to embark on a project of setting up
my own cluster in the hopes that by setting up, configuring, and optimizing a distributed computing system, I will better understand the underlying technology that I have been using for so long.
One of the primary goals of my project is to use “real steel” when building my cluster. There are plenty of platforms where you can spin up a virtual cluster, such as AWS. But I am also interested in
building my understanding how hardware configuration impacts a clusters performance. So actual computers it is. The second goal is to focus on the Spark computing platform. I have used a number of
platforms over the years, and I have found Spark to be the most elegant from a user perspective. It is also the focus of my current professional activity. So, I will set out to build a Spark cluster
on real computers.
I plan to execute this project in at least two phases. The first is to use low cost computer boards, such as the Raspberry Pi, to develop my initial understanding of how to set up and configure a
cluster. Once I have built my experience using low cost solutions, then I intend build a “personal cluster” that should be affordable but also large enough to reasonably handle Spark-based data
analysis work for data sets in the one to five terabyte range. Since I am doing this in my spare time, this project will likely take a while to execute. But do follow along, maybe we can learn
something together.
One thought on “DIY Big Data Project Goal”
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Identifying Characteristics Of Quadratic Functions Worksheet - Function Worksheets
Identifying Characteristics Of Quadratic Functions Worksheet
Identifying Characteristics Of Quadratic Functions Worksheet – The Quadratic Functions Worksheet will assist pupils to learn the qualities of quadratic features. This worksheet helps for detailing
the thought of quadratic operate and creating an Desk of Principles, and finding how to discover the Axis of Symmetry and Vertex from the work. Students may also have to distinguish By-Intercept/s as
well as Y-Intercept/s. A quadratic function worksheet that is well-designed could be used to start a lesson in algebra, or as part of a set of math resources. Identifying Characteristics Of Quadratic
Functions Worksheet.
Quadratic graphing
This Graphing Quadratic Capabilities workbook enables students to make use of graphs to represent quadratic equations. The worksheet can be purchased in vertex form along with regular form. It is
actually a requirement of college students to recognize both vertex along with the intercept by. They are also necessary to find out what is definitely the vertex for your formula as well as decide
the maximum and minimum values for your function. After they have determined these, they are able to transform the situation in to the typical kind.
Employing while using Graphing quadratic capabilities worksheet, college students can examination their understanding about graphing quadratic characteristics. The worksheets are present supplied in
pdf type and can be employed by each teachers and students. The worksheets may not have the necessary instruments found it necessary to teach the concept. The worksheets for college kids could have
errors or worksheets that don’t comply with the regular develop. It is actually a great concept to test the pupils’ comprehension with this concept through providing them an issue that they may
resolve with answers to the appropriate queries.
It is simple if you own an algebra worksheet. The graphing quadratic formulation is generally detailed inside a table with principles. The table’s values would be the minimum and maximum of the
situation. Variable x’s values should be positive or negative according to the sign of its value. When you’ve accomplished the task then you can attract this quadratic formula.
Standard type
A standard kind of an picture that is quadratic is a specific develop used to eliminate an equation. This sort of picture offers the equal form and employs the most important typical component (GCF)
of all the coefficients with nonzero principles since the solution. Pupils can learn to compose quadratic equations using the common form through the use of worksheets like this a single. Individuals
can also assessment their replies and determine regardless of whether they’re making use of the correct kind of formula.
The problem requires individuals to graph two distinct spots on charts to determine the highest earnings they are able to make from offering them at numerous rates. It is additionally possible to
resolve the problem by substituting a nicely-recognized point for every organize. In the same manner you are able to graph a quadratic formula using the classic type. Students need to find the
by-synchronize for that vertex along with the y-coordinate to the situation to find the by-intercept and obtain the most profit.
This assignment on the web consists of 12 Multiple Choice queries which are rated immediately. It may be utilized as studying or as being an evaluation. It is made up of parabolas in vertex kind and
transforms like horizontal and vertical stretching out, translations and also pressure and reflection. There exist 3 forms of issues that you can unravel utilizing that vertex in quadratic equations.
The worksheet can be obtained on Educators Pay Teachers. Educators Pay out Educators marketplace. You are able to download the worksheet at no cost to make use of in your type or at your house ..
If you are trying to solve a quadratic problem, you first need to determine which equation’s vertex. The vertex is the place which a quadratic formula is good. This is certainly also known as the
graph’s starting point. The vertex type is oftentimes referenced such as the parabola. It is able to wide open up because its vertex has a positive. To discover the issue of your parabola’s vertex
think about in the “a” element.
Gallery of Identifying Characteristics Of Quadratic Functions Worksheet
Algebra B Day 95 Part 1 Characteristics Of Quadratic Functions 9 2
Identifying Parts Of A Quadratic on A Graph Math Quadratic
Characteristics Of Linear Functions Practice Worksheet A Linear
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Volume of frustum of a cone - Definition, Formula, Solved Example Problems | Mensuration | Mathematics
Let H and h be the height of cone and frustum respectively, L and l be the slant height of the same.
Volume of frustum of a cone
Let H and h be the height of cone and frustum respectively, L and l be the slant height of the same.
If R, r are the radii of the circular bases of the frustum, then volume of the frustum of the cone is the difference of the volumes of the two cones.
Since the triangles ABC and ADE are similar, the ratio of their corresponding sides are proportional.
Example 7.23 If the radii of the circular ends of a frustum which is 45 cm high are 28 cm and 7 cm, find the volume of the frustum.
Solution :
Let h, r and R be the height, top and bottom radii of the frustum.
Given that, h = 45 cm, R = 28 cm, r = 7 cm
Therefore, volume of the frustum is 48510 cm^3
The adjacent figure represents an oblique frustum of a cylinder. Suppose this solid is cut by a plane through C, not parallel to the base AB, then
where h[1] and h[2] denote the greatest and least height of the frustum.
Then its volume =
Tags : Definition, Formula, Solved Example Problems | Mensuration | Mathematics , 10th Mathematics : UNIT 7 : Mensuration
Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail
10th Mathematics : UNIT 7 : Mensuration : Volume of frustum of a cone | Definition, Formula, Solved Example Problems | Mensuration | Mathematics
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CFA using MLR estimator and ordinal data
Nonnormality does not imply ordinality. Do your indicators measured using an ordinal (e.g., Likert) scale with only a few categories? If you have 7 or more categories, you can probably just treat
them as continuous and use MLR.
No one does, MLR assumes the data are continuous. Mplus misleadingly uses the command ESTIMATOR=MLR to trigger marginal maximum likelihood estimation, developed for IRT models. lavaan also has an
experimental estimator="MML" option to request that.
Regardless, the probit model implies a normally distributed latent item response underlying each observed ordinal response. There is no getting around that (latent) normality assumption using
frequentist estimators like MML or DWLS(MV). Using Bayesian MCMC estimation, you could specify an alternative to the normal latent item responses, such as a skew-t distribution that allows skew and
Yes they are. A 1-factor model is equivalent to a 2-factor model in which the 2 factor's correlation is fixed to 1 (easier to specify when identifying the model using
=TRUE). So you can use DWLS and compare those models using lavTestLRT(). You can verify that they are nested using the semTools function net(). That requires
installing the current development version
: devtools::install_github("simsem/semTools/semTools")
myData <- read.table("http://www.statmodel.com/usersguide/chap5/ex5.16.dat")
names(myData) <- c("u1","u2","u3","u4","u5","u6","x1","x2","x3","g")
model1 <- ' f1 =~ u1 + u2 + u3 + u4 + u5 + u6 '
model2 <- '
f1 =~ u1 + u2 + u3
f2 =~ u4 + u5 + u6
fit1 <- cfa(model1, data = myData, ordered = paste0("u", 1:6))
fit2 <- cfa(model2, data = myData, ordered = paste0("u", 1:6))
lavTestLRT(fit1, fit2) # compare fit
net(fit1, fit2) # check they are nested
Terrence D. Jorgensen
Assistant Professor, Methods and Statistics
Research Institute for Child Development and Education, the University of Amsterdam
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Cosparse regularization of physics-driven inverse problems
Cosparse regularization of physics-driven inverse problems (2015)
Solving inverse problems in room acoustics using physical models, sparse regularization and numerical optimization
Reverberation consists of a complex acoustic phenomenon that occurs inside rooms. Many audio signal processing methods, addressing source localization, signal enhancement and other tasks, often
assume absence of reverberation. Consequently, reverberant environments are considered challenging as state-ofthe-art methods can perform poorly. The acoustics of a room can be described using a
variety of mathematical models, among which, physical models are the most complete and accurate. The use of physical models in audio signal processing methods is often non-trivial since it can lead
to ill-posed inverse problems. These inverse problems require proper regularization to achieve meaningful results and involve the solution of computationally intensive large-scale optimization
problems. Recently, however, sparse regularization has been applied successfully to inverse problems arising in different scientific areas. The increased computational power of modern computers and
the development of new efficient optimization algorithms makes it possible ...
Antonello, Niccolò — KU Leuven
Sensing physical fields: Inverse problems for the diffusion equation and beyond
Due to significant advances made over the last few decades in the areas of (wireless) networking, communications and microprocessor fabrication, the use of sensor networks to observe physical
phenomena is rapidly becoming commonplace. Over this period, many aspects of sensor networks have been explored, yet a thorough understanding of how to analyse and process the vast amounts of sensor
data collected remains an open area of research. This work, therefore, aims to provide theoretical, as well as practical, advances this area. In particular, we consider the problem of inferring
certain underlying properties of the monitored phenomena, from our sensor measurements. Within mathematics, this is commonly formulated as an inverse problem; whereas in signal processing, it appears
as a (multidimensional) sampling and reconstruction problem. Indeed it is well known that inverse problems are notoriously ill-posed and very demanding to solve; meanwhile ...
Murray-Bruce, John — Imperial College London
Implementation of the radiation characteristics of musical instruments in wave field synthesis applications
In this thesis a method to implement the radiation characteristics of musical instruments in wave field synthesis systems is developed. It is applied and tested in two loudspeaker systems. Because
the loudspeaker systems have a comparably low number of loudspeakers the wave field is synthesized at discrete listening positions by solving a linear equation system. Thus, for every constellation
of listening and source position all loudspeakers can be used for the synthesis. The calculations are done in spectral domain, denying sound propagation velocity at first. This approach causes
artefacts in the loudspeaker signals and synthesis errors in the listening area which are compensated by means of psychoacoustic methods. With these methods the aliasing frequency is determined by
the extent of the listening area whereas in other wave field synthesis systems it is determined by the distance of adjacent loudspeakers. Musical ...
Ziemer, Tim — University of Hamburg
Group-Sparse Regression - With Applications in Spectral Analysis and Audio Signal Processing
This doctorate thesis focuses on sparse regression, a statistical modeling tool for selecting valuable predictors in underdetermined linear models. By imposing different constraints on the structure
of the variable vector in the regression problem, one obtains estimates which have sparse supports, i.e., where only a few of the elements in the response variable have non-zero values. The thesis
collects six papers which, to a varying extent, deals with the applications, implementations, modifications, translations, and other analysis of such problems. Sparse regression is often used to
approximate additive models with intricate, non-linear, non-smooth or otherwise problematic functions, by creating an underdetermined model consisting of candidate values for these functions, and
linear response variables which selects among the candidates. Sparse regression is therefore a widely used tool in applications such as, e.g., image processing, audio processing, seismological and
biomedical modeling, but is ...
Kronvall, Ted — Lund University
Cost functions for acoustic filters estimations in reverberant mixtures
This work is focused on the processing of multichannel and multisource audio signals. From an audio mixture of several audio sources recorded in a reverberant room, we wish to es- timate the acoustic
responses (a.k.a. mixing filters) between the sources and the microphones. To solve this inverse problem one need to take into account additional hypotheses on the nature of the acoustic responses.
Our approach consists in first identifying mathematically the neces- sary hypotheses on the acoustic responses for their estimation and then building cost functions and algorithms to effectively
estimate them. First, we considered the case where the source signals are known. We developed a method to estimate the acoustic responses based on a convex regularization which exploits both the
temporal sparsity of the filters and the exponentially decaying envelope. Real-world experi- ments confirmed the effectiveness of this method ...
Benichoux, Alexis — Université Rennes I
Cognitive Models for Acoustic and Audiovisual Sound Source Localization
Sound source localization algorithms have a long research history in the field of digital signal processing. Many common applications like intelligent personal assistants, teleconferencing systems
and methods for technical diagnosis in acoustics require an accurate localization of sound sources in the environment. However, dynamic environments entail a particular challenge for these systems.
For instance, voice controlled smart home applications, where the speaker, as well as potential noise sources, are moving within the room, are a typical example of dynamic environments. Classical
sound source localization systems only have limited capabilities to deal with dynamic acoustic scenarios. In this thesis, three novel approaches to sound source localization that extend existing
classical methods will be presented. The first system is proposed in the context of audiovisual source localization. Determining the position of sound sources in adverse acoustic conditions can be
improved by including ...
Schymura, Christopher — Ruhr University Bochum
Application of Sound Source Separation Methods to Advanced Spatial Audio Systems
This thesis is related to the field of Sound Source Separation (SSS). It addresses the development and evaluation of these techniques for their application in the resynthesis of high-realism sound
scenes by means of Wave Field Synthesis (WFS). Because the vast majority of audio recordings are preserved in two-channel stereo format, special up-converters are required to use advanced spatial
audio reproduction formats, such as WFS. This is due to the fact that WFS needs the original source signals to be available, in order to accurately synthesize the acoustic field inside an extended
listening area. Thus, an object-based mixing is required. Source separation problems in digital signal processing are those in which several signals have been mixed together and the objective is to
find out what the original signals were. Therefore, SSS algorithms can be applied to existing two-channel mixtures to ...
Cobos, Maximo — Universidad Politecnica de Valencia
Distributed Localization and Tracking of Acoustic Sources
Localization, separation and tracking of acoustic sources are ancient challenges that lots of animals and human beings are doing intuitively and sometimes with an impressive accuracy. Artificial
methods have been developed for various applications and conditions. The majority of those methods are centralized, meaning that all signals are processed together to produce the estimation results.
The concept of distributed sensor networks is becoming more realistic as technology advances in the fields of nano-technology, micro electro-mechanic systems (MEMS) and communication. A distributed
sensor network comprises scattered nodes which are autonomous, self-powered modules consisting of sensors, actuators and communication capabilities. A variety of layout and connectivity graphs are
usually used. Distributed sensor networks have a broad range of applications, which can be categorized in ecology, military, environment monitoring, medical, security and surveillance. In this
dissertation we develop algorithms for distributed sensor networks ...
Dorfan, Yuval — Bar Ilan University
General Approaches for Solving Inverse Problems with Arbitrary Signal Models
Ill-posed inverse problems appear in many signal and image processing applications, such as deblurring, super-resolution and compressed sensing. The common approach to address them is to design a
specific algorithm, or recently, a specific deep neural network, for each problem. Both signal processing and machine learning tactics have drawbacks: traditional reconstruction strategies exhibit
limited performance for complex signals, such as natural images, due to the hardness of their mathematical modeling; while modern works that circumvent signal modeling by training deep convolutional
neural networks (CNNs) suffer from a huge performance drop when the observation model used in training is inexact. In this work, we develop and analyze reconstruction algorithms that are not
restricted to a specific signal model and are able to handle different observation models. Our main contributions include: (a) We generalize the popular sparsity-based CoSaMP algorithm to any signal
Tirer, Tom — Tel Aviv University
Bayesian Compressed Sensing using Alpha-Stable Distributions
During the last decades, information is being gathered and processed at an explosive rate. This fact gives rise to a very important issue, that is, how to effectively and precisely describe the
information content of a given source signal or an ensemble of source signals, such that it can be stored, processed or transmitted by taking into consideration the limitations and capabilities of
the several digital devices. One of the fundamental principles of signal processing for decades is the Nyquist-Shannon sampling theorem, which states that the minimum number of samples needed to
reconstruct a signal without error is dictated by its bandwidth. However, there are many cases in our everyday life in which sampling at the Nyquist rate results in too many data and thus, demanding
an increased processing power, as well as storage requirements. A mathematical theory that emerged ...
Tzagkarakis, George — University of Crete
Regularized estimation of fractal attributes by convex minimization for texture segmentation: joint variational formulations, fast proximal algorithms and unsupervised selection of regularization
In this doctoral thesis several scale-free texture segmentation procedures based on two fractal attributes, the Hölder exponent, measuring the local regularity of a texture, and local variance, are
proposed.A piecewise homogeneous fractal texture model is built, along with a synthesis procedure, providing images composed of the aggregation of fractal texture patches with known attributes and
segmentation. This synthesis procedure is used to evaluate the proposed methods performance.A first method, based on the Total Variation regularization of a noisy estimate of local regularity, is
illustrated and refined thanks to a post-processing step consisting in an iterative thresholding and resulting in a segmentation.After evidencing the limitations of this first approach, deux
segmentation methods, with either "free" or "co-located" contours, are built, taking in account jointly the local regularity and the local variance.These two procedures are formulated as convex
nonsmooth functional minimization problems.We ...
Pascal, Barbara — École Normale Supérieure de Lyon
Modern Optimization Methods for Interpolation of Missing Sections in Audio Signals
Damage to audio signals is in practice common, yet undesirable. Information loss can occur due to improper recording (low sample rate or dynamic range), transmission error (sample dropout), media
damage, or because of noise. The removal of such disturbances is possible using inverse problems. Specifically, this work focuses on the situation where sections of an audio signal of length in the
order of tens of milliseconds are completely lost, and the goal is to interpolate the missing samples based on the unimpaired context and a suitable signal model. The first part of the dissertation
is devoted to convex and non-convex optimization methods, which are designed to find a solution to the interpolation problem based on the assumption of sparsity of the time-frequency spectrum. The
general background and some algorithms are taken from the literature and adapted to the interpolation problem, ...
Mokrý, Ondřej — Brno University of Technology
Motion Analysis and Modeling for Activity Recognition and 3-D Animation based on Geometrical and Video Processing Algorithms
The analysis of audiovisual data aims at extracting high level information, equivalent with the one(s) that can be extracted by a human. It is considered as a fundamental, unsolved (in its general
form) problem. Even though the inverse problem, the audiovisual (sound and animation) synthesis, is judged easier than the previous, it remains an unsolved problem. The systematic research on these
problems yields solutions that constitute the basis for a great number of continuously developing applications. In this thesis, we examine the two aforementioned fundamental problems. We propose
algorithms and models of analysis and synthesis of articulated motion and undulatory (snake) locomotion, using data from video sequences. The goal of this research is the multilevel information
extraction from video, like object tracking and activity recognition, and the 3-D animation synthesis in virtual environments based on the results of analysis. An ...
Panagiotakis, Costas — University of Crete
Dereverberation and noise reduction techniques based on acoustic multi-channel equalization
In many hands-free speech communication applications such as teleconferencing or voice-controlled applications, the recorded microphone signals do not only contain the desired speech signal, but also
attenuated and delayed copies of the desired speech signal due to reverberation as well as additive background noise. Reverberation and background noise cause a signal degradation which can impair
speech intelligibility and decrease the performance for many signal processing techniques. Acoustic multi-channel equalization techniques, which aim at inverting or reshaping the measured or
estimated room impulse responses between the speech source and the microphone array, comprise an attractive approach to speech dereverberation since in theory perfect dereverberation can be achieved.
However in practice, such techniques suffer from several drawbacks, such as uncontrolled perceptual effects, sensitivity to perturbations in the measured or estimated room impulse responses, and
background noise amplification. The aim of this thesis ...
Kodrasi, Ina — University of Oldenburg
From Blind to Semi-Blind Acoustic Source Separation based on Independent Component Analysis
Typical acoustic scenes consist of multiple superimposed sources, where some of them represent desired signals, but often many of them are undesired sources, e.g., interferers or noise. Hence, source
separation and extraction, i.e., the estimation of the desired source signals based on observed mixtures, is one of the central problems in audio signal processing. A promising class of approaches to
address such problems is based on Independent Component Analysis (ICA), an unsupervised machine learning technique. These methods enjoyed a lot of attention from the research community due to the
small number of assumptions that have to be made about the considered problem. Furthermore, the resulting generalization ability to unseen acoustic conditions, their mathematical rigor and the
simplicity of resulting algorithms have been appreciated by many researchers working in audio signal processing. However, knowledge about the acoustic scenario is often available ...
Brendel, Andreas — Friedrich-Alexander-Universität Erlangen-Nürnberg
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Online calculator. Distance from a point to a line - 3-Dimensional
This online calculator will help you to find distance from a point to a line in 3D.
Using this online calculator, you will receive a detailed step-by-step solution to your problem, which will help you understand the algorithm how to find distance from a point to a line in 3D.
Distance from a point to a line 3D calculator
Select the form of line representation:
Line equation:
x - = y - = z -
Point coordinates:
: (
Entering data into the distance from a point to a line 3D calculator
You can input only integer numbers or fractions in this online calculator. More in-depth information read at these rules.
Additional features of distance from a point to a line 3D calculator
• Use and keys on keyboard to move between field in calculator.
Theory. Distance from a point to a line 3D
Distance from a point to a line is equal to length of the perpendicular distance from the point to the line.
If M[0](x[0], y[0], z[0]) is point coordinates, s = {m; n; p} is directing vector of line l, M[1](x[1], y[1], z[1]) is coordinates of point on line l, then distance between point M[0](x[0], y[0], z
[0]) and line l , can be found using the following formula
d = |M[0]M[1]×s||s|
You can input only integer numbers, decimals or fractions in this online calculator (-2.4, 5/7, ...). More in-depth information read at these rules.
Add the comment
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Regularity of a general equilibrium in a model with infinite past and future
We develop easy-to-verify conditions to assure that a comparative statics exercise in a dynamic general equilibrium model is feasible, i.e., the implicit function theorem is applicable. Consider an
equilibrium equation, ϒ(k,E)=k of a model where an equilibrium variable (k) is a continuous bounded function of time, real line, and the policy parameter (E) is a locally integrable function of time.
The key conditions are time invariance of ϒ and the requirement that the Fourier transform of the derivative of ϒ with respect to k does not return unity. Further, in a general
constant-returns-to-scale production and homogeneous life-time-utility overlapping generations model we show that the first condition is satisfied at a balanced growth equilibrium and the second
condition is satisfied for “almost all” policies that give rise to such equilibria.
Bibliographical note
Publisher Copyright:
© 2017 Elsevier B.V.
• Comparative statics
• Determinacy
• Implicit function theorem
• Overlapping generations
• Time-invariance
ASJC Scopus subject areas
• Economics and Econometrics
• Applied Mathematics
Dive into the research topics of 'Regularity of a general equilibrium in a model with infinite past and future'. Together they form a unique fingerprint.
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Training a 3-Node Neural Network is NP-Complete
Training a 3-Node Neural Network is NP-Complete
Part of Advances in Neural Information Processing Systems 1 (NIPS 1988)
Avrim Blum, Ronald Rivest
We consider a 2-layer, 3-node, n-input neural network whose nodes compute linear threshold functions of their inputs. We show that it is NP-complete to decide whether there exist weights and
thresholds for the three nodes of this network so that it will produce output con(cid:173) sistent with a given set of training examples. We extend the result to other simple networks. This result
suggests that those looking for perfect training algorithms cannot escape inherent computational difficulties just by considering only simple or very regular networks. It also suggests the
importance, given a training problem, of finding an appropriate network and input encoding for that problem. It is left as an open problem to extend our result to nodes with non-linear functions such
as sigmoids.
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This is semi linear system and I don't know what to do
• Explanation from Alloprof
Explanation from Alloprof
This Explanation was submitted by a member of the Alloprof team.
Hello Red Ruby!
Thank you for asking us your question!
In this case, you can solve this as a system of equations. You can assume that y = y to compare both equations as below:
$$ y = y \Rightarrow 2x - 1 = (x+6)^2 + 11 $$
You can then isolate all the equations on one side to have an equation following the format \( 0 = ax^2 + bx + c \). You can then find the value for \( b^2 - 4ac \) to determine of the system
will admit one, two or no solutions.
You can follow the link below to look at similar examples:
Feel free to reach out if you have any other questions!
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Prime solution wows the math world!
WOW! :eek:
But can it dice, slice, and julienne?
I am such an asshole.
You know, I’m really bad at math and almost anything math-related, but I have an inordinate fondness for primes. (I have a tendency to recite primes in my head when I’m bored or trying to sleep.)
This is really…well…NEATO.
Thanks, astro.
What real-world implications does this have? anyone?
I’m sure the financial industry would love to know indivisible numbers, i.e. selling shares at whole dollars, rounding foreign currency conversions by specifying integer amounts. . .
Not to mention the nuclear physics. Wooo!
We haven’t even touched on the academics of it, yet.
I’d have to read up on cryptography algorithms, but it seems to me that if you could generate huge primes at will, one might be able to assign a unique RSA-type encryption to each message. The way it
stands now, if you lose your private RSA key (i.e. stolen by a angry ex-employee) all your past messages can be decrypted. Using randomly-selected huge primes for the individual messages means each
message will have to be decrypted individually.
This will have a little impact, but not much. The really sexy problem is figuring out how factor composite numbers quickly. That would have a much larger effect on cryptography (Bryan’s post is
accurate, btw).
Being able to generate primes faster may make RSA key-pair generation faster, but the encryption time would still be slow. Usually you just use RSA encryption to encrypt the key for a different
(faster) algorithm such as DES. You generate new DES keys for each message, but you keep the same RSA private key and keep it safe. The main advantage of public-private key pair schemes is
non-repudiation. That is, if you encrypt something with your private key, I can use your public key to decrypt it and I can be sure that it came from you. This is what makes digital certificates
Bah! Everybody knows P-time can’t hold a candle to P-Funk!
I’m pretty sure the financial industry can already easily figure out any prime numbers within the range of the total number of all shares of stock in the world.
My views:
I won’t believe it until someone like Gary Miller says it works. There are some lurking doubts about this claim that I’ve seen pop up in a few places.
We already know how to prove primaility in polynomial time if you’re willing to relax an issue. For example, the aforementioned Gary Miller already gave a poly time algorithm, assuming ERH. (Which is
why some wonder if the new result also assumes ERH and is nothing new.) There is also a method that provies prime/composite in “almost certainly” polynomial time. Note that since we need large primes
(and not large composites) there are algorithms (again due to Miller, not Rabin) that can provide them in short time with high probablility. There are what people use “in real life”. So no real
practical impact, but potentially very big theoretically. (Now, if factoring were poly time, we’d have a really important result, not just to RSA stockholders.)
So we can generate all the RSA keys we want day in and day out already. That’s the first problem with Bryan Esker’s post. The second problem ignores the whole point of public key cryptosystems.
Namely, key distribution. If key distribution were not a problem, then we’d all use one-time pads. If you’re generating new RSA keys for each message, you may as well save yourself the trouble and
generate one-time pads (which are far easier to create, encrypt and decrypt, but far harder to distribute). (I’m leaving out a lot of cryto-detail here. But think about it, someone sends you an RSA
key and says to encrypt the Big Plans using it and send it to them. How do you know this came from “them” since it is the only key you’ve got? Well, they sign it, but with what key? Cyclic stuff.)
(I’ll ask Miller about this the next time I have dinner with him. But since it was 3 years since the last dinner, don’t hold your breath. :))
Dude, yer not helping me here. . .
I may waste bandwidth, but sometimes I sound smart. . .
Never mind, I found it. ERH is the extended Riemann hypothesis, for those playing along at home.
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GCHQ Christmas Card Puzzle · Data Intellect
[image src=”http://www.gchq.gov.uk/SiteCollectionImages/grid-shading-puzzle.jpg” href=”http://www.gchq.gov.uk/press_and_media/news_and_features/Pages/Directors-Christmas-puzzle-2015.aspx” alt=”GCHQ
Christmas Card Puzzle”]
[custom_headline type=”center” style=”margin-top: 0;” level=”h4″ looks_like=”h4″ accent=”true”]Introduction[/custom_headline]
Rikesh posted a link to the GCHQ Christmas Card Puzzle on the K4 listbox, so I decided to take a look. Described as a grid shading problem, it consists of a 25 X 25 grid, on which a number of cells
are shaded black. The sequence of numbers against each row and column specifies the length of runs of unbroken black squares in the line, which may be separated by one or more white squares.
We’re going to model this problem by thinking of each cell as being in one of 3 states – known black, known white, and unknown. Given the inputs in the problem, using black/white/grey for the three
states, the current state of the grid is:
[image type=”thumbnail” src=”https://aquaq.co.uk/wp-content/uploads/2015/12/step1-e1450045545319.png” alt=”Initial State”]
Our first task is to try and use the information provided about the rows and columns to work out all valid lines which satisfy those constraints. It is helpful to think of the line consisting of N
black sections, with N+1 white sections between and around them.
blacks: | B1 | | B2 | … | Bn-1 | | Bn |
whites: | W1 | | W2 | | … | | Wn | | Wn+1 |
The first and last white sections may have zero length, while the sections which separate the black ones must be at least of length one.
[custom_headline type=”center” style=”margin-top: 0;” level=”h4″ looks_like=”h4″ accent=”true”]Permutations[/custom_headline]
p:{$[x>1;raze n,/:’.z.s'[x-1;y-n:til 1+y];enlist x#y]}
The function p returns all the possible distributions of y items across x slots, for example, if we have 2 items and 3 slots:
We can use this function to generate the different possible white run lengths, since we know how many there should be (n+1), and we know the total number of white squares by subtracting the total
number of black squares from the grid dimension.
[custom_headline type=”center” style=”margin-top: 0;” level=”h4″ looks_like=”h4″ accent=”true”]White run sizes[/custom_headline]
w:{(0,#[n-1;1],0)+/:p[1+n;y+1-sum[x]+n:count x]}
The function w returns all possible white run lengths given the black runs (x) and the line length (y). It creates an initial list with the minimum values – zero at each end and one otherwise, then
adds this to each permutation of the remaining whites, of which there are line length (y) less the ones already allocated (n-1) and the total number of blacks (sum x).
[custom_headline type=”center” style=”margin-top: 0;” level=”h4″ looks_like=”h4″ accent=”true”]Possible lines[/custom_headline]
j:{raze each (w[x;y]#”0b),’\:(x,0)#’1b}
With all these combinations, the next step is to reconstitute the run lengths into regular lines of 25 booleans – true for black and false for white. The function j does this. The example shown is
using the 5th row of the grid, which was chosen because there are only a few valid lines for that input. Some rows and columns have more than 10,000 valid line outputs.
q)j[1 3 1 5 2 1 3 1;25]
[custom_headline type=”center” style=”margin-top: 0;” level=”h4″ looks_like=”h4″ accent=”true”]Filtering[/custom_headline]
m:{x~x and y}/:
Given all the possibilities, we now want to use the information we already have about the line, to filter out potential lines which don’t fit the known values. We can use an integer matrix to hold
the grid state – +1 for known black, 0 for unknown and -1 for known white. As an example, we’ll look at the 17th row in the matrix, one of the ones for which some known blacks are marked in the
problem description.
q)count j[3 1 1 1 1 5 1;25]
For this row, there are 1716 potential lines which would fit the specified black run lengths. We can use the known blacks to filter out impossible candidates with the function m.
q)count where m[x[16]>0;j[3 1 1 1 1 5 1;25]]
Only 71 out of 1716 have all four given black squares in the correct place, so we can throw away the rest and see if any other inferences can be made from the new, smaller set.
At this point it’s appropriate to point out that if we have known white values, we can filter for those using the same function:
q)count where m[x[16]<0;not j[3 1 1 1 1 5 1;25]]
We don’t have any known whites to start, so this filter can’t help us restrict the set, but as we gain information about the grid, this check will help us reach the solution.
[custom_headline type=”center” style=”margin-top: 0;” level=”h4″ looks_like=”h4″ accent=”true”]Line state[/custom_headline]
f:{all[b]-all not b:r where m[y<0;not r] and m[y>0;r:j[x;count y]]}
When we use our filters to reduce the full set of possibilities to ones which are consistent with the currnt state, we then use the results to see whether any more conclusions can be drawn. If all
the possible lines have a black square at position N, which was previously unknown, then we can mark that square as a known black, likewise for squares where all remaining lines have a white at that
For an example, we’ll look at the 9th row:
q)h 8
q)count j[h 8;25]
q)count where m[x[8]>0;j[h 8;25]]
q)r where m[x[8]>0] r:j[h 8;25]
There are 55 possible configurations, restricted to 12 which match the puzzle input. Looking at the result, there is a section in the middle which is common to all, so we should be able to return a
new known state of the row with information which wasn’t in the input.
q)a:all r where m[x[8]>0] r:j[h 8;25] / all true
q)b:all not r where m[x[8]>0] r:j[h 8;25] / all false
0 0 0 1 0 0 1 1 0 -1 1 -1 1 -1 1 1 1 -1 1 -1 0 0 0 1 0i
Combining our permutation function, our two filters, and calculating the new state, we have function f, above.
[custom_headline type=”center” style=”margin-top: 0;” level=”h4″ looks_like=”h4″ accent=”true”]Rows and columns[/custom_headline]
g:{flip f'[y;flip f'[x;z]]}
Our function f takes a line constraint and line state, and returns the new line state. If we want to run f against each row, the resulting state looks like:
[image type=”thumbnail” src=”https://aquaq.co.uk/wp-content/uploads/2015/12/firstpass-e1450054975832.png” alt=”First pass”]
We also have information about the black run sequences down the columns. The function g combines a vertical scan down the rows with a scan across the columns on the row result by transposing the
matrix. After scanning down the rows then across columns, we have the following state:
[image type=”thumbnail” src=”https://aquaq.co.uk/wp-content/uploads/2015/12/step2-e1450055617488.png” alt=”First full scan”]
[custom_headline type=”center” style=”margin-top: 0;” level=”h4″ looks_like=”h4″ accent=”true”]Solution[/custom_headline]
To find a solution, our function g is called successively with the results of the previous run, until it converges on a result
[image type=”thumbnail” src=”https://aquaq.co.uk/wp-content/uploads/2015/12/output_TGJ5e11.gif” alt=”Iterating to reach solution”]
The result is a QR code which can be scanned by a QR code app on a mobile device to produce a link which leads to the next stage of the GCHQ Christmas Puzzle. To produce a QR code which you can scan,
download and run the q script, and open the qr.html file which it produces.
[icon type=”file-text-o”]
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Seminars and Colloquia by Series
Wednesday, August 21, 2013 - 16:30 for 1 hour (actually 50 minutes)
Klaus 1116W
Vijay V. Vazirani – School of Computer Science, Georgia Tech
Please Note: Hosted by School of Computer Science.
Equilibrium computation is among the most significant additions to the theory of algorithms and computational complexity in the last decade - it has its own character, quite distinct from the
computability of optimization problems. Our contribution to this evolving theory can be summarized in the following sentence: Natural equilibrium computation problems tend to exhibit striking
dichotomies. The dichotomy for Nash equilibrium, showing a qualitative difference between 2-Nash and k- Nash for k > 2, has been known for some time. We establish a dichotomy for market equilibrium.
For this purpose. we need to define the notion of Leontief-free functions which help capture the joint utility of a bundle of goods that are substitutes, e.g., bread and bagels. We note that when
goods are complements, e.g., bread and butter, the classical Leontief function does a splendid job. Surprisingly enough, for the former case, utility functions had been defined only for special cases
in economics, e.g., CES utility function. We were led to our notion from the high vantage point provided by an algorithmic approach to market equilibria. Note: Joint work with Jugal Garg and Ruta
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How to compute the correlation between two columns in a data frame in PySpark?
To compute the correlation between two columns in a data frame in PySpark, you can use the corr() method of the data frame.
Here is an example:
1 from pyspark.sql.functions import corr
3 # Create a data frame with two columns
4 data = [(1, 2), (2, 4), (3, 6), (4, 8)]
5 df = spark.createDataFrame(data, ["col1", "col2"])
7 # Compute the correlation between the two columns
8 correlation = df.corr("col1", "col2")
10 print("The correlation between col1 and col2 is:", correlation)
1 The correlation between col1 and col2 is: 1.0
Note that the corr() method returns the Pearson correlation coefficient between the two columns. The value of the correlation coefficient ranges from -1 to 1, where -1 indicates a perfect negative
correlation, 0 indicates no correlation, and 1 indicates a perfect positive correlation.
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Programming a Quantum Computer
Programming quantum computers is vastly different than programming conventional ones. To get a better idea of what it could be like, this article steps through the process of programming a quantum
computer, a D Wave System Two, using the direct embedded method to solve a relatively simple problem.
│ Glossary │
│ Coupler: A variable (b) that defines how two qubits affect each other. For example, coupler b[ij] determines how q[i] affects q[j]. │
│ │
│ Qubit: A variable (q) that has a value from the set {0, 1}. │
│ │
│ Quantum Machine Instruction (QMI):A restatement of the problem to be solved. The computer comes up with a distribution of qubit values that minimize the value of the QMI. │
│ │
│ Strength: Defines the coupler relationship between qubits and provides another way to influence qubits. A coupler connecting qubits q[i] and q[j] has a strength of b and is denoted b[ij]. │
│ │
│ Weight: In the QMI, each qubit (q) is given a weight (a) as one way of influencing qubits. For example, qubit, q[i] has a weight of a[i]. │
The problem
Map coloring is a type of combinatorial optimization problem. For this problem, the goal is to color the 13 territories and provinces of Canada so that no two regions sharing a border are the same
color, and regions touching only at one or more isolated points, such as Nunavit and Saskatchewan, are not considered to share borders. There are also a limited number of colors, C.
Next, choose a correspondence or encoding between colors for a region and the qubit values. After fixing the encoding, work out the form of a quantum machine instruction (QMI) that will give valid
colorings. This task is broken into four steps:
1. Turn on one of several qubits.
2. Map a single region to a unit cell.
3. Implement constraints using couplers.
4. Clone neighbors to meet similar constraints.
Using unary encoding for the possible colors for each region, assign C qubits to each region of the map. If the ith color is assigned to a region, then the ith qubit (q[i]) associated with that
region will have the value 1 in our samples (or results from the quantum computer) and the other (C – 1) qubits associated with that region will have the value 0.
Download this article in .PDF format
This file type includes high resolution graphics and schematics when applicable.
Turn on one of C qubits
First, solve the simpler problem of turning on just one qubit in a two-qubit system. For a two-qubit system, the objective becomes:
O(a, b; q) = a[1]q[1]+ a[2]q[2]+ b[12]q[1]q[2]
This table shows the possible qubit states and the objective in a two-qubit system.
The four possible states in the distribution are listed in the Two-Qubit System Table. Choose a[1], a[2]and b[12 ]values so that the distribution consists of the states with q[1]= 0 and q[2]= 1 and
the one in which q[1]= 1 and q[2]= 0. The other two states in which q[i’]s equal 0 or 1 should not be in the distribution.
To make encodings of either color equally likely in the distribution, a[1] must equal a[2]. These values also need to be less than 0 so that the state characterized by q[1]= 0 and q[2]= 0 will not
appear. Therefore:
a[1 ]= a[2 ]< 0
To eliminate the state in which q[1]and q[2 ]equal 1 from the distribution requires that:
a[1]+ a[2]+ b[12 ]= 0
A solution to these equations is:
a[1]= −1, a[2]= −1, and b[12]= 2
Substituting those values for a[1], a[2]and b[12]into the Two-Qubit System Table shows that the objective value is minimized for the two states in which one q[i]value is 1 and the other is 0. This
means samples from the distribution generated by the QMI will consist solely of these two states.
These coefficient values would be enough if the map had only two colors, but most maps require more colors. Therefore we must generalize to the case where C qubits represent the possible colors
assigned to a region. The goal is to find values of a[i]and b[ij]coefficients that will yield a distribution over those samples that have exactly one qubit turned on (equal to 1) and the other C − 1
qubits turned off (equal to 0).
To solve this problem, take a clue from the two-qubit problem. In that problem, two states needed exactly one qubit turned on to be equally represented in the distribution and thus we set a[1]= a[2].
The solution is symmetric if the two qubits are interchanged, as expected.
Now apply this principle to the case with three colors, along with a corresponding number of qubits to simplify the constraints.
If C = 3, then there are three qubits. The corresponding objective function is:
O(a, b ; q) = a[1]q[1]+ a[2]q[2]+ a[3]q[3]+ b[12]q[1]q[2]+ b[13]q[1]q[3]+ b[23]q[2]q[3]
Simplify this objective by applying the insight about the symmetry of the solutions, and require the three a[i]values equal one common value, a. Similarly, require the three b[ij]values equal a
common value b:
O(a, b ; q) = a(q[1 ] + q[2]+ q[3]) + b(q[1]q[2]+ q[1]q[3]+ q[2]q[3])
This table shows the possible qubit states and the objective in a three-qubit system.
Tabulate the eight states of this system (see the Three-Qubit System Table).
Taking a hint from our previous example, observe that setting a = −1 and b = 2 will give the three states with exactly one qubit turned on an objective value of −1. Among the other five states, four
will have objective values equal to 0 and one will have its objective equal to 3. This guarantees samples will consist only of qubit patterns with one of the three qubits is equal to 1.
A quick check confirms that this same symmetry argument can be applied to problems with C = 4 or higher. In all these cases, the distribution is influenced to contain only qubit patterns with exactly
one qubit turned on by choosing a[i]= −1 for all the weights and b[ij]= 2 for all the strengths.
Map a single region to a unit cell
In these diagrams, each vertex represents a qubit and each edge represents a coupler between qubits.
To finish the first step of transforming the map-coloring problem to a QMI, C qubits were introduced for each region in the map and weights and strengths for the qubits and couplers were initialized.
The connectivity pattern of the qubits and couplers is represented in the figure Connectivity of Qubits and Couplers. It illustrates how each graph vertex represents a qubit and each edge represents
a coupler between qubits. The Unit Cell figure depicts a small portion of the pattern of physical qubits and couplers corresponding to the unit cell. It is easy to find many instances of the complete
graph on two vertices, but there are no instances of the complete graph on three or four (or more) vertices. This poses the next challenge.
This figure depicts a small portion of the pattern of physical qubits and couplers in the D-Wave System corresponding to the unit cell.
To solve this problem, make a distinction between logical qubits and couplers and physical qubits and couplers inside the quantum computer. Each logical qubit corresponds (via an embedding) to one or
more connected physical qubits, which is called a chain. To implement a coupler between logical qubits, it is enough to find a physical coupler connecting any physical qubit in the chain for the
first logical qubit to another physical qubit in the chain for the second logical qubit.
This strategy makes it easy to map complete graphs on up to four vertices to the unit cell in the computer. The chain for the first logical qubit corresponds to the top two physical qubits in the
unit cell. Likewise, the chain for the nth logical qubit corresponds to the two physical qubits in the nth row of the unit cell. It is easy to confirm there is a physical coupler between each pair of
chains, yielding a unit-cell embedding for complete graphs of up to four vertices.
To ensure the physical qubits within one chain faithfully represent a single logical qubit, we must define weights and strengths for the physical qubits within a chain that keeps them aligned.
By referring to the table of objective values for the two-qubit system, it is easy to see that assigning a[1]= 1, a[2]= 1, and b[12]= −2 give the aligned chains an objective of 0 and misaligned
chains an objective of 1. Note that the weight and strengths necessary to implement this desired set of states are negative versions of the weights and strengths used to solve the “1 of 2” problem.
To complete this step, supply a rule to map weights and strengths for logical qubits and couplers to the physical qubits and couplers that represent them. Each logical qubit is represented by a chain
of length two so we divide the weight of -1 for the logical qubit in half and apply a weight of -1/2 to each physical qubit in the chain.
Likewise, we specified a strength of 2 for logical couplers. Because the chains for each pair of logical qubits are connected by two physical couplers, we also divide the logical strength of 2 into
two physical strengths of 1 each in the unit cell.
Implement constraints using couplers
At this point, colors are encoded for each region using logical qubits, and the logical qubits and couplers are mapped to physical unit cells. Now we need to enforce the neighbor constraints. With
some foresight, we chose the unary encoding and logical-to-physical mapping to simplify this next step (and the following one, too).
The task is to adjust the weights and strengths so that when, for example, the red qubits for British Columbia and Alberta both turn on, it increases the value of the objective function. On the other
hand, the objective should stay constant when both these red qubits are turned off or when one or the other (but not both) is turned on.
Referring to the Two-Qubit System Table, assume q[1] refers to British Columbia’s red qubit and q[2]refers to Alberta’s red qubit. By setting a[1]= 0 and a[2]= 0 and b[12]= 1, the objective is lifted
in the fourth state, which we need to penalize, and leaves the other three states at an objective of 0.
To implement this penalty, it will take a physical coupler that connects British Columbia’s and Alberta’s red qubits. This requires a look at a slightly larger portion of the fabric of physical
qubits and couplers that make up the D-Wave computer.
This diagram shows two neighboring unit cells (British Columbia and Alberta) and highlights the chains in each cell representing four logical qubits, one for each color.
The figure Neighboring cells and chains shows two neighboring unit cells and highlights the chains in each unit cell that represent the four logical qubits, one for each color. Most importantly, for
the current step, this figure also represents physical couplers connecting two adjacent unit cells as arcs.
It is clear these couplers are ideally positioned to implement the portion of the objective that ensures that if the chain representing color i is turned on in one unit cell and the chain
representing color i in the neighboring unit cell is also turned on, it will penalize this state. So the strength associated with the four arc-shaped couplers should be set to 1.
Clone regions as necessary
This last step in mapping the coloring problem to the quantum-programming model is necessitated by neighbor relations in the map and connectivity of unit cells. To appreciate this problem, note that
British Columbia, Alberta, and the Northwest Territories are all neighbors. Also note that unit cells are configured in a two-dimensional checkerboard array. The configuration of these three regions
cannot be mapped to unit cells while preserving the neighbor relation.
We could assign British Columbia and Alberta to unit cells that neighbor each other horizontally (see Neighboring cells and chains) and assign the Northwest Territories to a unit cell positioned
vertically above British Columbia. This configuration means Alberta is not a direct neighbor of the Northwest Territory in the unit-cell array and, hence, no physical couplers are available to ensure
the same color qubits will not be simultaneously activated in these two regions.
Using clones can solve this problem, though. Clones are analogous to chains of physical qubits representing a single logical qubit. In this case, several unit cells represent the color of a single
region. Just as with chains of physical qubits, cloning extends the footprint of a single region in the unit-cell array to provide more neighbors for a cloned region. This lets us enforce neighbor
constraints arising from Canada’s map that do not transfer directly to the unit-cell array.
This table represents a mapping of Canadaâ s 13 regions to a portion of a unit-cell array. Rows and column indices are 0-based. Regions are labeled using standard two-letter postal codes.
The Map of Canada in Unit Cell Array maps Canada’s 13 regions to unit cells with Alberta (AB), British Columbia (BC), and the Northwest Territories (NT) all cloned. It is easily checked by
referencing the Map of Canada that each neighbor relation from the map of Canada corresponds to some adjacent pair of unit cells in the cell array.
To complete this step, the strengths for the intercell couplers must be adjusted so that the color assigned to Alberta in the unit cell in row 0 and column 4 matches the color assigned to Alberta in
the unit cell in row 1 and column 4. This problem is solved in exactly the same way as solving for chains of physical qubits representing a logical qubit. For the physical coupler connecting the red
qubits in Alberta’s top and bottom cells, adjust the strength of the coupler to -2 and add a weight of 1 to the two physical qubits connected through the coupler. Repeat this for each of the other
colors as well as for British Columbia and Northwest Territories, which have also been cloned.
These four steps transformed the problem of generating a valid coloring for the regions of Canada into a single QMI for a quantum computer. They followed naturally from the decision to represent
colorings via a unary encoding scheme, which requires 13C qubits to represent the possible C colorings of the 13 regions of Canada. (Implementing the steps in a conventional programming language such
as C is shown in an appendix available to those who request it via email to mdeditor with quantum n the subject line.)
Regardless of language, standard constructs generate the weights and strengths of the QMI. Special library routines are used to pass the QMI to the D-Wave computer and retrieve samples from the
resulting distribution.
Three conclusions can be drawn immediately from this exercise. First, mapping this problem to a QMI could be simplified via routines which create embeddings of a logical formulation to a physical
QMI. Second, the size of a map that can be colored using this strategy is limited by the number of unit cells available in the computer. A more scalable strategy would include the ability to divide
larger maps into chunks that can be handled individually. Results from several chunks could be synthesized to create the coloring of larger maps. Finally, the D- Wave unit cell is large enough to
handle four colors with this encoding, but the scheme needs to be modified to handle more general mapping problems that use more than four colors.
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VE475 Introduction to Cryptography Assignment 6 solved
Ex. 1 — Application of the the DLP
Bob wants to prove his identity to Alice. Alice knows that Bob can compute logα β in Z/pZ, where α is
a generator of the group Z/pZ, and p is a known prime. Unfortunately Bob is not willing to share the
result with her, so he offers to apply the following strategy.
(i) Bob generates a random integer r and sends γ = α
r mod p to Alice;
(ii) Upon receiving γ Alice randomly requests r or x + r mod (p − 1);
(iii) Bob replies accordingly;
We now want to study Bod’s idea.
1. In the previous protocol,
a) Why are r and x + r considered modulo (p − 1)?
b) Prove that neither Bob nor Alice can cheat, while Bob can successfully prove his identity.
2. How many times should this be repeated for a
a) 128 bits security level?
b) 256 bits security level?
3. What type of protocol is this?
Ex. 2 — Pohlig-Hellman
Search and explain in details how the Pohlig-Hellman algorithm computes the discrete logarithm of an
element in a multiplicative group whose order can be completely factorized into small primes. As an
example calculate log3 3344 in G = U(Z/24389Z), knowing that 3 is generator of G.
Ex. 3 — Elgamal
1. Prove that the polynomial X
3 + 2X
2 + 1 is irreducible over F3[x], and conclude that it defines
the field F3
3 , which has 27 elements.
2. Explain how to define a simple map from the set of the letters of the alphabet into F3
3 .
3. What is the order of the subgroup generated by X?
4. If we set the ecret key to be 11, determine the public key.
5. Encrypt the message “goodmorning”, and then decrypt the ciphertext.
Ex. 4 — Simple questions
1. Let n be the product two large primes, p and q. We define h(x) ≡ x
2 mod n. Is h (i) pre-image
resistant, (ii) second pre-image resistant, and (iii) collision resistant?
2. Supposed a message m is divided into blocks of 160 bits: m = m1km2k · · · kml
. Which properties
of a hash function does the function h(m) = m1 ⊕ m2 ⊕ · · · ⊕ ml verify?
Ex. 5 — Merkle-Damg˚ard construction
The Merkle-Damg˚ard construction provided in the slides is only valid when t ≤ 2, therefore we now use
the same notations as in the slides to provide an alternative construction for t = 1.
Let g be a compression function from {0, 1}
m+1 −→ {0, 1}
m, and f be the function defined by f (0) = 0
and f (1) = 01. The map from x to y is defined by y = 11kf (x1)kf (x2)k … kf (x|x|), where xi represents
the i-th bit of x. Assuming |y| = k, compute
z1 = g(0mky1)
zi+1 = g(zikyi+1), 1 ≤ i ≤ k − 1,
and define h(x) as zk .
1. Check that
a) The map s from x to y is injective.
b) There is no strings x 6= x
0 and z such that s(x) = zks(x
2. Explain why the two previous conditions are of a major importance.
3. Following a similar strategy as in the case t ≥ 2, prove that h is a collision resistant hash function.
Ex. 6 — Programming
Implement the Pollard-rho factorization algorithm.
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This tutorial is generated from a Jupyter notebook that can be found here.
In practice inference problems often have a complicated and computationally heavy simulator, and one simply cannot run it for millions of times. The Bayesian Optimization for Likelihood-Free
Inference BOLFI framework is likely to prove useful in such situation: a statistical model (usually Gaussian process, GP) is created for the discrepancy, and its minimum is inferred with Bayesian
optimization. This approach typically reduces the number of required simulator calls by several orders of magnitude.
This tutorial demonstrates how to use BOLFI to do LFI in ELFI.
import numpy as np
import scipy.stats
import matplotlib
import matplotlib.pyplot as plt
%matplotlib inline
%precision 2
import logging
# Set an arbitrary global seed to keep the randomly generated quantities the same
seed = 1
import elfi
Although BOLFI is best used with complicated simulators, for demonstration purposes we will use the familiar MA2 model introduced in the basic tutorial, and load it from ready-made examples:
from elfi.examples import ma2
model = ma2.get_model(seed_obs=seed)
Fitting the surrogate model
Now we can immediately proceed with the inference. However, when dealing with a Gaussian process, it may be beneficial to take a logarithm of the discrepancies in order to reduce the effect that high
discrepancies have on the GP. (Sometimes you may want to add a small constant to avoid very negative or even -Inf distances occurring especially if it is likely that there can be exact matches
between simulated and observed data.) In ELFI such transformed node can be created easily:
log_d = elfi.Operation(np.log, model['d'])
As BOLFI is a more advanced inference method, its interface is also a bit more involved as compared to for example rejection sampling. But not much: Using the same graphical model as earlier, the
inference could begin by defining a Gaussian process (GP) model, for which ELFI uses the GPy library. This could be given as an elfi.GPyRegression object via the keyword argument target_model. In
this case, we are happy with the default that ELFI creates for us when we just give it each parameter some bounds as a dictionary.
Other notable arguments include the initial_evidence, which gives the number of initialization points sampled straight from the priors before starting to optimize the acquisition of points,
update_interval which defines how often the GP hyperparameters are optimized, and acq_noise_var which defines the diagonal covariance of noise added to the acquired points. Note that in general BOLFI
does not benefit from a batch_size higher than one, since the acquisition surface is updated after each batch (especially so if the noise is 0!).
bolfi = elfi.BOLFI(log_d, batch_size=1, initial_evidence=20, update_interval=10,
bounds={'t1':(-2, 2), 't2':(-1, 1)}, acq_noise_var={'t1':0.1, 't2':0.1}, seed=seed)
Sometimes you may have some samples readily available. You could then initialize the GP model with a dictionary of previous results by giving initial_evidence=result.outputs.
The BOLFI class can now try to fit the surrogate model (the GP) to the relationship between parameter values and the resulting discrepancies. We’ll request only 100 evidence points (including the
initial_evidence defined above).
%time post = bolfi.fit(n_evidence=200)
INFO:elfi.methods.parameter_inference:BOLFI: Fitting the surrogate model...
INFO:elfi.methods.posteriors:Using optimized minimum value (-1.6146) of the GP discrepancy mean function as a threshold
CPU times: user 1min 48s, sys: 1.29 s, total: 1min 50s
Wall time: 1min
(More on the returned BolfiPosterior object below.)
Note that in spite of the very few simulator runs, fitting the model took longer than any of the previous methods. Indeed, BOLFI is intended for scenarios where the simulator takes a lot of time to
The fitted target_model uses the GPy library, and can be investigated further:
Name : GP regression
Objective : 151.86636065302943
Number of Parameters : 4
Number of Optimization Parameters : 4
Updates : True
[1mGP_regression. [0;0m | value | constraints | priors
[1msum.rbf.variance [0;0m | 0.321697451372 | +ve | Ga(0.024, 1)
[1msum.rbf.lengthscale [0;0m | 0.541352150083 | +ve | Ga(1.3, 1)
[1msum.bias.variance [0;0m | 0.021827430988 | +ve | Ga(0.006, 1)
[1mGaussian_noise.variance[0;0m | 0.183562040169 | +ve |
<matplotlib.figure.Figure at 0x11b2b2ba8>
It may be useful to see the acquired parameter values and the resulting discrepancies:
There could be an unnecessarily high number of points at parameter bounds. These could probably be decreased by lowering the covariance of the noise added to acquired points, defined by the optional
acq_noise_var argument for the BOLFI constructor. Another possibility could be to add virtual derivative observations at the borders, though not yet implemented in ELFI.
BOLFI Posterior
Above, the fit method returned a BolfiPosterior object representing a BOLFI posterior (please see the paper for details). The fit method accepts a threshold parameter; if none is given, ELFI will use
the minimum value of discrepancy estimate mean. Afterwards, one may request for a posterior with a different threshold:
post2 = bolfi.extract_posterior(-1.)
One can visualize a posterior directly (remember that the priors form a triangle):
Finally, samples from the posterior can be acquired with an MCMC sampler. By default it runs 4 chains, and half of the requested samples are spent in adaptation/warmup. Note that depending on the
smoothness of the GP approximation, the number of priors, their gradients etc., this may be slow.
%time result_BOLFI = bolfi.sample(1000, info_freq=1000)
INFO:elfi.methods.posteriors:Using optimized minimum value (-1.6146) of the GP discrepancy mean function as a threshold
INFO:elfi.methods.mcmc:NUTS: Performing 1000 iterations with 500 adaptation steps.
INFO:elfi.methods.mcmc:NUTS: Adaptation/warmup finished. Sampling...
INFO:elfi.methods.mcmc:NUTS: Acceptance ratio: 0.423. After warmup 68 proposals were outside of the region allowed by priors and rejected, decreasing acceptance ratio.
INFO:elfi.methods.mcmc:NUTS: Performing 1000 iterations with 500 adaptation steps.
INFO:elfi.methods.mcmc:NUTS: Adaptation/warmup finished. Sampling...
INFO:elfi.methods.mcmc:NUTS: Acceptance ratio: 0.422. After warmup 71 proposals were outside of the region allowed by priors and rejected, decreasing acceptance ratio.
INFO:elfi.methods.mcmc:NUTS: Performing 1000 iterations with 500 adaptation steps.
INFO:elfi.methods.mcmc:NUTS: Adaptation/warmup finished. Sampling...
INFO:elfi.methods.mcmc:NUTS: Acceptance ratio: 0.419. After warmup 65 proposals were outside of the region allowed by priors and rejected, decreasing acceptance ratio.
INFO:elfi.methods.mcmc:NUTS: Performing 1000 iterations with 500 adaptation steps.
INFO:elfi.methods.mcmc:NUTS: Adaptation/warmup finished. Sampling...
INFO:elfi.methods.mcmc:NUTS: Acceptance ratio: 0.439. After warmup 66 proposals were outside of the region allowed by priors and rejected, decreasing acceptance ratio.
4 chains of 1000 iterations acquired. Effective sample size and Rhat for each parameter:
t1 2222.1197791 1.00106816947
t2 2256.93599184 1.0003364409
CPU times: user 1min 45s, sys: 1.29 s, total: 1min 47s
Wall time: 55.1 s
The sampling algorithms may be fine-tuned with some parameters. The default No-U-Turn-Sampler is a sophisticated algorithm, and in some cases one may get warnings about diverged proposals, which are
signs that something may be wrong and should be investigated. It is good to understand the cause of these warnings although they don’t automatically mean that the results are unreliable. You could
try rerunning the sample method with a higher target probability target_prob during adaptation, as its default 0.6 may be inadequate for a non-smooth posteriors, but this will slow down the sampling.
Note also that since MCMC proposals outside the region allowed by either the model priors or GP bounds are rejected, a tight domain may lead to suboptimal overall acceptance ratio. In our MA2 case
the prior defines a triangle-shaped uniform support for the posterior, making it a good example of a difficult model for the NUTS algorithm.
Now we finally have a Sample object again, which has several convenience methods:
Method: BOLFI
Number of samples: 2000
Number of simulations: 200
Threshold: -1.61
Sample means: t1: 0.429, t2: 0.0277
The black vertical lines indicate the end of warmup, which by default is half of the number of iterations.
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binary radian to diameter part
angle units conversion
Amount: 1 binary radian (brad) of angle
Equals: 1.47 diameter parts (Ø dia- part) in angle
Converting binary radian to diameter parts value in the angle units scale.
TOGGLE : from diameter parts into binary radians in the other way around.
CONVERT : between other angle measuring units - complete list.
How many diameter parts are in 1 binary radian? The answer is: 1 brad equals 1.47 Ø dia- part
1.47 Ø dia- part is converted to 1 of what?
The diameter parts unit number 1.47 Ø dia- part converts to 1 brad, one binary radian. It is the EQUAL angle value of 1 binary radian but in the diameter parts angle unit alternative.
brad/Ø dia- part angle conversion result
From Symbol Equals Result Symbol
1 brad = 1.47 Ø dia- part
Conversion chart - binary radians to diameter parts
1 binary radian to diameter parts = 1.47 Ø dia- part
2 binary radians to diameter parts = 2.95 Ø dia- part
3 binary radians to diameter parts = 4.42 Ø dia- part
4 binary radians to diameter parts = 5.89 Ø dia- part
5 binary radians to diameter parts = 7.36 Ø dia- part
6 binary radians to diameter parts = 8.84 Ø dia- part
7 binary radians to diameter parts = 10.31 Ø dia- part
8 binary radians to diameter parts = 11.78 Ø dia- part
9 binary radians to diameter parts = 13.25 Ø dia- part
10 binary radians to diameter parts = 14.73 Ø dia- part
11 binary radians to diameter parts = 16.20 Ø dia- part
12 binary radians to diameter parts = 17.67 Ø dia- part
13 binary radians to diameter parts = 19.14 Ø dia- part
14 binary radians to diameter parts = 20.62 Ø dia- part
15 binary radians to diameter parts = 22.09 Ø dia- part
Category: main menu • angle menu • Binary radians
Convert angle of binary radian (brad) and diameter parts (Ø dia- part) units in reverse from diameter parts into binary radians.
This calculator is based on conversion of two angle units. An angle consists of two rays (as in sides of an angle sharing a common vertex or else called the endpoint.) Some belong to rotation
measurements - spherical angles measured by arcs' lengths, pointing from the center, plus the radius. For a whole set of multiple units of angle on one page, try that Multiunit converter tool which
has built in all angle unit-variations. Page with individual angle units.
Converter type: angle units
First unit: binary radian (brad) is used for measuring angle.
Second: diameter part (Ø dia- part) is unit of angle.
15 brad = ? Ø dia- part
15 brad = 22.09 Ø dia- part
Abbreviation, or prefix, for binary radian is:
Abbreviation for diameter part is:
Ø dia- part
Other applications for this angle calculator ...
With the above mentioned two-units calculating service it provides, this angle converter proved to be useful also as a teaching tool:
1. in practicing binary radians and diameter parts ( brad vs. Ø dia- part ) measures exchange.
2. for conversion factors between unit pairs.
3. work with angle's values and properties.
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Probability of error in bandlimited spread spectrum binary communication systems
The performance of a pseudonoise-modulated spread spectrum binary communication system is analyzed to determine the probability of error in the presence of bandlimiting, Gaussian noise, and
synchronization errors. It is shown that the correlation receiver is an optimum receiver in the presence of white gaussian noise with no bandlimiting. Synchronization errors are introduced into the
analysis in the form of a timing delay between the received code and local code. Bandlimiting is modeled by a causal low-pass filter at the input of the correlation receiver. Probability of error in
detecting the binary message symbols is analyzed first by developing upper bounds based on the correlation properties of maximal length pseudonoise sequences and the decaying response typical of
causal filters. Then, the average probability of error is computed using a series expansion of the characteristic function of the intersymbol interference. Results show that for large spreading
ratios, the effects of inter-symbol interference are negligible. Synchronization errors in addition to bandlimiting further degrade the performance, but less dramatically than in the case of the
infinite-bandwidth system.
Ph.D. Thesis
Pub Date:
December 1975
□ Digital Systems;
□ Error Analysis;
□ Probability Theory;
□ Telecommunication;
□ Pulse Modulation;
□ Random Noise;
□ Synchronism;
□ Time Lag;
□ Communications and Radar
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Online Basic Trigonometry Midterm Exam
The questions for the exam were as follows:
Online Trigonometry Midterm Exam Name:
Directions: Show a complete solution to each problem in the space provided. You may use a calculator and your book on this exam. You have 2 hours to complete this test.
Define the complement and supplement of an angle. Find the complement and supplement of the angle 18°.
Angles are generally measured in either degrees or radians. Explain the difference between 2° and 2 radians. Which is larger?
Use the Unit Circle to find the following values:
cos p/4=
sin 4p/3=
tan p/2=
Use your calculator to find the following values: (Hint: Make sure you are set in the proper mode for each problem – degrees or radians!)
Find the value of x for the diagram shown on page 161, number 64 of the textbook.
Sketch a right triangle corresponding to ?=3/4 , where ? is an acute angle, and find the value of the other five trigonometric functions for ?.
Use the Circular Functions to find the exact value of the six trigonometric functions for the angle ? which passes through the point (7, -9).
Determine the quadrant where the angle ? lies if cos?<0 and sin???>0?.
Use the Unit Circle to find two values of ? (in degrees) which satisfy ?=1/2 .
Use your calculator to find 2 different angles between 0^° and ?360?^° that satisfy ?=0.4565 .
List one basic identity that we have studied in this course and justify why it is true.
Suppose that ? is an angle lying in quadrant IV with ?=3/5 . Use the identity
?cos?^2 ??+sin?^2 ?=1 to find the value of sin?.
Suppose that a 40 foot ladder leans against the side of a building. Find the distance h, up the side of the building if the angle of elevation of the ladder is 52° .
Suppose that a surveyor wishes to find the distance across a river. Use the information in the diagram on page 162, problem #70 to find the distance w.
Suppose that you are standing due west of the Eiffel Tower at an unknown distance and, at that position, the angle of elevation to the top of the tower is 54° . Next you walk 100 meters due west of
your first position and calculate the new angle of elevation to be 44°. Use this information to determine the height of the Eiffel Tower. You must show all of your work to receive full credit.
About the Solutions The answers for the first question are reproduced below. If you wish to see the solutions for the remaining questions, you can purchase the full solutions.
Define the complement and supplement of an angle. Find the complement and supplement of the angle 18°.
The complement of an angle x (0° < x < 90°) is the angle whose measure is 90° - x. For example, the complement of ?40?^0 is ?50?^0.
The supplement of an angle x (0° < x < 180°) is the angle whose measure is 180° - x. For example, the supplement of ?40?^0 is ?140?^0.
The complement of the angle 180 is equal to ?90?^0-?18?^0=?72?^0
The supplement of the angle 180 is equal to ?180?^0-?18?^0=?162?^0
Other Details about the Project/Assignment Subjects
Mathematics -> Trigonometry Topic
Trigonometry Midterm Exam Level
College / University Tags
Online Exam, Trigonometry, Midterm
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Modeling a Vehicle Dynamics System
This example shows nonlinear grey-box modeling of vehicle dynamics. Many new vehicle features (like Electronic Stability Programs (ESP), indirect Tire Pressure Monitoring Systems (TPMS), road-tire
friction monitoring systems, and so forth) rely on models of the underlying vehicle dynamics. The so-called bicycle vehicle model is a rather simple model structure that is frequently being used in
the vehicle dynamics literature. In this example we will start-off with this model structure and try to estimate the longitudinal and the lateral stiffness of a tire. The actual modeling work was
originally carried out by Erik Narby in his MSc work at NIRA Dynamics AB, Sweden.
Vehicle Dynamics Modeling
The following figure illustrates the vehicle modeling situation to be considered.
Figure 1: Schematic view of a vehicle dynamics system.
By the use of Newton's law of motion and some basic geometric relationships, the longitudinal velocity v_x(t), the lateral velocity v_y(t) and the yaw rate r(t) measured around the Center Of Gravity
(COG) of the vehicle can be described by the following three differential equations:
-- v_x(t) = v_y(t)*r(t) + 1/m*( (F_x_FL(t) + F_x_FR(t))*cos(delta(t))
dt - (F_y_FL(t) + F_y_FR(t))*sin(delta(t))
+ F_x_RL(t) + F_x_RR(t)
- C_A*v_x(t)^2)
-- v_y(t) = -v_x(t)*r(t) + 1/m*( (F_x_FL(t) + F_x_FR(t))*sin(delta(t))
dt + (F_y_FL(t) + F_y_FR(t))*cos(delta(t))
+ F_y_RL(t) + F_y_RR(t))
-- r(t) = 1/J*( a*( (F_x_FL(t) + F_x_FR(t))*sin(delta(t))
dt + (F_y_FL(t) + F_y_FR(t))*cos(delta(t)))
- b*(F_y_RL(t) + F_y_RR(t)))
where subscript x is used to denote that a force F acts in the longitudinal direction and y that it acts in the lateral direction. The abbreviations FL, FR, RL and RR label the tires: Front Left,
Front Right, Rear Left and Rear Right, respectively. The first equation describing the longitudinal acceleration also contains an air resistance term that is assumed to be a quadratic function of the
longitudinal vehicle velocity v_x(t). In addition, delta(t) (an input) is the steering angle, J a moment of inertia, and a and b the distances from the center of gravity to the front and rear axles,
Let us assume that the tire forces can be modeled through the following linear approximations:
F_x_i(t) = C_x*s_i(t)
F_y_i(t) = C_y*alpha_i(t) for i = {FL, FR, RL, RR}
where C_x and C_y are the longitudinal and lateral tire stiffness, respectively. Here we have assumed that these stiffness parameters are the same for all 4 tires. s_i(t) is the so-called
(longitudinal) slip of tire i and alpha_i(t) a tire slip angle. For a front-wheel driven vehicle (as considered here), the slips s_FL(t) and s_FR(t) are derived from the individual wheel speeds
(measured) by assuming that the rear wheels do not show any slip (i.e., s_RL(t) = s_RR(t) = 0). Hence the slips are inputs to our model structure. For the front wheels, the tire slip angles alpha_Fj
(t) can be approximated by (when v_x(t) > 0)
alpha_Fj(t) = delta(t) - arctan((v_y(t) + a*r(t))/v_x(t))
~ delta(t) - (v_y(t) + a*r(t))/v_x(t) for j = {L, R}
For the rear wheels, the tire slip angles alpha_Rj(t) are similarly derived and computed as
alpha_Rj(t) = - arctan((v_y(t) - b*r(t))/v_x(t))
~ - (v_y(t) - b*r(t))/v_x(t) for j = {L, R}
With J = 1/((0.5*(a+b))^2*m) we can next set up a state-space structure describing the vehicle dynamics. Introduce the states:
x1(t) = v_x(t) Longitudinal velocity [m/s].
x2(t) = v_y(t) Lateral velocity [m/s].
x3(t) = r(t) Yaw rate [rad/s].
the five measured or derived input signals
u1(t) = s_FL(t) Slip of Front Left tire [ratio].
u2(t) = s_FR(t) Slip of Front Right tire [ratio].
u3(t) = s_RL(t) Slip of Rear Left tire [ratio].
u4(t) = s_RR(t) Slip of Rear Right tire [ratio].
u5(t) = delta(t) Steering angle [rad].
and the model parameters:
m Mass of the vehicle [kg].
a Distance from front axle to COG [m].
b Distance from rear axle to COG [m].
Cx Longitudinal tire stiffness [N].
Cy Lateral tire stiffness [N/rad].
CA Air resistance coefficient [1/m].
The outputs of the system are the longitudinal vehicle velocity y1(t) = x1(t), the lateral vehicle acceleration (measured by an accelerometer):
y2(t) = a_y(t) = 1/m*( (F_x_FL(t) + F_x_FR(t))*sin(delta(t))
+ (F_y_FL(t) + F_y_FR(t))*cos(delta(t))
+ F_y_RL(t) + F_y_RR(t))
and the yaw rate y3(t) = r(t) (measured by a gyro).
Put together, we arrive at the following state-space model structure:
-- x1(t) = x2(t)*x3(t) + 1/m*( Cx*(u1(t)+u2(t))*cos(u5(t))
dt - 2*Cy*(u5(t)-(x2(t)+a*x3(t))/x1(t))*sin(u5(t))
+ Cx*(u3(t)+u4(t))
- CA*x1(t)^2)
-- x2(t) = -x1(t)*x3(t) + 1/m*( Cx*(u1(t)+u2(t))*sin(u5(t))
dt + 2*Cy*(u5(t)-(x2(t)+a*x3(t))/x1(t))*cos(u5(t))
+ 2*Cy*(b*x3(t)-x2(t))/x1(t))
-- x3(t) = 1/((0.5*(a+b))^2)*m)*( a*( Cx*(u1(t)+u2(t)*sin(u5(t))
dt + 2*Cy*(u5(t) - (x2(t)+a*x3(t))/x1(t))*cos(u5(t)))
- 2*b*Cy*(b*x3(t)-x2(t))/x1(t))
y1(t) = x1(t)
y2(t) = 1/m*( Cx*(u1(t)+u2(t))*sin(u5(t))
+ 2*Cy*(u5(t)-(x2(t)+a*x3(t))/x1(t))*cos(u5(t))
+ 2*Cy*(b*x3(t)-x2(t))/x1(t))
y3(t) = x3(t)
IDNLGREY Vehicle Model
As a basis for our vehicle identification experiments we first need to create an IDNLGREY model file describing these vehicle equations. Here we rely on C-MEX modeling and create a vehicle_c.c model
file, in which NY is set to 3. The state and output update functions of vehicle_c.c, compute_dx and compute_y, are somewhat involved and includes several standard C-defined mathematical functions,
like cos(.) and sin(.) as well as pow(.) for computing the power of its argument.
The state update function compute_dx returns dx (argument 1) and uses 3 input arguments: the state vector x, the input vector u, and the six scalar parameters encoded in p (t and auxvar of the
template C-MEX model file have been removed here):
/* State equations. */
void compute_dx(double *dx, double *x, double *u, double **p)
/* Retrieve model parameters. */
double *m, *a, *b, *Cx, *Cy, *CA;
m = p[0]; /* Vehicle mass. */
a = p[1]; /* Distance from front axle to COG. */
b = p[2]; /* Distance from rear axle to COG. */
Cx = p[3]; /* Longitudinal tire stiffness. */
Cy = p[4]; /* Lateral tire stiffness. */
CA = p[5]; /* Air resistance coefficient. */
/* x[0]: Longitudinal vehicle velocity. */
/* x[1]: Lateral vehicle velocity. */
/* x[2]: Yaw rate. */
dx[0] = x[1]*x[2]+1/m[0]*(Cx[0]*(u[0]+u[1])*cos(u[4])
dx[1] = -x[0]*x[2]+1/m[0]*(Cx[0]*(u[0]+u[1])*sin(u[4])
dx[2] = 1/(pow(((a[0]+b[0])/2),2)*m[0])
The output update function compute_y returns y (argument 1) and uses 3 input arguments: the state vector x, the input vector u, and five of the six parameters (the air resistance CA is not needed)
encoded in p:
/* Output equations. */
void compute_y(double *y, double *x, double *u, double **p)
/* Retrieve model parameters. */
double *m = p[0]; /* Vehicle mass. */
double *a = p[1]; /* Distance from front axle to COG. */
double *b = p[2]; /* Distance from rear axle to COG. */
double *Cx = p[3]; /* Longitudinal tire stiffness. */
double *Cy = p[4]; /* Lateral tire stiffness. */
/* y[0]: Longitudinal vehicle velocity. */
/* y[1]: Lateral vehicle acceleration. */
/* y[2]: Yaw rate. */
y[0] = x[0];
y[1] = 1/m[0]*(Cx[0]*(u[0]+u[1])*sin(u[4])
y[2] = x[2];
Having a proper model structure file, the next step is to create an IDNLGREY object reflecting the modeling situation. For ease of bookkeeping, we also specify the names and units of the inputs and
FileName = 'vehicle_c'; % File describing the model structure.
Order = [3 5 3]; % Model orders [ny nx nu].
Parameters = [1700; 1.5; 1.5; 1.5e5; 4e4; 0.5]; % Initial parameters.
InitialStates = [1; 0; 0]; % Initial value of initial states.
Ts = 0; % Time-continuous system.
nlgr = idnlgrey(FileName, Order, Parameters, InitialStates, Ts, ...
'Name', 'Bicycle vehicle model', 'TimeUnit', 's');
nlgr.InputName = {'Slip on front left tire'; ... % u(1).
'Slip on front right tire'; ... % u(2).
'Slip on rear left tire'; ... % u(3).
'Slip on rear right tire'; ... % u(4).
'Steering angle'}; ... % u(5).
nlgr.InputUnit = {'ratio'; 'ratio'; 'ratio'; 'ratio'; 'rad'};
nlgr.OutputName = {'Long. velocity'; ... % y(1); Longitudinal vehicle velocity
'Lat. accel.'; ... % y(2); Lateral vehicle acceleration
'Yaw rate'}; ... % y(3).
nlgr.OutputUnit = {'m/s'; 'm/s^2'; 'rad/s'};
The names and the units of the (initial) states and the model parameters are specified via SETINIT. We also use this command to specify that the first initial state (the longitudinal velocity) ought
to be strictly positive for the model to be valid and to specify that all model parameters should be strictly positive. These constraints will subsequently be honored when performing initial state
and/or model parameter estimation.
nlgr = setinit(nlgr, 'Name', {'Longitudinal vehicle velocity' ... % x(1).
'Lateral vehicle velocity' ... % x(2).
'Yaw rate'}); ... % x(3).
nlgr = setinit(nlgr, 'Unit', {'m/s'; 'm/s'; 'rad/s'});
nlgr.InitialStates(1).Minimum = eps(0); % Longitudinal velocity > 0 for the model to be valid.
nlgr = setpar(nlgr, 'Name', {'Vehicle mass'; ... % m.
'Distance from front axle to COG'; ... % a
'Distance from rear axle to COG'; ... % b.
'Longitudinal tire stiffness'; ... % Cx.
'Lateral tire stiffness'; ... % Cy.
'Air resistance coefficient'}); ... % CA.
nlgr = setpar(nlgr, 'Unit', {'kg'; 'm'; 'm'; 'N'; 'N/rad'; '1/m'});
nlgr = setpar(nlgr, 'Minimum', num2cell(eps(0)*ones(6, 1))); % All parameters > 0!
Four of the six parameters of this model structure can readily be obtained through the data sheet of the vehicle in question:
m = 1700 kg
a = 1.5 m
b = 1.5 m
CA = 0.5 or 0.7 1/m (see below)
Hence we will not estimate these parameters:
nlgr.Parameters(1).Fixed = true;
nlgr.Parameters(2).Fixed = true;
nlgr.Parameters(3).Fixed = true;
nlgr.Parameters(6).Fixed = true;
With this, a textual summary of the entered IDNLGREY model structure is obtained through PRESENT as follows.
nlgr =
Continuous-time nonlinear grey-box model defined by 'vehicle_c' (MEX-file):
dx/dt = F(t, x(t), u(t), p1, ..., p6)
y(t) = H(t, x(t), u(t), p1, ..., p6) + e(t)
with 5 input(s), 3 state(s), 3 output(s), and 2 free parameter(s) (out of 6).
u(1) Slip on front left tire(t) [ratio]
u(2) Slip on front right tire(t) [ratio]
u(3) Slip on rear left tire(t) [ratio]
u(4) Slip on rear right tire(t) [ratio]
u(5) Steering angle(t) [rad]
States: Initial value
x(1) Longitudinal vehicle velocity(t) [m/s] xinit@exp1 1 (fixed) in ]0, Inf]
x(2) Lateral vehicle velocity(t) [m/s] xinit@exp1 0 (fixed) in [-Inf, Inf]
x(3) Yaw rate(t) [rad/s] xinit@exp1 0 (fixed) in [-Inf, Inf]
y(1) Long. velocity(t) [m/s]
y(2) Lat. accel.(t) [m/s^2]
y(3) Yaw rate(t) [rad/s]
Parameters: Value
p1 Vehicle mass [kg] 1700 (fixed) in ]0, Inf]
p2 Distance from front axle to COG [m] 1.5 (fixed) in ]0, Inf]
p3 Distance from rear axle to COG [m] 1.5 (fixed) in ]0, Inf]
p4 Longitudinal tire stiffness [N] 150000 (estimated) in ]0, Inf]
p5 Lateral tire stiffness [N/rad] 40000 (estimated) in ]0, Inf]
p6 Air resistance coefficient [1/m] 0.5 (fixed) in ]0, Inf]
Name: Bicycle vehicle model
Created by direct construction or transformation. Not estimated.
More information in model's "Report" property.
Input-Output Data
At this point, we load the available input-output data. This file contains data from three different experiments:
A. Simulated data with high stiffness tires [y1 u1].
B. Simulated data with low stiffness tires [y2 u2].
C. Measured data from a Volvo V70 [y3 u3].
In all cases, the sample time Ts = 0.1 seconds.
A. System Identification Using Simulated High Tire Stiffness Data
In our first vehicle identification experiment we consider simulated high tire stiffness data. A copy of the model structure nlgr and an IDDATA object z1 reflecting this particular modeling situation
is first created. The 5 input signals are stored in u1 and the 3 output signals in y1. The slip inputs (generated from the wheel speed signals) for the front wheels were chosen to be sinusoidal with
a constant offset; the yaw rate was also sinusoidal but with a different amplitude and frequency. In reality, this is a somewhat artificial situation, because one rarely excites the vehicle so much
in the lateral direction.
nlgr1 = nlgr;
nlgr1.Name = 'Bicycle vehicle model with high tire stiffness';
z1 = iddata(y1, u1, 0.1, 'Name', 'Simulated high tire stiffness vehicle data');
z1.InputName = nlgr1.InputName;
z1.InputUnit = nlgr1.InputUnit;
z1.OutputName = nlgr1.OutputName;
z1.OutputUnit = nlgr1.OutputUnit;
z1.Tstart = 0;
z1.TimeUnit = 's';
The inputs and outputs are shown in two plot figures.
h_gcf = gcf;
h_gcf.Position = [100 100 795 634];
for i = 1:z1.Nu
subplot(z1.Nu, 1, i);
plot(z1.SamplingInstants, z1.InputData(:,i));
title(['Input #' num2str(i) ': ' z1.InputName{i}]);
axis tight;
xlabel([z1.Domain ' (' z1.TimeUnit ')']);
Figure 2: Inputs to a vehicle system with high tire stiffness.
for i = 1:z1.Ny
subplot(z1.Ny, 1, i);
plot(z1.SamplingInstants, z1.OutputData(:,i));
title(['Output #' num2str(i) ': ' z1.OutputName{i}]);
axis tight;
xlabel([z1.Domain ' (' z1.TimeUnit ')']);
Figure 3: Outputs from a vehicle system with high tire stiffness.
The next step is to investigate the performance of the initial model and for this we perform a simulation. Notice that the initial state has been fixed to a non-zero value as the first state (the
longitudinal vehicle velocity) is used as denominator in the model structure. A comparison between the true and the simulated outputs (with the initial model) is shown in a plot window.
compare(z1, nlgr1, [], compareOptions('InitialCondition', 'model'));
Figure 4: Comparison between true outputs and the simulated outputs of the initial vehicle model with high tire stiffness.
In order to improve the model fit, the two tire stiffness parameters Cx and Cy are next estimated, and a new simulation with the estimated model is carried out.
nlgr1 = nlgreyest(z1, nlgr1);
A comparison between the true and the simulated outputs (with the estimated model) is shown in a plot window.
compare(z1, nlgr1, [], compareOptions('InitialCondition', 'model'));
Figure 5: Comparison between true outputs and the simulated outputs of the estimated vehicle model with high tire stiffness.
The simulation performance of the estimated model is quite good. The estimated stiffness parameters are also close to the ones used in Simulink® to generate the true output data:
disp(' True Estimated');
fprintf('Longitudinal stiffness: %6.0f %6.0f\n', 2e5, nlgr1.Parameters(4).Value);
fprintf('Lateral stiffness : %6.0f %6.0f\n', 5e4, nlgr1.Parameters(5).Value);
True Estimated
Longitudinal stiffness: 200000 198517
Lateral stiffness : 50000 53752
B. System Identification Using Simulated Low Tire Stiffness Data
In the second experiment we repeat the modeling from the first experiment, but now with simulated low tire stiffness data.
nlgr2 = nlgr;
nlgr2.Name = 'Bicycle vehicle model with low tire stiffness';
z2 = iddata(y2, u2, 0.1, 'Name', 'Simulated low tire stiffness vehicle data');
z2.InputName = nlgr2.InputName;
z2.InputUnit = nlgr2.InputUnit;
z2.OutputName = nlgr2.OutputName;
z2.OutputUnit = nlgr2.OutputUnit;
z2.Tstart = 0;
z2.TimeUnit = 's';
The inputs and outputs are shown in two plot figures.
for i = 1:z2.Nu
subplot(z2.Nu, 1, i);
plot(z2.SamplingInstants, z2.InputData(:,i));
title(['Input #' num2str(i) ': ' z2.InputName{i}]);
axis tight;
xlabel([z2.Domain ' (' z2.TimeUnit ')']);
Figure 6: Inputs to a vehicle system with low tire stiffness.
for i = 1:z2.Ny
subplot(z2.Ny, 1, i);
plot(z2.SamplingInstants, z2.OutputData(:,i));
title(['Output #' num2str(i) ': ' z2.OutputName{i}]);
axis tight;
xlabel([z2.Domain ' (' z2.TimeUnit ')']);
Figure 7: Outputs from a vehicle system with low tire stiffness.
Next we investigate the performance of the initial model (which has the same parameters as the initial high tire stiffness model). A comparison between the true and the simulated outputs (with the
initial model) is shown in a plot window.
compare(z2, nlgr2, [], compareOptions('InitialCondition', 'model'));
Figure 8: Comparison between true outputs and the simulated outputs of the initial vehicle model with low tire stiffness.
The two stiffness parameters are next estimated.
nlgr2 = nlgreyest(z2, nlgr2);
A comparison between the true and the simulated outputs (with the estimated model) is shown in a plot window.
compare(z2, nlgr2, [], compareOptions('InitialCondition', 'model'));
Figure 9: Comparison between true outputs and the simulated outputs of the estimated vehicle model with low tire stiffness.
The simulation performance of the estimated model is again really good. Even with the same parameter starting point as was used in the high tire stiffness case, the estimated stiffness parameters are
also here close to the ones used in Simulink to generate the true output data:
disp(' True Estimated');
fprintf('Longitudinal stiffness: %6.0f %6.0f\n', 1e5, nlgr2.Parameters(4).Value);
fprintf('Lateral stiffness : %6.0f %6.0f\n', 2.5e4, nlgr2.Parameters(5).Value);
True Estimated
Longitudinal stiffness: 100000 99573
Lateral stiffness : 25000 26117
C. System Identification Using Measured Volvo V70 Data
In the final experiment we consider data collected in a Volvo V70. As above, we make a copy of the generic vehicle model object nlgr and create a new IDDATA object containing the measured data. Here
we have also increased the air resistance coefficient from 0.50 to 0.70 to better reflect the Volvo V70 situation.
nlgr3 = nlgr;
nlgr3.Name = 'Volvo V70 vehicle model';
nlgr3.Parameters(6).Value = 0.70; % Use another initial CA for the Volvo data.
z3 = iddata(y3, u3, 0.1, 'Name', 'Volvo V70 data');
z3.InputName = nlgr3.InputName;
z3.InputUnit = nlgr3.InputUnit;
z3.OutputName = nlgr3.OutputName;
z3.OutputUnit = nlgr3.OutputUnit;
z3.Tstart = 0;
z3.TimeUnit = 's';
The inputs and outputs are shown in two plot figures. As can be seen, the measured data is rather noisy.
for i = 1:z3.Nu
subplot(z3.Nu, 1, i);
plot(z3.SamplingInstants, z3.InputData(:,i));
title(['Input #' num2str(i) ': ' z3.InputName{i}]);
axis tight;
xlabel([z3.Domain ' (' z3.TimeUnit ')']);
Figure 10: Measured inputs from a Volvo V70 vehicle.
for i = 1:z3.Ny
subplot(z3.Ny, 1, i);
plot(z3.SamplingInstants, z3.OutputData(:,i));
title(['Output #' num2str(i) ': ' z3.OutputName{i}]);
axis tight;
xlabel([z3.Domain ' (' z3.TimeUnit ')']);
Figure 11: Measured outputs from a Volvo V70 vehicle.
Next we investigate the performance of the initial model with the initial states being estimated. A comparison between the true and the simulated outputs (with the initial model) is shown in a plot
nlgr3 = setinit(nlgr3, 'Value', {18.7; 0; 0}); % Initial value of initial states.
compare(z3, nlgr3);
Figure 12: Comparison between measured outputs and the simulated outputs of the initial Volvo V70 vehicle model.
The tire stiffness parameters Cx and Cy are next estimated, in this case using the Levenberg-Marquardt search method, whereupon a new simulation with the estimated model is performed. In addition, we
here estimate the initial value of the longitudinal velocity, whereas the initial values of the lateral velocity and the yaw rate are kept fixed.
nlgr3 = setinit(nlgr3, 'Fixed', {false; true; true});
nlgr3 = nlgreyest(z3, nlgr3, nlgreyestOptions('SearchMethod', 'lm'));
A comparison between the true and the simulated outputs (with the estimated model) is shown in a plot window.
Figure 13: Comparison between measured outputs and the simulated outputs of the first estimated Volvo V70 vehicle model.
The estimated stiffness parameters of the final Volvo V70 model are reasonable, yet it is here unknown what their real values are.
disp(' Estimated');
fprintf('Longitudinal stiffness: %6.0f\n', nlgr3.Parameters(4).Value);
fprintf('Lateral stiffness : %6.0f\n', nlgr3.Parameters(5).Value);
Longitudinal stiffness: 108873
Lateral stiffness : 29964
Further information about the estimated Volvo V70 vehicle model is obtained through PRESENT. It is here interesting to note that the uncertainty related to the estimated lateral tire stiffness is
quite high (and significantly higher than for the longitudinal tire stiffness). This uncertainty originates partly from that the lateral acceleration is varied so little during the test drive.
nlgr3 =
Continuous-time nonlinear grey-box model defined by 'vehicle_c' (MEX-file):
dx/dt = F(t, x(t), u(t), p1, ..., p6)
y(t) = H(t, x(t), u(t), p1, ..., p6) + e(t)
with 5 input(s), 3 state(s), 3 output(s), and 2 free parameter(s) (out of 6).
u(1) Slip on front left tire(t) [ratio]
u(2) Slip on front right tire(t) [ratio]
u(3) Slip on rear left tire(t) [ratio]
u(4) Slip on rear right tire(t) [ratio]
u(5) Steering angle(t) [rad]
States: Initial value
x(1) Longitudinal vehicle velocity(t) [m/s] xinit@exp1 17.6049 (estimated) in ]0, Inf]
x(2) Lateral vehicle velocity(t) [m/s] xinit@exp1 0 (fixed) in [-Inf, Inf]
x(3) Yaw rate(t) [rad/s] xinit@exp1 0 (fixed) in [-Inf, Inf]
y(1) Long. velocity(t) [m/s]
y(2) Lat. accel.(t) [m/s^2]
y(3) Yaw rate(t) [rad/s]
Parameters: ValueStandard Deviation
p1 Vehicle mass [kg] 1700 0 (fixed) in ]0, Inf]
p2 Distance from front axle to COG [m] 1.5 0 (fixed) in ]0, Inf]
p3 Distance from rear axle to COG [m] 1.5 0 (fixed) in ]0, Inf]
p4 Longitudinal tire stiffness [N] 108873 26.8501 (estimated) in ]0, Inf]
p5 Lateral tire stiffness [N/rad] 29963.5 217.877 (estimated) in ]0, Inf]
p6 Air resistance coefficient [1/m] 0.7 0 (fixed) in ]0, Inf]
Name: Volvo V70 vehicle model
Termination condition: Maximum number of iterations reached..
Number of iterations: 20, Number of function evaluations: 41
Estimated using Solver: ode45; Search: lm on time domain data "Volvo V70 data".
Fit to estimation data: [-374.2;29.74;34.46]%
FPE: 2.362e-07, MSE: 0.3106
More information in model's "Report" property.
Concluding Remarks
Estimating the tire stiffness parameters is in practice a rather intricate problem. First, the approximations introduced in the model structure above are valid for a rather narrow operation region,
and data during high accelerations, braking, etc., cannot be used. The stiffness also varies with the environmental condition, e.g., the surrounding temperature, the temperature in the tires and the
road surface conditions, which are not accounted for in the used model structure. Secondly, the estimation of the stiffness parameters relies heavily on the driving style. When mostly going straight
ahead as in the third identification experiment, it becomes hard to estimate the stiffness parameters (especially the lateral one), or put another way, the parameter uncertainties become rather high.
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Merge Sort Explained: A Data Scientist
Merge Sort Explained: A Data Scientist’s Algorithm Guide
Data Scientists deal with algorithms daily. However, the data science discipline as a whole has developed into a role that does not involve implementation of sophisticated algorithms. Nonetheless,
practitioners can still benefit from building an understanding and repertoire of algorithms.
In this article, the sorting algorithm merge sort is introduced, explained, evaluated, and implemented. The aim of this post is to provide you with robust background information on the merge sort
algorithm, which acts as foundational knowledge for more complicated algorithms.
Although merge sort is not considered to be complex, understanding this algorithm will help you recognize what factors to consider when choosing the most efficient algorithm to perform data-related
tasks. Created in 1945, John Von Neumann developed the merge sort algorithm using the divide-and-conquer approach.
Divide and conquer
To understand the merge sort algorithm, you must be familiar with the divide and conquer paradigm, alongside the programming concept of recursion. Recursion within the computer science domain is when
a method defined to solve a problem involves an invocation of itself within its implementation body.
In other words, the function calls itself repeatedly.
Figure 1. Visual illustration of recursion – Image by author.
Divide and conquer algorithms (which merge sort is a type of) employ recursion within its approach to solve specific problems. Divide and conquer algorithms decompose complex problems into smaller
sub-parts, where a defined solution is applied recursively to each sub-part. Each sub-part is then solved separately, and the solutions are recombined to solve the original problem.
The divide-and-conquer approach to algorithm design combines three primary elements:
• Decomposition of the larger problem into smaller subproblems. (Divide)
• Recursive utilization of functions to solve each of the smaller subproblems. (Conquer)
• The final solution is a composition of the solution to the smaller subproblems of the larger problem. (Combine)
Other algorithms use the divide-and-conquer paradigm, such as Quicksort, Binary Search, and Strassen’s algorithm.
Merge sort
In the context of sorting elements in a list and in ascending order, the merge sort method divides the list into halves, then iterates through the new halves, continually dividing them down further
to their smaller parts.
Subsequently, a comparison of smaller halves is conducted, and the results are combined together to form the final sorted list.
Steps and implementation
Implementation of the merge sort algorithm is a three-step procedure. Divide, conquer, and combine.
The divide component of the divide-and-conquer approach is the first step. This initial step separates the overall list into two smaller halves. Then, the lists are broken down further until they can
no longer be divided, leaving only one element item in each halved list.
The recursive loop in merge sort’s second phase is concerned with the list’s elements being sorted in a particular order. For this scenario, the initial array is sorted in ascending order.
In the following illustration, you can see the division, comparison, and combination steps involved in the merge sort algorithm.
Figure 2. Divide component illustration of the Merge sort algorithm—Image by Author.
Figure 3. Conquer and combine components—Image by author.
To implement this yourself:
• Create a function called merge_sort that accepts a list of integers as its argument. All following instructions presented are within this function.
• Start by dividing the list into halves. Record the initial length of the list.
• Check that the recorded length is equal to 1. If the condition evaluates to true, return the list as this means that there is just one element within the list. Therefore, there is no requirement
to divide the list.
• Obtain the midpoint for a list with a number of elements greater than 1. When using the Python language, the // performs division with no remainder. It rounds the division result to the nearest
whole number. This is also known as floor division.
• Using the midpoint as a reference point, split the list into two halves. This is the divide aspect of the divide-and-conquer algorithm paradigm.
• Recursion is leveraged at this step to facilitate the division of lists into halved components. The variables ‘left_half’ and ‘right_half’ are assigned to the invocation of the ‘merge_sort’
function, accepting the two halves of the initial list as parameters.
• The ‘merge_sort’ function returns the invocation of a function that merges two lists to return one combined, sorted list.
def merge_sort(list: [int]):
list_length = len(list)
if list_length == 1:
return list
mid_point = list_length // 2
left_half = merge_sort(list[:mid_point])
right_half = merge_sort(list[mid_point:])
return merge(left_half, right_half)
• Create a ‘merge’ function that accepts two lists of integers as its arguments. This function contains the conquer and combine aspects of the divide-and-conquer algorithm paradigm. All following
steps are executed within the body of this function.
• Assign an empty list to the variable ‘output’ that holds the sorted integers.
• The pointers ‘i’ and ‘j’ are used to index the left and right lists, respectively.
• Within the while loop, there is a comparison between the elements of both the left and right lists. After each comparison, the output list is populated within the two compared elements. The
pointer of the list of the appended element is incremented.
• The remaining elements to be added to the sorted list are elements obtained from the current pointer value to the end of the respective list.
def merge(left, right):
output = []
i = j = 0
while (i
Performance and complexity
Big O notation is a standard for defining and organizing the performance of algorithms in terms of their space requirement and execution time.
Merge sort algorithm time complexity is the same for its best, worst, and average scenarios. For a list of size n, the expected number of steps, minimum number of steps, and maximum number of steps
for the merge sort algorithm to complete, are all the same.
As noted earlier in this article, the merge sort algorithm is a three-step process: divide, conquer, and combine. The ‘divide’ step involves the computation of the midpoint of the list, which,
regardless of the list size, takes a single operational step. Therefore the notation for this operation is denoted as O(1).
The ‘conquer’ step involves dividing and recursively solving subarrays–the notation log n denotes this. The ‘combine’ step consists of combining the results into a final list; this operation
execution time is dependent on the list size and denoted as O(n).
The merge sort notation for its average, best, and worst time complexity is log n * n * O(1). In Big O notation, low-order terms and constants are negligible, meaning the final notation for the merge
sort algorithm is O(n log n). For a detailed analysis of the merge sort algorithm, refer to this article.
Merge sort performs well when sorting large lists, but its operation time is slower than other sorting solutions when used on smaller lists. Another disadvantage of merge sort is that it will execute
the operational steps even if the initial list is already sorted. In the use case of sorting linked lists, merge sort is one of the fastest sorting algorithms to use. Merge sort can be used in file
sorting within external storage systems, such as hard drives.
Key takeaways
This article describes the merge sort technique by breaking it down in terms of its constituent operations and step-by-step processes.
Merge sort algorithm is commonly used and the intuition and implementation behind the algorithm is rather straightforward in comparison to other sorting algorithms. This article includes the
implementation step of the merge sort algorithm in Python.
You should also know that the time complexity of the merge sort method’s execution time in different situations, remains the same for best, worst, and average scenarios. It is recommended that merge
sort algorithm is applied in the following scenarios:
• When dealing with larger sets of data, use the merge sort algorithm. Merge sort performs poorly on small arrays when compared to other sorting algorithms.
• Elements within a linked list have a reference to the next element within the list. This means that within the merge sort algorithm operation, the pointers are modifiable, making the comparison
and insertion of elements have a constant time and space complexity.
• Have some form of certainty that the array is unsorted. Merge sort will execute its operations even on sorted arrays, a waste of computing resources.
• Use merge sort when there is a consideration for the stability of data. Stable sorting involves maintaining the order of identical values within an array. When compared with the unsorted data
input, the order of identical values throughout an array in a stable sort is kept in the same position in the sorted output.
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Dispatching experts to do maintenance
Below a poster about Dispatching experts to do maintenance which was made for mathematics exhibition IMAGINARY. The poster was made by Stella Kapodistria, assistant professor in the section
Stochastics of the Department of Mathematics and Computer Science in the Eindhoven University of Technology, and Peter Verleijsdonk, Doctoral Candidate in the Eindhoven University of Technology.
Dispatching experts to do maintenance.
Nowadays surgical operations require advanced robotic equipment. Such equipment can help save lives. Unfortunately, such equipment deteriorates with usage and it can eventually fail. When it fails,
it requires maintenance from an expert engineer. Until it is maintained, it cannot be used and hospital operation is disrupted.
Thankfully such equipment is mounted with several sensors that collect data about the condition of the equipment in real time. Using data analytic techniques and Artificial Intelligence, we analyse
the data and discover hidden patterns that allow us to often predict (within a margin of accuracy) the failures before they happen. When a failure is predicted, we issue an alert and we plan for
preventive maintenance by an expert engineer.
Predicting failures and treating them preventively is cost effective, as maintenance is now planned causing minimal disruption to the hospital operation. Planning the maintenance of the equipment
(upon failure or preventively) is a very complicated mathematical optimization problem: At every instant of time, given the available information on the condition of the equipment, we need to decide
which engineer to send to treat which issue. However, over time, the available information changes as new data becomes available. So, from one instance of time to the next, as new information becomes
available, the problem changes and then we need to solve the problem anew. This makes the problem complicated, but this is not the only complication.
Note that as it stands, the solution to the optimization problem only assigns experts to maintenance issues, but it does not take into account information about the future that is hidden in the data.
Analyzing the data, we can often predict (within a margin of accuracy) when an issue will occur. E.g., how long we have in our disposal before an alert is issued or a failure happens. But if we know
how long we have in our disposal, we can then strategically reposition the idle experts to a different city so as to ensure they are close to issues when these issues happen. Such repositioning is
extremely effective as experts do not wait for an issue to happen but they proactively travel and get close to a city where they will need to perform maintenance in the future. So, by repositioning
the experts, we can achieve large coverage with a small response time.
Combining mathematics and Artificial Intelligence, we combine predictions for the future with smart maintenance strategies for the expert engineers. Our solution algorithms consider all available
information (current issues and future predictions) and, based on that information, they determine the best way to dispatch and to reposition the experts ensuring as few and as short as possible
disruptions at a low cost.
Our poster demonstrates some key insights of the solution algorithm on a small instance of only 5 experts and 18 pieces of equipment. However, keep in mind that in a realistic instance, there are
typically hundreds of experts and thousands of equipment. Due to the large number of experts and equipment, it is very challenging to design effective algorithms that can quickly provide good
insights. In order to design effective algorithms, we combine knowledge from three mathematical fields: mathematical modeling (we take a real problem and we formulate it into a mathematical problem
that captures all its essential elements), data analytics and Artificial Intelligence (we extract hidden patterns from the data), and mathematical optimization (we design algorithms that solve the
mathematical problem).
Related articles
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LDraw to Web
LDraw to Web 2011-09-09, 19:32 (This post was last modified: 2011-11-27, 12:54 by Steffen.)
Hi- I've been working on a converter to take ldraw dat files and parse them so they can be displayed in 3d in a canvas element on any web page. So far, I've had no problem doing this with any single
parts (it works great), but I'm having trouble with assemblies (single dat files, with other referenced dat files). What's tripping me up is the 3d transformations.
I'm trying to understand the calculations that are at:
In the calculation:
u' = (a * u) + (b * v) + (c * w) + x;
v' = (d * u) + (e * v) + (f * w) +y;
w' = (g *u) + (h * v) + (i * w) + z;
Is u, v, and w the X, Y, Z coords of the vertices that we're transforming under the part itself, and the x, y, z come from the line type 1, where it gives the origin of the entire part?
My results that I seem to be getting are somewhat of a smeared part in one direction. This tells me that some of the vertices are transforming, but not all. I just need to double check that I'm doing
this in the correct order.
Re: LDraw to Web
2011-09-09, 21:32
Are you multiplying the reference type 1 line (4x4) matrix recursively with the parent's matrices?
And if so, are you multiplying in the right order? (backwards)
Re: LDraw to Web
2011-09-11, 15:22
Roland Melkert Wrote:
> Are you multiplying the reference type 1 line
> (4x4) matrix recursively with the parent's
> matrices?
> And if so, are you multiplying in the right order?
> (backwards)
I'm not sure I entirely understand what you're asking, let me further illustrate what I'm exactly doing...
Here's some quick pseudo code of what I'm trying to do:
//Parent File one-liner
1 0 -20 -36 0 0 1 0 0 0 1 1 0 0 fullbeam.dat
// x y z a b c d e f g h i
//fullbeam.dat sub-file one-liner
// u v w u v w u v w
foreach VertexPoint in fullbeam.dat{
u' = (a * u) + (b * v) + (c * w) + x;
v' = (d * u) + (e * v) + (f * w) + y;
w' = (g * u) + (h * v) + (i * w) + z;
So, with the 3 vertices that are listed in the fullbeam.dat:
Loop 1
u' = (0*44) + (1*0) + (0*4) + -20
v' = (0*44) + (0*0) + (1*4) + -36
w' = (1*44) + (0*0) + (0*4) + 0
Loop 2
u' = (0*44) + (1*2) + (0*4) + -20
v' = (0*44) + (0*2) + (1*4) + -36
w' = (1*44) + (0*2) + (0*4) + 0
Loop 3
u' = (0*44) + (1*0) + (0*10) + -20
v' = (0*44) + (0*0) + (1*10) + -36
w' = (1*44) + (0*0) + (0*10) + 0
So my resulting transformed coords for line 3 in fullbeam.dat would be:
1 0 -20 -32 44 -18 -32 44 -20 -26 44
Re: LDraw to Web
2011-09-11, 17:21
Nick McBride Wrote:
> I'm not sure I entirely understand what you're
> asking
> For example:
> Say the line reads 1 10 -40 -10 -20 1 0 0 0 1 0 0
> 1 1 x.dat
> x y z
> a b c d e f g h i file
> I'm running through x.dat and pulling all the
> vertexes from each line and using this calculation
> on them:
> u' = (a * u) + (b * v) + (c * w) + x;
> v' = (d * u) + (e * v) + (f * w) +y;
> w' = (g *u) + (h * v) + (i * w) + z;
> where u, v, and w = the x,y,z coords from each of
> the the vertexes in x.dat. The a,b,c,d,e,f,g,h,
> and i come from the line 1, and so does x,y,z
> the u',v',and w' would then replace the vertices
> that were coming from x.dat
I take it you haven't worked with matrices before?
The type 1 line tells you how to transform the vertices of the referenced part in order to place it in the calling document's model space. This is done by a rotation (incl optional scaling) and
translation. The rotation/scaling is given by a..i and the translation by x, y and z. But LDraw files are highly recursive so any file within the called file has to ALSO apply the transformation of
higher level references.
To do this the 'easy way' you must use the 12 rotation and translation numbers to construct a 4x4 matrix which can be multiplied with the higher level matrices and then applied in one step to the
The formula you using above is actually a simplified matrix transformation for a 1x4 with a 4x4 matrix, which in it's 'pure' form looks like:
abs ver. vertex ref matrix
/ x'\ / x \ / a b c x \
| y'| | y | | d e f y |
| z'| = | z | * | g h i z |
\ 1 / \ 1 / \ 0 0 0 1 /
But it's only the one type 1 line matrix so it will only work for one level deep files. You need to multiply all level matrices with each other in the 4x4 matrix and use a full transformation formula
(this one ignores the bottom line, cause it always results in zero)
For example you have the following file structure
AB -> b.dat
AC -> c.dat
CD -> d.dat
CE -> e.dat
To render e.g. e.dat absolutely you need to apply:
abs ver. vertex CE Matrix AC Matrix
/ x'\ / x \ / a b c x \ / a b c x \
| y'| | y | | d e f y | | d e f y |
| z'| = | z | * | g h i z | * | g h i z |
\ 1 / \ 1 / \ 0 0 0 1 / \ 0 0 0 1 /
So in short you need to do some research on (rotation) matrices and get your hands on a matrix math library for your programming language or write one your self (you actually only need a
multiplication and a transformation routine).
Hope this helps you, or at least gets you started. I'm by no means an expert so maybe another forum user can explain it better.
Anyway, I know matrix math can be hard to grasp. When I started with LDraw it was all magic to me too (it still kinda is)
[edit] -> forgot to add the '=' part of the formulas.
Re: LDraw to Web
2011-09-11, 20:11
Thanks for the good lesson. In the specific way I'm planning on using my converter, the files will only be one level deep, so I don't have to worry about recursion this time around (but always good
to know). So I guess I'm going to have to do a little more research on the whole matrix thing. I thought my calculations were correct, due to it only being one level deep. I see your point with
having to apply it at the higher levels (from the bottom up), but if we're only talking 1 level deep, my
u' = (a * u) + (b * v) + (c * w) + x;
v' = (d * u) + (e * v) + (f * w) +y;
w' = (g *u) + (h * v) + (i * w) + z;
formula is still correct, right? But then again, I might be a little thick :-)
Just wanted to add a picture of what's happening:
http://www.altoonalights.com/block.gif It's a custom part, but you can see that it would seem that one of the sides is intact, while the rest is "smeared" out of whack.
Re: LDraw to Web
2011-09-13, 15:58
So I fixed the issue after realizing how thick I was.
I passed the translation variables into a function that was doing the conversion for me. The translation variables were named letters a-i.
Within my function I had a loop, which used "i" as an increment and counter. It simply was over-writing the "i" that came from the transformation matrix. I renamed the variables, and it works like a
Special thanks to Roland though, who helped clarify my math and give me a little more understanding what was going on.
Re: LDraw to Web
2011-09-17, 14:37
Cool! A canvas-compatible LDraw renderer would be quite useful.
You might be interested in this very simple LDraw renderer written in Processing (a graphics programming system based on Java). After setting it up, I tried using it to make an LDraw applet you could
embed in a web page. You can try it here. I haven't done anything with it since, though, mainly because Java applets are kind of finicky and other technologies (like what you're working on) seem
better suited for today's web.
why not use JavaScript ?
2011-09-27, 17:44
While reading the above interesting thread,
it just jumped to my mind that we could think of using JavaScript to render a *.dat or *.ldr input
to the browser's canvas.
As this will probably not be a lot of code, we could bear with the ugly JavaScript syntax
and at the same time profit from the recent speedup in JavaScript engines (Google's V8, Firefox's etc.),
and at the same time save users from having to install Java first.
Being a big Iron browser fan (the de-googled version of Chrome,
see http://www.srware.net/en/software_srware_iron.php ),
I start liking that idea more and more:
Have a look at
to see what is possible with JavaScript nowadays
(I am told that there are even translator tools which translate nice Java code into JavaScript code,
but I have to find out more about this first)
that solution will also work for pad computers
2011-09-27, 17:50
using that suggestion would also allow to view LDRAW models on tablet (pad) computers
running either Android _OR_ iOS.
so no special Apps need to be developed for each target platform,
just simple JavaScript painting on a browser canvas - this is something that all current browsers
on all platforms can do
(for example Opera on some mobile devices etc.)
see also here
Re: that solution will also work for pad computers
2011-09-27, 19:27
Interesting, but I won't be fast and it probably needs alot of mem for starters.
And where does the script get the needed dat files? I'm still not sure why the original topic starter thinks he doesn't need recursion. Cause that means you can only render primitives which isn't
very exiting imho
Using the clients local library raises security issues (most browsers won't allow it by default). Using some remote server might get slow very fast (not too mention needed bandwidth etc).
An alternative would be to put every thing in a mpd so no further dependencies are needed, but that needs additional coding on the site serving the canvas script (a.g. attach a model to a post will
automatically generate a (potential gigantic) standalone mpd). But if the server needs to do this, it could just as easily generate a png or something instead using e.g. a custum ldview version.
Just my 2cts
Re: that solution will also work for pad computers
2011-11-04, 19:20
On the parts angle, I don't think it would be that bad. Parts are already HTTP accessible from LDraw.org, although I doubt it's set-up to handle this kind of load.
An app can cache the parts using one of the several HTML5 storage options, as well as using HTTP caches. Additionally, parts average 10k each and subparts average 5k (very compressible). Include
procedurally-generated subparts (a la LDView), and I would consider this reasonable downloads.
As for the speed of JavaScript, Chrome and Firefox are both JIT compiling now, so it should be comparable in speed to Java.
Re: that solution will also work for pad computers
2011-11-04, 19:43
Jamie B. Wrote:
> As for the speed of JavaScript, Chrome and Firefox
> are both JIT compiling now, so it should be
> comparable in speed to Java.
I was more thinking about the canvas speed, I haven't done much on this subject lately so I don't know if you can easily get a fully hardware accelerated OpenGL context.
WebGL in action in a similar application
2011-11-04, 21:08
this is a 3D building application, loading parts on demand.
you can construct a robot, first from basic parts, then add smaller ones,
colorize it, apply stickers etc.
here are more WebGL demos
I think that all major browsers will soon have that technology & performance
Re: that solution will also work for pad computers
2011-11-05, 0:01
Jamie B. Wrote:
> On the parts angle, I don't think it would be that
> bad. Parts are already HTTP accessible from
> LDraw.org, although I doubt it's set-up to handle
> this kind of load.
I ask that this not be tried for the reason above. Our web space is donated by Peeron and I don't want to increase Dan's traffic.
Re: that solution will also work for pad computers
2011-11-09, 2:42
Roland Melkert Wrote:
> I was more thinking about the canvas speed, I
> haven't done much on this subject lately so I
> don't know if you can easily get a fully hardware
> accelerated OpenGL context.
WebGL is exactly that. While function calls won't be as fast as C, the hard work is done by hardware.
For practical tests, try http://lawriecape.co.uk/threejs/webGL.html. On my puny little chromebook, it clocks in at 5-7FPS. (I consider this acceptable for a web viewer.)
question to forum admins: merge 3 threads?
2011-11-27, 12:54
Suggestion: I would like to ask the three threads
merged into a single thread "LDraw App / LDraw Web Renderer"
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We define the independence ratio and the chromatic number for bounded, self-adjoint operators on an L^2-space by extending the definitions for the adjacency matrix of finite graphs. In analogy to the
Hoffman bounds for finite graphs, we give bounds for these parameters in terms of the numerical range of the operator. This provides a theoretical … Read more
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...Wiley series in probability and mathematical statistics. Probability and mathematical statistics...
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Yang, Xiangqun
Changsha : Chichester ; New York : Hunan Science and Technology Pub. House ; J. Wiley, 1990
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The Autocorrelation Function - Alan Zucconi
The purpose of this tutorial is to show a simple technique to estimate periodicity in time series, called autocorrelation.
This tutorial is part of a longer series that focuses on how to analyse time series.
In the previous part of this tutorial, Time Series Decomposition, we have seen how is possible to decompose sales in their original components. One of the inputs of this process, is knowing the exact
periodicity of the seasonal components. When it comes to real data, this is rarely the case.
The Correlation Coefficient
The first step is to find a way of measuring how similar two time series are. There are countless way of doing this, depending on the underlying assumptions of your data. The most used one for those
applications is called correlation. The correlation between two functions (or time series) is a measure of how similarly they behave. It can be expressed as:
with standard deviation and the mean of
The mean is simply the average of the whole time series. The standard deviation, instead, indicates how much the points of the series tends distance themselves from the mean. This quantity is often
associated with variance, defined as:
When the variance is zero, all the points in the series are equal to the mean. A high variance indicates that the points are scattered around.
The term covariance between
The covariance is calculated as follow:
and is easy to see that indeed
Looking back to the definition of correlation, it is now easy to understand what is trying to capture. It’s a measure of how similarly
Autocorrelation Function
The idea behind the concept of autocorrelation is to calculate the correlation coefficient of a time series with itself, shifted in time. If the data has a periodicity, the correlation coefficient
will be higher when those two periods resonate with each other.
The first step is to define an operator to shift a time series in time, causing a delay of lag operator:
The autocorrelation of a time series with lag
which can also be expressed as:
The code
The above mentioned form is amenable to be written as code. The easiest function is surely the one that calculates the mean of a time series:
public float Mean (float [] x)
float sum = 0;
for (int i = 0; i < x.length; i ++)
sum += x[i];
return sum / x.length;
A little bit complicates is the case for the autocorrelation function. It creates an array which will contain the final result. Each t-th element contains
public float [] Autocorrelation (float [] x)
float mean = Mean(x);
float [] autocorrelation = new float[x.length/2];
for (int t = 0; t < autocorrelation.length; t ++)
float n = 0; // Numerator
float d = 0; // Denominator
for (int i = 0; i < x.length; i ++)
float xim = x[i] - mean;
n += xim * (x[(i + t) % x.length] - mean);
d += xim * xim;
autocorrelation[t] = n / d;
return autocorrelation;
Line 14 implements an inline lag operator. It shifts i by t, and uses the modulo operator so that the time series loops. If this is not the desired case, then you should only loop up to x.length -t.
The Correlogram
Autocorrelation is a relatively robust technique, which doesn’t come with strong assumptions on how the data has been created. If in the previous post we have used a synthetic sales data, this time
we can confidently use real analytics:
This is the plot for the autocorrelation function, also known as correlogram:
All correlograms start at
Because of this resonance, interpreting correlograms is not always easy. There are several improvements on this technique which can help to extract actual cycles. Partial autocorrelation functions
controls for the values of the time series at all shorter lags. This removes interference and resonance with multiple cycles, highlighting a more clear periodicity. A more advanced technique, called
Power Spectral Density, performs a Fourier analysis on the correlogram to find its main component.
This tutorial concludes the series on time series analysis. We have explored valuable techniques to extract information from temporal data, focusing on their potential and limitations.
Other resources
5 responses to “The Autocorrelation Function”
Very clear explanation, Thank you for the efforts
You’re welcome!
Hi Alan, thanks for the nice tutorial! Should the Autocorrelation function’s for loop end condition (line 6) be ‘autocorrelation.length’ instead of ‘autocorrelation.length/2’ since you’ve already
halved the input array (x) in line 5? Thanks!
You’re right!
Thank you!
Nice tutorial. Really enjoyed reading this series.
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mreg: To perform regression when discrete outcome variables are... in mreg: Fits Regression Models When the Outcome is Partially Missing
This software was created for the paper referred to below. If a longitudinal data base has regularly updated explanatory variables, but whose outcome variable is only intermittently collected then we
can still perform exact maximum likelihood estimation of a regression model if the outcome variable is discrete.
mreg( formula, data, patid, start.theta = NULL, modify = unity, modify.p = 0, mod.formula = ~1, density.name = "negbin", link = "log", iterlim = 100, gradtol = 1e-06, steptol = 1e-06, na.action =
NULL, print.level = 2, zero.start = FALSE )
formula This is a formula object e.g. Y~A+B to describe the location parameter
data This is a data frame in which the variables are recorded
patid In a longitudinal context this indexes the individuals. Note that the observations within each patient is assumed to be ordered according the timing of the observations.
start.theta Optional vector of starting values for location and nuisance parameters
We may wish to let the location depend on functions of the previous outcomes. Since these may be missing, we have to provide a function that can cope with all the potential values the
modify outcome may have taken. See paper
modify.p This is the dimension of the parameters associated with the modify function.
If we require other variables to interact with the previous observation we must create a set of variables to use. This is a one-sided formula e.g. ~X+Z, if we wanted to use those
mod.formula variables.
density.name This is the density the increment in outcome is assumed to follow. It can be one of three values: negbin, poisson, geometric.
This is the link function g(\mu)=\eta. Where \eta is a linear combination of covariates, and \mu is the expected value of the outcome. The link function can be one of four values:
link identity, log, logit, hyper.
iterlim The maximum number of iterations allowed for the nlm function.
gradtol The parameter gradtol for the nlm function.
steptol The parameter steptol for the nlm function
na.action Parameter is not used: If any covariates are missing the function will return an error.
print.level The parameter print.level for the nlm function. Set to the maximum, verbose level.
zero.start It may be the case that it is known that the first value of the outcome was zero for all individuals, in which case invoke this TRUE/FALSE option.
This is a formula object e.g. Y~A+B to describe the location parameter
This is a data frame in which the variables are recorded
In a longitudinal context this indexes the individuals. Note that the observations within each patient is assumed to be ordered according the timing of the observations.
Optional vector of starting values for location and nuisance parameters
We may wish to let the location depend on functions of the previous outcomes. Since these may be missing, we have to provide a function that can cope with all the potential values the outcome may
have taken. See paper
This is the dimension of the parameters associated with the modify function.
If we require other variables to interact with the previous observation we must create a set of variables to use. This is a one-sided formula e.g. ~X+Z, if we wanted to use those variables.
This is the density the increment in outcome is assumed to follow. It can be one of three values: negbin, poisson, geometric.
This is the link function g(\mu)=\eta. Where \eta is a linear combination of covariates, and \mu is the expected value of the outcome. The link function can be one of four values: identity, log,
logit, hyper.
The maximum number of iterations allowed for the nlm function.
Parameter is not used: If any covariates are missing the function will return an error.
The parameter print.level for the nlm function. Set to the maximum, verbose level.
It may be the case that it is known that the first value of the outcome was zero for all individuals, in which case invoke this TRUE/FALSE option.
It returns an object of class mreg which is similar to a lm object. It has print and summary methods to display the fitted parameters and standard errors.
Bond S, Farewell V, 2006, Exact Likelihood Estimation for a Negative Binomial Regression Model with Missing Outcomes, Biometrics
data(public) ## Not run: mod1 <- mreg( damaged~offset(log(intervisit.time))+esr.init, data=public,patid=ptno,print.level=2, iterlim=1000 ) mod.ncar <-mreg(damaged ~ offset(log(intervisit.time)) +
esr.init + tender + effused + clinic.time, data = public, patid = ptno, modify = paper, modify.p = 5, mod.formula = ~art.dur.init, density.name = "negbin.ncar", iterlim = 1000, print.level = 2) ##
End(Not run)
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Calculating a person's age?
How can I get someone’s age from their birthday and today’s date? Subtracting dates gives a number of days, and I haven’t been able to find a formatting option or formula to convert to years.
Hi @Jean_Goodwin
Have you tried some thing like this -
Round( Today() - birthday)/365, 0)
2 Likes
But leap year! Although, 365.24… whatever should therefore work. Thanks!
Accurately calculating age in all cases is a surprisingly tricky challenge. For example, take a look at this Java example - it’s way bigger than you would expect!
Since we just need age in years, not in years + months + days, so it will be just the result of years subtraction. And minus one if there was no birthday this year yet.
Today().Year() - thisRow.Birthday.Year() - If(Date(Today().Year(), thisRow.Birthday.Month(), thisRow.Birthday.Day()) > Today(), 1, 0)
7 Likes
Nice, thanks! Not only does it solve the problem, it aids my continuing education in formula-language.
What is the advantage of doing this versus what was suggested above?
Round((Today() - birthday)/365.24, 0)
Asking because I’m always wanting to simplify my formulas and keep things clean, but striving for accuracy of course.
Round is not right formula because it rounds to closest. I am 29, my birthday is 23rd of November, 1989. Your formula gives me 30 because I am closer to 30 than 29.
Okay, maybe we should switch to Floor?
Floor((Today() - birthday)/365.24, 1)
However, there are still some pairs of dates which will give not right answer. E.g.: today is the 8th of July, 2019. If today is my birthday, and I am born in 1989, this formula still gives me 29,
when I am technically already 30. It happens because the average number of days in year depends on specific start and end year we take.
Sometimes you just have to write the way you do it mentally.
@Denis_Peshekhonov any guesses?
It’s strange! Can you share the doc?
I can’t because of the sensitive information that’s involved. My birthday column is formatted as a Date if that helps. Otherwise I copied and pasted your formula into my doc.
So, try to rewirite this fomula by hand
1 Like
I ended up with:
If(thisRow.Birthday.IsNotBlank(),Today().Year()-thisRow.Birthday.Year() - If(Date(Today().Year(),thisRow.Birthday.Month(),thisRow.Birthday.Day())>Today(),1,0) ,"")
For future reference: I wrapped the entire formula in an “if” statement to leave the field blank if the input data is blank, in this case “Birthday.” Otherwise for blank fields the output would be
3 Likes
Cool, thanks! Yes, I forgot about blank values.
Excellent thread! I’m curious if someone could help me with this formula (recommended above) with the addition of a “Died on” date column?
If(thisRow.Birthday.IsNotBlank(),Today().Year()-thisRow.Birthday.Year() - If(Date(Today().Year(),thisRow.Birthday.Month(),thisRow.Birthday.Day())>Today(),1,0) ,"")
I’m unsure where in the statement it would be best to include that factor.
1 Like
Hi @Alice_Packard,
I guess that a good solution would be having a Date column Last Day containing the formula:
Died on.ifBlank(Today())
(assuming you also have a Died On column, of course)
Then, replace all the "Today()"s in the given formula with Last Day.
I hope this helps.
1 Like
Excellent! Thank you so so much Federico, much appreciated. Having both a “Died on” and “Last Day” column is really clever, thanks for the tip. The formula works as expected
1 Like
This was good, but I added “Need DOB” Rather than leave it blank just for fun.
This makes no sense unless it’s written out. Too confusing.
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Who can help me with matrix error propagation analysis in R programming? | Pay Someone To Take My R Programming Assignment
Who can help me with matrix error propagation analysis in R programming? First off, I’m using matrix package from e-Science, not R1. The problem is that rather than trying to make the difference in a
macro with a known function, you’ll end up with a matrix of the form $$\textsc{E}(\rho) := \left( \begin{bmatrix} E^{(1)}_{e} \\ \vdots \\ E^{(2)}_{e}\\ \vdots \end{bmatrix} \right)$$ If the function
has one column and column positions, nothing has changed. If the function has an additional column, the matrix of degrees is the same as a column matrix. In this case, we can just take the next
column and apply the same steps using either of them. The reason for this is that when you first apply the matrix structure to any number of matrices, it becomes a vectorized product. This means that
you have to have a vectorized product to represent e.g. the number of rows or columns. During a bito here, we have the same thing. The number of rows your matrix can be represented by is not the
quantity that doesn’t change because of the matrix class. What other classes are there? It’s a piece in some machinery between matrix package. It’s a simple matter to have them passed through a
classifier or other entity and then they’ve been fitted with the name-type, or number-type, of the classifier to be used. It’s not appropriate for matrix to itself be a vectorized product; it doesn’t
include a fixed quantity. But something along the lines of vectors without data types and where it fits has to happen that it’s taking in some number of elements as the right idea of how a given
object is stored in a data structure. I actually think it’s really the same reasoning as this case. The way we do it is differently: we treat the number of elements as the number of entries in a
matrix, so that a matrix of the form we have here is entirely related to the factor that generates them. Then we’re not referring to the fact that matrices are always column tuples–you want a column
matrix to be treated as a row matrix. If that’s the case, then we have to treat them as column tuples. A: I believe there are multiple ways to translate and transform data. One possible approach
involves vectorized vectors, but it’s usually better to vectorize the representation in a more combthy way.
Course Help 911 Reviews
The vectorization of data into a linear format is hard to do, but the idea of vectorizing arrays is surprisingly good. Instead, vectorized data is to be represented as views of more helpful hints
vector of vectors. Here are two illustrative examples that illustrate the idea. Recall that each of the columns of your matrix is represented as an *n*-dimensionalWho can help me with matrix error
propagation analysis in R programming? This is the source code #ubuntu-Desktop 2007-06-07 @waze85: I think you can use %{name}, but a namespace needs to be linked by parenthesis and you need the name
as well. i think that you need to make it fully qualified name for your variable in each package you read. @waze85: (defun %{name}: i18n “\\.\” \\n:” @waze85: oops error here I could google… but I
don’t like to learn the Java language. @waze85: that’ll make all of %{alias}, but that’s not it right? oops error here I just understand it too. Yes, but that’s a bad practice we don’t like to
install now so we didn’t have to do that when we were learning it for a while for 1 year. Oh yeah, I forgot Odd_Bloke: \ \| ## Test program : /usr/bin/python/something >appname\s+(%{name}: /path/to/
package/setup_json_s2.py): ,\n\a\n\c[ __init__-\s+ ..- .vnd.class: \ .vnd.class: 1-\2,\n1-\4,\n\n\*\2+ ;\n4-4\3\4,\n8\5\6\7\1\2-\1,\n4-8\5\1\3-\1\n8\3\7\2-\1\1\4,\n8\6\1-\1\3 __getattr__: \6+ /usr/
Online Classwork
module ..\1\4 5- :/usr/local/lib/python2.7/dist-packages “-e:”= \4 @Ulf, :/usr/bin/python # $.@:3 : 1.3 \0/ ?? @Ulf, :/usr/bin/python1.8 Who can help me with matrix error propagation analysis in R
programming? As Jadhav said, “This is not a field for everyone, and not a field for everyone on top of the hierarchy.” “This is a field for everyone,” jadhav said, “but a field for everyone who’s
worked with R, is not a good fit for everyone right now.” If you’ve made such an mistake you can simply look at it and reply, saying: “Thank you.” And he was still saying, “Thank you.” This is NOT a
field for anyone, nor is this a lot of words. Jadhav replied, “Pardon.” Well, the next time you make mistakes, please try not to create “holes” in the main text. Also, don’t wait for the correct
answer to your mistakes. They’ll do you a favor. I understand the urgency in making these mistakes, which you all have to keep in mind. Jadhav was still a good Christian and good in his own right, so
have to maintain that for the centuries to come. Now, in case you haven’t realized, this was written on some great paper. This was just your average book, and it’s not common knowledge that this type
of book is designed to be used on a church. That would have been good to use, I have some “lovely” books that were published in Canada.
Take My College Algebra Class For Me
This is something that I often read with two-people, why not two like it. If a book were dedicated solely by the author, I would have been more than happy to purchase it. As the author of this one, I
can say this, not so much as I can talk about the church, so in essence will you buy it? Not sure I’ve spoken to them at dinner last night, but after you check out this post, you’d quickly figure out
that the author is definitely a conservative, probably socialist, individual. I mean, it reads very strange, right? Someone who reads the book, they’ll be told he means “excellent” in such a way that
he actually wishes he didn’t read it. Sorry dude, but don’t you have to wait for the correct answer? I don’t like the term “good guy”. I’m not trying to make a joke. I’m just pointing out that I
don’t actually like anyone. I may be wrong, but just need to know that it would be nice to have some sort of clue… It was written by two guys, when the original book was on the back, I couldn’t go to
copy it. I posted it in the comments, all of the same ones. official statement Bower’s (Thesis?) I really need to realize I’ve been lied to. Nothing just happened, and now I am so deep in the mire
that I am holding everything on fire. Why was that? I am not allowed to say any more about it, though… but I am being educated since it is not enough to report it, so what? Just because something is
so right, I don’t understand why the author of the book could be “good.” Isn’t that how it works? As long as you follow where I was pointing you to she is absolutely free to go forward. She doesn’t
have to worry about who you are next, but before you go forward on the topics, tell her crack the r coding assignment your being the only one is standing in that same ground.
Can I Pay Someone To Write My Paper?
Okay, some really good things are happening to you from time to time, but a weak “princessed” woman, and maybe an I-imagine-A-neighbor-like-soul-coming-to-you-takes-all-from-that-fem arrived and is
now “loved” in that same place; and then you only make her angry. Hey
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From Encyclopedia of Mathematics
A projective transformation (projective isomorphism) of a projective space Perspective); if
The projective collineations exhaust all the projective transformations if and only if every automorphism of the skew-field
There is a large amount of confusion in the literature on projective geometry about the terminology for the different kinds of transformations. The transformation defined above, a projective
collineation, sometimes also called a projectivity, is usually defined as
[a1] M. Berger, "Geometry" , I , Springer (1987)
How to Cite This Entry:
Collineation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Collineation&oldid=17653
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article
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Quilting Pi
When John Sims contemplates a number, he sees color and shape. And an intriguing, enigmatic number such as pi, the ratio of a circle’s circumference to its diameter, conjures up vivid patterns that
belong on quilts.
Courtesy of John Sims
Photo by Tobey Albright
Photo by Tobey Albright
Photo by Tobey Albright
Starting with 3.14159265, the decimal digits of pi run on forever, and there’s no discernible pattern to ease the task of compiling (or memorizing) these digits. Computer scientists have so far
succeeded in computing 1,241,100,000,000 decimal digits of pi.
Both a mathematician and an artist, Sims taught for many years at the Ringling School of Art and Design in Sarasota, Fla. He’s passionately interested in the collision of mathematical ideas and
visual culture.
Pi is one of the few mathematical constants that have successfully entered the pop-culture psyche, Sims notes. Pi has appeared as the title of movie, for instance, and as the name of a perfume.
Several years ago, Sims created a visualization of pi’s digits in a digital video format—with music by Frank Rothkamm and the participation of Paul D. Miller, who is better known on the New York City
scene and elsewhere as Dj Spooky. You can find this creation at the John Sims Project Web site. In this visualization, each of the digits from 0 to 9 is represented by its own color on a vast grid of
Working in base 2 and using the colors black and white, Sims then created “Black White Pi”. In base 3, using red, white, and blue, he made “American Pi.”
A second pi-based project involved a collaboration with conceptual artist Sol LeWitt. LeWitt’s instructions were to put 1,000 straight lines inside a square. Sims achieved that result by dividing
each side of the square into 10 parts (like the axes of a graph), labeling the divisions from 0 to 9, and drawing lines from a division on one side to a division on an adjacent side. The lines
followed successive digits of pi from side to side, starting at the top and moving in a clockwise direction until the wall drawing had 1,000 lines. Sims’ former student, Brandon Styza, actually drew
the lines. The result formed the basis for a LeWitt wall drawing in the math lounge at Wesleyan University in Middletown, Conn.
This year, before heading for New York City, Sims completed a number of pi works, including several quilts that were constructed by an Amish quilting group in Sarasota. These artworks were on display
at Sarasota’s mack b gallery.
Sims started out with a drawing of pi’s decimal digits on a square grid, with successive digits forming a clockwise spiral from the center.
In the gallery, this drawing was displayed with a phonograph that played a recording of Sims reciting the digits of pi in order. A second track presented the digits in German.
With each digit from 0 to 9 mapped to a different color (but not black or white), the central portion of the drawing was then converted into a striking, square quilt of colored patches, with a black
border. Sims calls the creation “Pi sans Salt and Pepper.”
In a variation on this pi-based theme, another quilt designed by Sims features several, differently color-coded representations of pi. It’s called “Civil Pi Movement.”
“The mathematical art that I seek to develop combines mathematical language and analysis with the expressiveness and creativity of the process to make expressive visual theorems,” Sims says. “To see
mathematically, one draws from creativity and intuition, as in the case with the art process itself.”
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Why am I getting different results when using the law of total expectation?
• Thread starter adnaps1
• Start date
In summary, the conversation discusses calculating the expected number of red marbles in a box after 3 selections, given that there was initially 1 red and 1 black marble. Two different methods are
used, but one method appears to be incorrect due to not considering the probability of the initial condition. The correct method uses the law of total expectation and considers the probability of the
initial condition.
Suppose a box initially contains 1 red marble and 1 black marble and that, at each time n = 1, 2, ..., we randomly select a marble from the box and replace it with one additional marble of the same
color. Let X_n denote the number of red marbles in the box at time n (note that X_0 = 1). What is E(X_3)?
In solving this problem, I would like to calculate E(X_3 | X_2 = 1). If there is 1 red marble at time 2 (X_2 = 1), that means the first 2 selections resulted in black marbles. So at the time of the
third selection, there are 3 black marbles in the box and 1 red marble. Therefore, E(X_3 | X_2 = 1) = 1(3/4) + 2(1/4) = 5/4 (that is, 1 with probability 3/4 and 2 with probability 1/4).
However, if I would like to calculate E(X_3 | X_2 = 1) differently, using the law of total expectation, I can write E(X_3 | X_2 = 1) = E(X_3 | X_2 = 1, X_1 = 1) P(X_1 = 1). (There are no other values
of X_1 to condition on, because if we know there is 1 red marble at time n = 2, there cannot be 2 red marbles at time n = 1.) However, this simplifies to [1(3/4) + 2(1/4)](1/2) = 5/8.
Why am I getting different results? I think the problem has something to do with the following: When I condition on X_1 = 1, I already know X_1 = 1 with probability 1 because X_2 = 1; however, then I
say P(X_1 = 1) = 1/2, which is also true.
adnaps1 said:
However, if I would like to calculate E(X_3 | X_2 = 1) differently, using the law of total expectation, I can write E(X_3 | X_2 = 1) = E(X_3 | X_2 = 1, X_1 = 1) P(X_1 = 1).
Shouldn't that be E(X_3 | X_2 = 1) = E(X_3 | X_2 =1, X_1=1) P(X_1=1 | X_2 = 1) ?
Yes, you're right. Thank you very much.
FAQ: Why am I getting different results when using the law of total expectation?
What is the "Law of Total Expectation"?
The Law of Total Expectation states that the expected value of a random variable can be calculated by taking the sum of the expected values of its conditional distributions, weighted by their
respective probabilities.
Why is the "Law of Total Expectation" important in probability and statistics?
The Law of Total Expectation allows us to calculate the expected value of a complex random variable by breaking it down into simpler conditional distributions. This is useful in various fields such
as finance, economics, and engineering, where we often deal with multiple variables and their relationships.
How is the "Law of Total Expectation" used in real-world applications?
The Law of Total Expectation is used in many real-world applications, such as forecasting stock prices, predicting weather patterns, and estimating insurance premiums. It allows us to make informed
decisions based on the expected value of a variable, taking into account all possible outcomes.
What is the difference between the "Law of Total Expectation" and the "Law of Total Probability"?
The Law of Total Expectation deals with the expected value of a random variable, while the Law of Total Probability deals with the probability of an event. The former is used to calculate the
expected value of a complex random variable, while the latter is used to calculate the probability of an event given multiple possible outcomes.
Can the "Law of Total Expectation" be applied to continuous random variables?
Yes, the Law of Total Expectation can be applied to both discrete and continuous random variables. In the case of a continuous random variable, the sum is replaced by an integral, and the
probabilities are represented by a probability density function.
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Abstracts for Mini-Workshop on "Geometric Group Theory in Bonn", January 31 - February 1, 2019
University of Oxford
Fri, 01/02/2019 - 09:30 - 10:30
Cross ratios naturally arise on boundaries of negatively curved spaces
and are a valuable tool in their study. If one however slightly relaxes the
curvature assumption, simply requiring it to be *non-positive*, things
tend to get more complicated. Even the mere definition of a cross ratio becomes a
more delicate matter.
Restricting to the context of CAT(0) cube complexes $X$, we observe that
most issues disappear if one considers the $\ell^1$ metric on $X$, rather than the
CAT(0) metric. We obtain a canonical cross ratio on the horoboundary of the $\ell^1$
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T-Test Add-on for the SQL Statistics Package
By: Rick Dobson | Updated: 2018-09-25 | Comments | Related: > TSQL
MSSQLTips.com previously published a helpful introduction for doing statistics with SQL Server T-SQL code. Please update the SQL statistics package to allow more and different kinds of t tests. Also,
enable the package to lookup automatically the statistical significance of computed t values. Finally, demonstrate how to compute an F test to assess if the variance in one group is different than
An important element of many data science projects is the capability to perform statistics. As the popularity of data science projects within organizations continues to grow, it becomes increasingly
beneficial for SQL Server professionals to have an integrated set of stored procedures for implementing analyses of data in SQL Server instances. These stored procedures should be easy to use, add on
to, and expand as data science requirements evolve. This tip builds on several prior tips in moving towards the fulfillment of these goals.
Here's some history on the topic.
• The T-SQL Starter Statistics Package for SQL Server tip initially presented a SQL starter statistics package available from MSSQLTips.com.
• This general topic was revisited in a follow-up overview tip on the role of statistics in data science projects and how to implement statistics in SQL Server.
• This tip extends the starter statistics package by adding new functionality relating primarily to t tests. A t test is very commonly used to assess whether the average of one group of values is
statistically different from another group of values or a standard value.
This tip extends the SQL starter statistics package with new stored procedures, tables, and additional capabilities that remove the need to reference external resources for looking up the statistical
significance of computed t values.
• Some new SQL Server tables based on Excel built-in statistics functions remove the need for manually looking up the statistical significance of computed t values. Understanding the derivation of
these tables can grow your knowledge of Excel functions used to populate critical values in the lookup tables and generally enhance your understanding of how to perform statistics with SQL.
• You will discover within this tip, SQL code for implementing five different variations of a t test. Each section for a t test begins with a short statement explaining when to use it. Each t test
presentation includes a stored procedure for implementing the t test, sample data for demonstrating the t test, and some SQL code illustrating the running of the stored procedure with the sample
□ The sections for two previously presented stored procedures highlight how to add on the capability for automatically looking up whether computed t values are statistically different.
□ This tip also describes three additional types of t tests not included in the SQL starter statistics package.
□ The stored procedures for the freshly demonstrated t tests along with the original two t tests equip you with a wide range of statistical tools for evaluating the differences of means with
the SQL statistics package.
• Finally, this tip introduces the package's first F test. In the context of this tip, the F test allows you to statistically verify if the variances for two sample groups of scores are from
populations with the same or different variances. This capability is important because it can help you decide which of two different t tests to use for evaluating the differences between two
means from distinct groups.
A framework for the statistics package
The initial release of the SQL statistics package featured two SQL Server objects.
• A temporary table for holding the input data for which the package generates a statistical value, such as a computed t value or the median for a group of scores. The layout of the temporary table
can vary from one statistical analysis technique to the next.
• A stored procedure for performing the computations for a statistic. The stored procedure performs one or more computations to calculate a computed value based on the data in a temporary table.
With these two objects, any SQL professional should be easily able to use the SQL statistics package to perform statistics. Just follow these steps.
• Discover the stored procedure for the statistical technique that needs to be implemented.
• Populate the temporary table for the stored procedure that implements the desired statistical method.
• Run the stored procedure and use or examine the output.
Here's some pseudo code for the stored procedures used to compute statistics in this tip. The original statistics package did not have any input parameters, but all the stored procedures in this
add-on to the original package require one or more input parameters. This add-on to the original SQL statistics package performs some of its calculations by
• embedding a stored procedure from the original package into a new stored procedure or
• creating a new stored procedure from scratch that enables additional features not available with the SQL starter statistics package
create procedure stored_procedure_name input parameter(s)
set nocount on;
Place your debugged code here for a statistic and whatever associated values you want to
output from your stored procedure.
The package relies on the population of global temporary tables with sample values for demonstrating how to run a t test by invoking a stored procedure. A global temporary table facilitates
development and testing because it permits debugging a script in one SSMS tab and then copying the finished code into another tab with a collection of stored procedures for performing different kinds
of statistical tests. A professional and/or enterprise SQL statistics package may benefit from using another SQL Server object for storing the data for a stored procedure. For example, you can
consider local temporary tables, table variables, common table expressions, or permanent SQL Server tables.
The original SQL statistics package tip and this add-on tip were tested and developed inside the AllNasdaqTickerPricesfrom2014into2017 database; see here and here for more about this database. At
present, you will need to manually copy stored procedures and some lookup tables to another database if you want to run the code from within another database. As the scope of the SQL statistics
package grows, future tips may address using the package with any database without the need for copying stored procedures and tables between databases.
Computing and saving lookup tables containing critical values
The degrees of freedom for a computed t value or a computed F value can be looked up in a table of critical t or F values to assess if the computed t or F value is statistically significant in a
one-tailed test or a two-tailed test. The type of test you use (two-tailed or one-tailed) depends on your null and alternative hypotheses.
• You should use a two-tailed test if it just matters that the two means are different, but it does not matter that one mean is either greater or less than the other mean. In this kind of test:
□ The null hypothesis is that the means are the same.
□ The alternative hypothesis is that the means are not the same, and it does not matter which mean is greater than the other.
□ This type of test is common when doing exploratory data analysis, and you have no a priori reason for believing that one group's mean is greater than another.
• You should use a one-tailed test if it does matter that one mean is greater than the other mean.
□ For example, if you are testing two different web page designs (a new one versus an old one), you may care which one generates the largest dollar amount of orders. In this case, it matters
critically for whether to switch to the new page design if the dollar volume for the new page is greater than the dollar volume for the original page.
□ The null hypothesis is that the new page does not generate a larger dollar volume gain.
□ The alternative hypothesis is the new pages does generate a larger dollar volume.
• In addition to one-tailed and two-tailed t tests, this tip also demonstrates two-tailed F tests for assessing if the variance for one group is different than the variance for another group.
□ The null hypothesis is usually that the variances are the same in both groups.
□ The alternative hypothesis is that the variances are different, but it does not matter which variance is greater than the other.
□ This test for the equality of variances between two samples can indicate which t test to use for assessing if sample means are the same or different.
The following screen shot shows three tables of critical test values in the AllNasdaqTickerPricesfrom2014into2017 database. By comparing a computed t or F value to a critical t or F value based on
the degrees of freedom of the computed test value, you can confirm if a computed t or F is statistically significant at some probability level. The first table is for critical F values. The second
and third tables are for critical t values for one-tailed and two-tailed tests, respectively.
The design of the tables of critical values may differ depending on the type of test and how you want to assert the probability of a result being statistically significant.
• The two tables of critical values for evaluating computed t values have four columns each.
□ The first column is for degrees of freedom (df) values, which is typically based on the sample size for one or two groups of data values.
□ The three remaining columns are for different probability levels of assessing the difference between one sample mean and another or between a sample mean and a standard value.
☆ A common minimum probability level is .05, which means that there are 5 chances in one hundred that a difference of this size could occur without there being a difference between a sample
mean and a comparison value.
☆ If a difference is significant at .01 or .001 probability levels, then the likelihood of obtaining that computed t value by chance is correspondingly rarer.
• The table of critical F test values is designed specifically for testing if there is a difference in the variances between two groups.
□ The computed F value for this kind of test is based on the ratio of the variance within one group to the variance within another group. Each sample or group can have the same or different
sample sizes.
☆ The degrees of freedom for the numerator in a ratio is denoted by df_num, and the degrees of freedom for the denominator in a ratio is denoted by df_dnom.
☆ In either degrees of freedom, the sample size less one designates the degrees of freedom for a sample.
□ The F probability distribution, unlike the t probability distribution, is asymmetrical.
☆ The absolute value of a computed t value can be compared to the same critical t value no matter which group has the larger mean.
☆ In contrast, the computed F values need to be compared to different critical F values for each side of a two-tailed test. This is why the Critical_Fs_for_variance_test_table object has
two separate columns of critical values.
○ If the computed F value is either less than or equal to the critical left-tail-f value or greater than or equal to the critical right-tail-f value, then the variances are different at
the .05 probability level.
○ The critical F values table does not offer different probability value levels because if the variances are different at the .05 probability level, then that is sufficient to use one t
test versus the other for comparing the difference between the means.
The next screen shot shows a pair of excerpts from the one-tailed and two-tailed critical t value tables. There are two hundred fifty rows in each of the tables, but just the first twenty-two rows
are excerpted for the screen shots.
• The critical t values decline as the degrees of freedom (df) increase. It is easier to detect differences as the sample sizes for compared groups grow.
• The critical t values increase as the probability level rises from .05 through .001. It takes bigger differences to confirm a statistically difference at the .001 probability level than at the
.05 probability level.
• For the same degrees of freedom (df) and probability level, the two-tailed critical t values are larger than the critical t values for a one-tailed test. In this sense, it is easier to confirm
that group means are different for a one-tailed test than for a two-tailed test.
The next screen shot shows an excerpt from the F critical values lookup table in the database named AllNasdaqTickerPricesfrom2014into2017. Computed F values based on sample data are compared to
critical F values to assess if an outcome is statistically significant. For this tip, the computed F value derives from the ratio of two sample variances.
• Recall that the df_num column value and df_dnom column value represent, respectively, the degrees of freedom for a F computed value.
• The left-tail-f column value represents a critical F value. Computed F values less than or equal to the left-tail-f column value qualify as statistically significant at the .05 level. In other
words, the null hypothesis is rejected.
• The right-tail-f value represents another critical F value. Computed F values greater than or equal to the right-tail-f column value qualify as statistically significant at the .05 level. In
other words, the null hypothesis is rejected.
• Computed F values that are greater than the left-tail-f column value and less than the right-tail-f value result in the acceptance of the null hypothesis, which for this tip corresponds to there
being no significant difference in the variances between the two samples.
The expressions for computing critical t and F values can be complex. Beyond that, fully understanding the expressions for computing critical t and F values involves developing an appreciation of the
distinction between probability density functions and cumulative probability density functions (for example, see here, here, here, and here).
Happily, there is an easier way for users of Microsoft software to derive critical t and F values; this tip relies on the T.INV.2T Excel function for calculating critical t values. Microsoft Excel
includes multiple built-in functions for computing critical values for multiple kinds of statistical functions. For example, the following spreadsheet shows an excerpt with critical t values from a
worksheet tab named one-tailed.
• Start your review by confirming that the values in this table correspond to those for the one-tailed critical t values shown above.
• Therefore, the column A values represent df values and column B, C, and D values represent critical t values for .05, .01, and .001 probability levels.
• The cursor rests in cell B2, which denotes a critical t value for a computed t value with a df value of one and a probability level of .05.
• The expression in cell B2 is =ABS(T.INV(B$1,$A2)). This expression links the cell to a df value from cell A2 and the probability level in cell B1.
• The full worksheet tab has critical t values based on the formula for df values from one through two hundred fifty at probability levels of .05, .01, and .001.
• By switching the expression in cell B2 to =T.INV.2T(B$1,$A2), you can compute the comparable critical t values for a two-tailed t test. The two-tailed critical t values for df values one through
two hundred fifty at .05, .01, and .001 probability levels are in another worksheet tab named two-tailed.
• Both the one-tailed and two-tailed tabs reside in an Excel worksheet file named "critical t values from excel functions". This file is available for download with this tip.
• After computing critical t values in Excel, you can import them into a table within a SQL Server database. This is how the Critical_ts_for_mean_difference_test_1_tail_table and
Critical_ts_for_mean_difference_test_2_tail_table objects described above were populated in the AllNasdaqTickerPricesfrom2014into2017 database.
A comparable worksheet tab was developed for critical F values with df_num and df_dnom from three through one thousand. The Excel workbook file has the name critical F values from excel functions,
and it is available as a download with this tip. The F distribution is defined by two degrees of freedom values (df_num and df_dnom). Each critical F value is defined by the intersection of values
from these two different degrees of freedom. For this tip, we are only interested in the two-tailed critical F values at the .05 probability level. This probability level is used to assess if
variances between two groups are the same or different. If the variances are different, it does not matter which one is greater.
The following screen shot shows an excerpt from the Excel workbook file that served as the original source for the Critical_Fs_for_variance_test_table object with critical F values in the SQL
statistics package. For each pair of df_num and df_dnom values there are a pair of critical F values. Within the screen shot, the cursor rests in cell C7989. The critical F value within this cell is
for df_num 11 and df_dnom 6. The function used to return the critical F value is =F.INV(0.025,A7989,B7989) for the left side of the critical F function values. The limit value is 0.257689. Cell D7989
contains the F function expression of =F.INV(0.975,A7989,B7989) for the limiting right tail value of 5.409761. After computing the values in Excel, you can use your favorite technique for
transferring the values to SQL Server.
Comparing two means from samples with the same size
This first t test is meant for determining if the difference between two sample means is statistically significant when the means for both samples are from the same population of entities. The sample
means can reflect the outcome from two different treatments applied to the members of each sample. This test depends on both samples having the same sample size. Additionally, the population variance
is assumed to be the same for both samples. MSSQLTips.com has a prior tip covering this type of t test without automatic lookup of the significance of a computed t value.
This t test depends on two equations, which are described in Wikipedia. One equation is for a pooled standard deviation, which is the square root of the average variance across the two random samples
from the same population. The S[p] term references the pooled standard deviation. The sample variance for the members in each sample X[1] and X[2], are simply divided by two to derive the pooled
sample variance, which is the average variance across both samples.
After the pooled standard deviation is computed, you can divide the difference between the means from each sample (X-bar[1] and X-bar[2]) by the pooled standard deviation weighted by the number of
members (n) in each sample across both samples. The degrees of freedom for the computed t value is 2n - 2 where n is sum of the sample size for each group. Therefore, if the sample size for each
group is six, then the degrees of freedom for the computed t value is ten.
These equations were programmed in SQL for the SQL starter statistics package. The starter package includes a stored procedure named compute_t_between_2_groups. The stored procedure implements the
preceding two equations. This add-on tip to the starter package embeds the compute_t_between_2_groups stored procedure in a new one named compute_t_between_2_groups_with_lookup. The new stored
procedure takes an input parameter designating whether to lookup the computed t value in a critical t values set for a one-tailed test or a two-tailed test.
The following script shows the code for the compute_t_between_2_groups_with_lookup stored procedure. The create procedure statement has an @tail_num input parameter. The code expects this parameter
to be 1 for a one-tailed test or 2 for a two-tailed test.
The beginning part of the compute_t_between_2_groups_with_lookup stored procedure sets up for and actually invokes the compute_t_between_2_groups stored procedure. The output from the
compute_t_between_2_groups stored procedure is inserted into the #two_group_ouput_base_temp table, which consists of a single row of values that include among other values the computed t and the df
for the computed t value.
Then, SQL code inner joins the #two_group_ouput_base_temp table with either the critical table of one-tailed t values (Critical_ts_for_mean_difference_test_1_tail_table) or the critical table of
two-tailed values (Critical_ts_for_mean_difference_test_2_tail_table). The inner join matches the two sources on df from the #two_group_ouput_base_temp table and the table of critical t values. The
@tail_num input parameter guides the join to the critical t values for a one-tailed or a two-tailed test.
A case statement with output named probability_of_significance returns one of four probability levels for the computed t value. The probability_of_significance values are:
• significant beyond .001
• significant beyond .01
• significant beyond .05
• not significant beyond .05
create procedure compute_t_between_2_groups_with_lookup @tail_num int
-- create #two_group_ouput_base_temp
begin try
drop table #two_group_ouput_base_temp
end try
begin catch
print '#two_group_ouput_base_temp not available to drop'
end catch
create table #two_group_ouput_base_temp
avg_group_1 float
,avg_group_2 float
,mean_difference float
,computed_t float
,df int
-- invoke base compute_t_between_2_groups sp
-- and save output in #two_group_ouput_base_temp
insert into #two_group_ouput_base_temp
exec compute_t_between_2_groups
if @tail_num = 1
-- join critical t values to outtput from sp (#two_group_ouput_base_temp)
-- use 1_tailed critical t values
when abs(computed_t) > t_001 then 'significant beyond .001 level for one-tailed test'
when abs(computed_t) > t_01 then 'significant beyond .01 level for one-tailed test'
when abs(computed_t) > t_05 then 'significant beyond .05 level for one-tailed test'
else 'not significant beyond .05 level'
end probability_of_significance
,@tail_num [test tail type]
from #two_group_ouput_base_temp c
inner join [dbo].[Critical_ts_for_mean_difference_test_1_tail_table] t
on c.df = t.df
if @tail_num = 2
-- join critical t values to outtput from sp (#two_group_ouput_base_temp)
-- use 2_tailed critical t values
top 1
when abs(computed_t) > t_001 then 'significant beyond .001 level for two-tailed test'
when abs(computed_t) > t_01 then 'significant beyond .01 level for two-tailed test'
when abs(computed_t) > t_05 then 'significant beyond .05 level for two-tailed test'
else 'not significant beyond .05 level'
end probability_of_significance
,@tail_num [test tail type]
from #two_group_ouput_base_temp c
inner join [dbo].[Critical_ts_for_mean_difference_test_2_tail_table] t
on c.df = t.df
The next code segment shows the input and output from the compute_t_between_2_groups_with_lookup stored procedure. The steps in the code are as follows.
• A global temporary table named ##temp_group_scores is created to hold the sample values from the two sets of input sample data.
• Next, six values each from the first and second samples are inserted into the global temporary table. Sample identifier values of Group_1 and Group_2 indicate whether a score is from the first or
second sample.
• Then, the code invokes the compute_t_between_2_groups_with_lookup stored procedure twice - initially for a one-tailed test and a second time for a two-tailed test.
• Finally, the code displays the source data for the computed t value.
-- prepare data for stored proc
-- create and populate ##temp_group_scores
begin try
drop table ##temp_group_scores
end try
begin catch
print '##temp_group_scores not available to drop'
end catch
create table ##temp_group_scores
group_id varchar(10)
,score float
-- from --http://www.stat.columbia.edu/~martin/W2024/R2.pdf
-- Two-sample t-tests section
insert into ##temp_group_scores values ('Group_1', 91)
insert into ##temp_group_scores values ('Group_1', 87)
insert into ##temp_group_scores values ('Group_1', 99)
insert into ##temp_group_scores values ('Group_1', 77)
insert into ##temp_group_scores values ('Group_1', 88)
insert into ##temp_group_scores values ('Group_1', 91)
insert into ##temp_group_scores values ('Group_2', 101)
insert into ##temp_group_scores values ('Group_2', 110)
insert into ##temp_group_scores values ('Group_2', 103)
insert into ##temp_group_scores values ('Group_2', 93)
insert into ##temp_group_scores values ('Group_2', 99)
insert into ##temp_group_scores values ('Group_2', 104)
-- end of data prep for stored proc
-- invoke compute_t_between_2_groups_with_lookup sp
exec compute_t_between_2_groups_with_lookup @tail_num = 1
exec compute_t_between_2_groups_with_lookup @tail_num = 2
-- show data for t test
select * from ##temp_group_scores
The next screen shot shows the outcome from the preceding script. The first two lines are for the one-tailed test and two-tailed test, respectively. The last twelve lines are for the sample scores
from each of two samples. The scores represent reaction times to a stimulus, such as turning on a light. Group 1 is for subjects given a placebo, and Group 2 is for subjects given an active
ingredient. The results confirm that the active ingredient slowed reaction times in both a one-tailed test as well as a two-tailed test.
Comparing two sets scores based on different treatments from one sample
Another t test previously addressed in the SQL starter statistics package is one to assess the difference between two sets of measurements on sample members from a single group; this kind of t test
is often called a paired sample t test. When using a design for this t test, a data science project reduces sources of variance relative to a two-sample design by re-using the same sample members for
two separate rounds of measurements. For example, you could answer the question: is gas mileage better with premium or regular gasoline for a single set of cars? Each car can have its miles per
gallon measured twice - once with regular gas and a second time with premium gas.
Like the preceding section, the goal of this section is to develop an adaptation of the starter package code that automatically looks up the statistical significance of a computed t value. Because
this type of t test examines two sets of measures for a single group of sample members, it computes a difference between the two measurements for each member. The t test compares the mean size of the
differences to the weighted sample standard deviation of the differences between the two sets of measurements.
Wikipedia presents the following equation for computing a t value for this kind of test. The X-bar[D] term is for the mean of the differences between the measurements for the sample members. The S[D]
term is for the sample standard deviation of the differences. The number of differences is n, and the reciprocal of the square root of n is the weight for the standard deviation. The degrees of
freedom for looking up the statistical significance of the computed t value is n - 1.
The value of mu[0] is frequently omitted, and it is also omitted in this implementation of the test. When the sole goal of the test is just to assess if there is no difference between two sets of
measurements, there is no need for a mu[0] term. For this demonstration of the test, we keep a focus on the comparison of two sets of measurements to assess if they are the same or different.
The SQL starter statistics package uses a stored procedure named compute_paired_sample_t for computing this kind of t test. The add-on code re-uses the same stored procedure, but it joins the
computed t value and degrees of freedom from the stored procedure to critical t values for either one-tailed or two-tailed tests. The sole advantage of the add-on code for the t test is the automatic
lookup of the statistical significance of the computed t value.
The add-on code for the paired sample t test relies on a stored procedure named compute_paired_sample_t_with_lookup. Again, an input parameter named @tail_num allows a user to specify results for
either a one-tailed test or a two-tailed test. An If…Else statement facilitates the joining of output from the compute_paired_sample_t stored procedure with one-tailed or two-tailed critical t
values. As with the t test from the preceding section, the code reports statistical significance (or its absence) at one of four different levels (not signification at .05, significant at .05,
significant at .01, or significant at .001). Here's the code for creating the compute_paired_sample_t_with_lookup stored procedure.
create procedure compute_paired_sample_t_with_lookup @tail_num int
-- create #paired_sample_ouput_base_temp
begin try
drop table #paired_sample_ouput_base_temp
end try
begin catch
print '#paired_sample_ouput_base_temp not available to drop'
end catch
create table #paired_sample_ouput_base_temp
avg_difference float
,computed_t float
,df int
-- invoke base compute_paired_sample_t sp
-- and save output in #paired_sample_ouput_base_temp
insert into #paired_sample_ouput_base_temp
exec compute_paired_sample_t
if @tail_num = 1
-- creates new output column based on probability
-- level of significance for one-tailed and two-tailed tests
when abs(computed_t) > t_001 then 'significant beyond .001 level for one-tailed test'
when abs(computed_t) > t_01 then 'significant beyond .01 level for one-tailed test'
when abs(computed_t) > t_05 then 'significant beyond .05 level for one-tailed test'
else 'not significant beyond .05 level'
end probability_of_significance
,@tail_num [test tail type]
from #paired_sample_ouput_base_temp c
inner join [dbo].[Critical_ts_for_mean_difference_test_1_tail_table] t
on c.df = t.df
if @tail_num = 2
when abs(computed_t) > t_001 then 'significant beyond .001 level for two-tailed test'
when abs(computed_t) > t_01 then 'significant beyond .01 level for two-tailed test'
when abs(computed_t) > t_05 then 'significant beyond .05 level for two-tailed test'
else 'not significant beyond .05 level'
end probability_of_significance
,@tail_num [test tail type]
from #paired_sample_ouput_base_temp c
inner join [dbo].[Critical_ts_for_mean_difference_test_2_tail_table] t
on c.df = t.df
The sample code for this tip also includes a demonstration of the use of the compute_paired_sample_t_with_lookup stored procedure with some sample data. Below is the code for the demonstration; you
can view its result sets in following screen shot.
• The ##temp_subject_id_results global temporary table stores the sample data.
• This table has one column for identifying members of each sample data member and two more columns for each set of measurements.
□ The names for first column row entries is arbitrary, but each row should have a unique name. Each of these names should correspond to a single sample member.
□ The names for second and third column row entries are for measurements. This kind of t test requires two measurements for each sample member. In this demonstration, column apply_1_result is
for miles per gallon from premium gas and apply_2_result is for miles per gallon from regular gas.
• The output shows the outcome for a one-tailed test followed by a two-tailed test. As you can see, the average difference of two miles per gallon is significant at beyond the .001 level for a
one-tailed test, but the same difference is significant at beyond the .01 level for a two-tailed test.
• The final set of output rows show the input data for the computed t values.
-- prepare data for stored proc
-- create and populate ##temp_subject_id_results
begin try
drop table ##temp_subject_id_results
end try
begin catch
print '##temp_subject_id_results not available to drop'
end catch
create table ##temp_subject_id_results
subject_id varchar(10)
,apply_1_result float
,apply_2_result float
--delete from ##temp_subject_id_results
-- from --http://www.stat.columbia.edu/~martin/W2024/R2.pdf
-- Paired t-tests section
insert into ##temp_subject_id_results values ('subject_1', 19, 16)
insert into ##temp_subject_id_results values ('subject_2', 22, 20)
insert into ##temp_subject_id_results values ('subject_3', 24, 21)
insert into ##temp_subject_id_results values ('subject_4', 24, 22)
insert into ##temp_subject_id_results values ('subject_5', 25, 23)
insert into ##temp_subject_id_results values ('subject_6', 25, 22)
insert into ##temp_subject_id_results values ('subject_7', 26, 27)
insert into ##temp_subject_id_results values ('subject_8', 26, 25)
insert into ##temp_subject_id_results values ('subject_9', 28, 27)
insert into ##temp_subject_id_results values ('subject_10', 32, 28)
-- end of data prep for stored proc
-- invoke compute_paired_sample_t_with_lookup sp
exec compute_paired_sample_t_with_lookup @tail_num = 1
exec compute_paired_sample_t_with_lookup @tail_num = 2
-- show data for t test
select * from ##temp_subject_id_results
Comparing one set of scores from one sample to a standard value
This is the first section of three in this tip dealing with t tests for which the computed t value is calculated from scratch. In other words, there is a no dependence on a prior tip for this t test.
The t test for this section is unique from other tests in this tip in that it depends on just one sample and just one set of measurements. The computed t value assesses via a two-tailed significance
test whether a set of scores for members from a single sample are the same or different from a standard value. You can use a one-tailed version of this test to assess if a set of scores from a sample
is greater (or less) than a standard value.
Wikipedia offers the following formula for computing the t value for this kind of test. The test is named the one-sample t test in Wikipedia. As you can see, the equation for this computed t value is
very similar to the one in the preceding section for assessing the difference between two sets of measurements from a single sample. The main difference is that this test does not require two sets of
measurements for a single sample. In fact, just one set of measurements are required for this test.
The x-bar term is for the mean of the sample scores. The S term is for the sample standard deviation of the scores in the sample. The n term represents the number of scores in the sample. The term mu
[0] represents the standard value to which the mean of the sample scores are compared; this term is used in the demonstration for this t test. The degrees of freedom are n - 1 for looking up the
statistical significance of the computed t value.
The stored procedure for this t test has the name compute_one_sample_t_with_lookup. This stored procedure takes two input parameter values. The first of these values is for the @tail_num parameter
that lets a user specify whether to lookup the statistical significance of the computed t value for a one-tailed test or a two-tailed test. This stored procedure also has an input parameter named
@mu_value, which is for the value of mu[0].
The code for creating the stored procedure appears next.
• A pair of queries, one nested within the other, populate the #one_sample_ouput_base_temp table with values for calculating the computed t value.
□ The for_one_sample_computed_t inner query computes the mean, standard deviation, and sample size based on the members in the sample. The input sample scores reside in the ##temp_group_scores
global table, which is populated from the code invoking the compute_one_sample_t_with_lookup stored procedure.
□ The outer query populates the #one_sample_ouput_base_temp temporary table with the inherited values, the value of @mu_value, and the computed t value and its degrees of freedom. The @mu_value
is included when evaluating the computed t value.
• Next, an If…Else statement facilitates joining the #one_sample_ouput_base_temp table with either one-tailed or two-tailed critical t values for assessing the statistical significance of the
computed t value. The @tail_num input parameter determines which set of critical values are used for assessing the statistical significance of the computed t value.
• Statistical significance is evaluated to one of four distinct levels (not significant beyond .05, significant beyond .05, significant beyond .01, or significant beyond .001).
create procedure compute_one_sample_t_with_lookup @tail_num int, @mu_value float
declare @mu float = @mu_value
-- create #one_sample_ouput_base_temp
begin try
drop table #one_sample_ouput_base_temp
end try
begin catch
print '#one_sample_ouput_base_temp not available to drop'
end catch
create table #one_sample_ouput_base_temp
x_bar float
,mu float
,s float
,computed_t float
,df int
insert into #one_sample_ouput_base_temp
,@mu mu
,(x_bar-@mu)/(s/sqrt(n)) computed_t
,df = n - 1
avg(score) x_bar
,stdev(score) s
,count(*) n
from ##temp_group_scores
) for_one_sample_computed_t
if @tail_num = 1
when abs(computed_t) > t_001 then 'significant beyond .001 level for two-tailed test'
when abs(computed_t) > t_01 then 'significant beyond .01 level for two-tailed test'
when abs(computed_t) > t_05 then 'significant beyond .05 level for two-tailed test'
else 'not significant beyond .05 level'
end probability_of_significance
,@tail_num [test tail type]
from #one_sample_ouput_base_temp c
inner join [dbo].[Critical_ts_for_mean_difference_test_1_tail_table] t
on c.df = t.df
if @tail_num = 2
when abs(computed_t) > t_001 then 'significant beyond .001 level for two-tailed test'
when abs(computed_t) > t_01 then 'significant beyond .01 level for two-tailed test'
when abs(computed_t) > t_05 then 'significant beyond .05 level for two-tailed test'
else 'not significant beyond .05 level'
end probability_of_significance
,@tail_num [test tail type]
from #one_sample_ouput_base_temp c
inner join [dbo].[Critical_ts_for_mean_difference_test_2_tail_table] t
on c.df = t.df
The sample data used for demonstrating the computation of the one-sample computed t value is derived from data and a worked example by professor Martin A. Lindquist when he served in the Statistics
department at Columbia university. The url for the web resource appears with the sample data and worked example in a comment line within the code below.
• As you can see, the code for invoking the compute_one_sample_t_with_lookup stored procedure follows the same outline as for the preceding three code examples on running the stored procedure for
invoking code to compute a t value and lookup its statistical significance.
• The fact that all members contributing data to the computed t value are from a single population is underscored by each row having a group_ID value of Group_1 in the ##temp_group_scores table.
• The @mu_value parameter is .3. In the sample of measurements used for this demonstration, this is the standard salmonella count for ice cream batches that are counted as safe. Salmonella count
values greater than .3 are not safe.
• The MPN/g count for a sample of 9 ice cream batches is 0.4564, which is clearly greater than the upper limit for a safe value of .3.
• The null hypothesis tested is whether the MPN/g count less than or equal to .3. Consequently, the alternative hypothesis is whether the MPN/g count is greater than .3.
• Because testing is for greater than a standard value, a one-tailed t test is required.
-- prepare data for stored proc
-- create and populate ##temp_group_scores
begin try
drop table ##temp_group_scores
end try
begin catch
print '##temp_group_scores not available to drop'
end catch
create table ##temp_group_scores
group_id varchar(10)
,score float
-- from --http://www.stat.columbia.edu/~martin/W2024/R2.pdf
-- Two-sample t-tests section
insert into ##temp_group_scores values ('Group_1', 0.593)
insert into ##temp_group_scores values ('Group_1', 0.142)
insert into ##temp_group_scores values ('Group_1', 0.329)
insert into ##temp_group_scores values ('Group_1', 0.691)
insert into ##temp_group_scores values ('Group_1', 0.231)
insert into ##temp_group_scores values ('Group_1', 0.793)
insert into ##temp_group_scores values ('Group_1', 0.519)
insert into ##temp_group_scores values ('Group_1', 0.392)
insert into ##temp_group_scores values ('Group_1', 0.418)
-- end of data prep for stored proc
-- invoke stored procedures for one sample t test
exec compute_one_sample_t_with_lookup @tail_num = 1, @mu_value = .3
exec compute_one_sample_t_with_lookup @tail_num = 2, @mu_value = .3
-- show data for t test
select * from ##temp_group_scores
The output from the preceding code appears in the following screen shot.
• Results are presented for one-tailed and two-tailed tests, but only the one-tailed test results apply to the hypothesis being tested in this case.
• The results are significant at the .05 probability level for a one-tailed test. This test is the appropriate one for the sample data.
• The counts for individual batches of ice cream appear below the .05 significance level for a two-tailed test. This test is only run to confirm the operation of the code.
Comparing two means from samples with equal or unequal sample sizes and equal variance
Up to this point in the tip, we considered one two-sample t test. Comparing the difference between the means for two independent samples is a common data science requirement. However, there are
several slight variations among two-sample t tests.
• The first t test in this tip is for two samples with the same sample size and variance. This is the t test previously demonstrated in the "Comparing two means from samples with the same size"
• The t test for this section is for the case where the sample sizes can be different between the two samples, but the variance is assumed to be the same.
• The t test for the next section is for the case where the sample sizes can be different, but the variances are assumed to be different.
I found a couple of highly useful resources for calculating a computed t value for the differences between two sample means with unequal sample sizes (Wikipedia and StatsDirect). The StatsDirect site
was particularly helpful because it offered a worked example and helpful intermediate result sets. The code for this section relies on the data from the StatsDirect sample demonstration and the
equations for Wikipedia.
The Wikipedia site offers two equations for computing a t value and degrees of freedom when computing a t value between two samples which can have unequal sample sizes.
• The top-level equation is for the computed t value. The two terms X-bar[1] and X-bar[2] refer, respectively, to the means for the first and second samples. The S[p] term is for the pooled
standard deviation across the two samples. The terms n[1] and n[2] are, respectively, for the sample sizes of the first and second samples.
• The equation for the pooled standard deviation appears below. The S^2-sub-X[1] term and the S^2-sub-X[2] terms are, respectively, the sample variance for the first and second samples. You can use
the built-in T-SQL functions for computing the values of these terms.
• The degrees of freedom (df) for looking up the statistical significance of the computed t value is n1 + n2 - 2. This is an implicit equation for the degrees of freedom.
The following script shows the create procedure statement for computing a t value and interpreting it relative to a one-tailed or two-tailed type of t test. This interpretation includes calculating
the appropriate degrees of freedom for the sample data that serves as a source for the computed t value and matching those degrees of freedom with corresponding critical t values.
• The name of the stored procedure is compute_t_between_2_groups_with_unequal_sizes_with_lookup.
• The @tail_num input parameter allows a user to designate either a one-tailed or two-tailed comparison.
• An If…Else statement allows one of two blocks of code to execute.
□ Within each code block, the computed t value is calculated and then matched to appropriate critical t values.
□ A case statement assigns a probability level for the computed t value relative to the appropriate critical t values.
□ A collection of nested queries within the If and Else code blocks derives computed t values.
☆ The innermost nested subqueries have the names Group_1 and Group_2; these subqueries compute the mean, variance, and sample size within each group.
☆ Next, a subquery with a cross join brings the mean, variance, and sample size for each group into a single-line result set for the for_computed_t subquery. In addition, the subquery with
the cross join computes the pooled standard deviation and the difference between the means from the two samples.
☆ The subquery containing the cross join subquery develops the computed t value and the degrees of freedom for looking up appropriate critical t values.
☆ Finally, the outermost query within each If…Else code block computes the probability level of obtaining a computed t value relative to the appropriate critical t values.
create procedure compute_t_between_2_groups_with_unequal_sizes_with_lookup @tail_num int
if @tail_num = 1
when abs(computed_t) > t_001 then 'significant beyond .001 level for one-tailed test'
when abs(computed_t) > t_01 then 'significant beyond .01 level for one-tailed test'
when abs(computed_t) > t_05 then 'significant beyond .05 level for one-tailed test'
else 'not significant beyond .05 level'
end probability_of_significance
,@tail_num [test tail type]
mean_Group_1 avg_group_1
,mean_Group_2 avg_group_2
,mean_difference/t_dnom computed_t
, (n1-1) + (n2-1) df
-- Combined results for Group_1 and Group_2
-- with inputs for computed_t
,sqrt(((n1-1)*variance_Group_1 + (n2-1)*variance_Group_2)/((n1-1)+(n2-1))) sp
,sqrt(1/cast(n1 as float) + 1/cast(n2 as float)) sp_multiplier
,sqrt(((n1-1)*variance_Group_1 + (n2-1)*variance_Group_2)/((n1-1)+(n2-1)))
sqrt(1/cast(n1 as float) + 1/cast(n2 as float)) t_dnom
,(mean_Group_1 - mean_Group_2) mean_difference
-- Group_1 results
count(*) n1
,avg(score) mean_Group_1
,var(score) variance_Group_1
from ##temp_group_scores where group_id = 'Group_1'
) Group_1
cross join
-- Group_2 results
count(*) n2
,avg(score) mean_Group_2
,var(score) variance_Group_2
from ##temp_group_scores where group_id = 'Group_2'
) Group_2
) for_computed_t
) c
inner join [dbo].[Critical_ts_for_mean_difference_test_1_tail_table] t
on c.df = t.df
if @tail_num = 2
when abs(computed_t) > t_001 then 'significant beyond .001 level for two-tailed test'
when abs(computed_t) > t_01 then 'significant beyond .01 level for two-tailed test'
when abs(computed_t) > t_05 then 'significant beyond .05 level for two-tailed test'
else 'not significant beyond .05 level'
end probability_of_significance
,@tail_num [test tail type]
mean_Group_1 avg_group_1
,mean_Group_2 avg_group_2
,mean_difference/t_dnom computed_t
, (n1-1) + (n2-1) df
-- Combined results for Group_1 and Group_2
-- with inputs for computed_t
,sqrt(((n1-1)*variance_Group_1 + (n2-1)*variance_Group_2)/((n1-1)+(n2-1))) sp
,sqrt(1/cast(n1 as float) + 1/cast(n2 as float)) sp_multiplier
,sqrt(((n1-1)*variance_Group_1 + (n2-1)*variance_Group_2)/((n1-1)+(n2-1)))
sqrt(1/cast(n1 as float) + 1/cast(n2 as float)) t_dnom
,(mean_Group_1 - mean_Group_2) mean_difference
-- Group_1 results
count(*) n1
,avg(score) mean_Group_1
,var(score) variance_Group_1
from ##temp_group_scores where group_id = 'Group_1'
) Group_1
cross join
-- Group_2 results
count(*) n2
,avg(score) mean_Group_2
,var(score) variance_Group_2
from ##temp_group_scores where group_id = 'Group_2'
) Group_2
) for_computed_t
) c
inner join [dbo].[Critical_ts_for_mean_difference_test_2_tail_table] t
on c.df = t.df
The next code block shows the SQL for loading sample data into the ##temp_group_scores table. This table holds the data for a computed t value.
• The sample data comes from the StatsDirect web site. The sample is for female rat weights based on a high protein diet versus a low protein diet.
• There is a total of 19 data points:
□ 12 data points for the first group (high protein)
□ 7 additional data points for the second group (low protein)
• Two exec statements run both one-tailed and two-tailed tests for the sample data, but the one-tailed test is the appropriate one for assessing if the high protein diet results in greater weights
among the rats than a low protein diet. The two-tailed test is included merely to confirm the operation of the Else block in the If…Else statement for the
compute_t_between_2_groups_with_unequal_sizes_with_lookup stored procedure.
-- prepare data for stored proc
-- create and populate ##temp_group_scores
begin try
drop table ##temp_group_scores
end try
begin catch
print '##temp_group_scores not available to drop'
end catch
create table ##temp_group_scores
group_id varchar(10)
,score float
-- from --https://www.statsdirect.co.uk/help/parametric_methods/utt.htm
-- Two-sample t-tests section with unequal sample sizes
insert into ##temp_group_scores values ('Group_1', 134)
insert into ##temp_group_scores values ('Group_1', 146)
insert into ##temp_group_scores values ('Group_1', 104)
insert into ##temp_group_scores values ('Group_1', 119)
insert into ##temp_group_scores values ('Group_1', 124)
insert into ##temp_group_scores values ('Group_1', 161)
insert into ##temp_group_scores values ('Group_1', 107)
insert into ##temp_group_scores values ('Group_1', 83)
insert into ##temp_group_scores values ('Group_1', 113)
insert into ##temp_group_scores values ('Group_1', 129)
insert into ##temp_group_scores values ('Group_1', 97)
insert into ##temp_group_scores values ('Group_1', 123)
insert into ##temp_group_scores values ('Group_2', 70)
insert into ##temp_group_scores values ('Group_2', 118)
insert into ##temp_group_scores values ('Group_2', 101)
insert into ##temp_group_scores values ('Group_2', 85)
insert into ##temp_group_scores values ('Group_2', 107)
insert into ##temp_group_scores values ('Group_2', 132)
insert into ##temp_group_scores values ('Group_2', 94)
-- end of data prep for stored proc
exec compute_t_between_2_groups_with_unequal_sizes_with_lookup @tail_num = 1
exec compute_t_between_2_groups_with_unequal_sizes_with_lookup @tail_num = 2
-- show data for t test
select * from ##temp_group_scores
The output from the preceding script appears in following screen shot.
• The results are statistically significant for a one-tailed test at beyond the .05 probability level.
• The two-tailed test outcome is shown to confirm the operation of code for that path in an If…Else block.
• The display of input scores verifies the values used to derive a computed t value.
Comparing two means from samples with unequal variances
When comparing two means where the samples are known (or suspected) to have unequal variances, you can use the t test described in this section. At a top-line conceptual level, there are three
equations to consider for the computed t value: the difference between the sample means, the divisor term for the difference between the sample means, and a process for arriving at a degrees of
freedom for discovering appropriate critical t values for the t test.
• The first equation is the same as for the t test in the preceding section. It is merely the mean for the first sample less the mean for the second sample.
• The second and third equations are substantially different from the preceding t test, and they are more complicated from a computational perspective.
In addition to Wikipedia, I found a couple of other highly informative references for the t test in this section. Because of the computational complexity of this t test, it is especially useful to
have worked examples as references. The Real Statistics Using Excel website had an especially rich explanation of the theory behind this test as well as good coverage of best practices for
simplifying the computations for this kind of t test. Adjunct professor Marcus John Hamilton of the University of New Mexico offers a detailed worked example for computing a t test for two samples
with unequal variances. This tip uses professor Hamilton's sample data because it was so easy to perform unit tests with the help of his published example, and it allows a cross check of the
computing framework from the Real Statistics Using Excel website.
The Real Statistics Using Excel website offers five equations that were coded in SQL within this tip for developing a computed t value and its matching degrees of freedom. The following is the first
of the equations for the computed t value. The x-bar term is for the mean of the first sample. The y-bar term is for the mean of the second sample. The S^2-sub-X terms and the S^2-sub-Y terms are,
respectively, the sample variances for the first and second samples. The n[x] and n[y] terms are, respectively, the number of observations in the first and second samples. The equation below is
modified slightly from the original source for this tip to omit unused terms (mu[x] and mu[y]) in the source web page. This action additionally helps to keep the focus for the example on the
difference between group means.
The degrees of freedom for the computed t value is represented by m in the following equation. The other terms in the equation are defined in the preceding equation. Notice that the equation for the
degrees of freedom is considerably more elaborate than any of the preceding t tests that do not account for unequal variances between two samples.
Because of the complexity of the above expression, Satterthwaite's correction is widely used for computing the degrees of freedom for the t test. I present three additional equations based on
Satterthwaite's correction as described in the website that facilitate the computation of degrees of freedom for samples with unequal variance.
• The degrees of freedom can be represented by following more simple equation. The n terms are defined as above, and the c terms are given in the following two equations.
• The c[x] term is specified by the following equation. Again, the S and n terms are defined as in the equation for the computed t value.
• The c[y] term is defined, in turn, as one less c[x].
Given these equations for processing a computed t value from two samples with unequal variance, we can use the same general SQL approach for assessing statistical significance as described in the
preceding section. The detailed adapted code appears below.
• As you can see, the name of the stored procedure is compute_t_between_2_groups_with_unequal_variances_with_lookup.
• The @tail_num input parameter allows a user to specify either a one-tailed test or a two-tailed test.
• An If…Else statement runs the one-tailed test when the value of @tail_num equals one or the two-tailed test when the value of @tail_num equals two.
• Within the SQL query statement for the If and Else clauses of the If…Else statement are a series of nested subqueries.
□ The innermost subqueries named Group_1 and Group_2 compute the sample mean, variance, and size for each of the two groups.
□ A cross join of the result set for the two innermost subqueries returns a single row with means, variances, and sizes for each of the two groups.
• The c subquery containing the cross joined results computes intermediate values based on the preceding equations for the computed t value and degrees of freedom.
• The outermost query within If and Else code blocks return final results for display from the stored procedure. For example, the outermost query performs a join to select the appropriate critical
t values for assessing the statistical significance of the computed t value. The outermost query also uses a case statement to report the probability level for rejecting the null hypothesis or
that it is not appropriate to reject the null hypothesis.
create procedure compute_t_between_2_groups_with_unequal_variances_with_lookup @tail_num int
if @tail_num = 1
when abs(computed_t) > t_001 then 'significant beyond .001 level for one-tailed test'
when abs(computed_t) > t_01 then 'significant beyond .01 level for one-tailed test'
when abs(computed_t) > t_05 then 'significant beyond .05 level for one-tailed test'
else 'not significant beyond .05 level'
end probability_of_significance
,@tail_num [test tail type]
-- detailed results
,sqrt((Group_1_var/n1) + (Group_2_var/n2)) t_div
,(Group_1_mean - Group_2_mean) mean_difference
-- computation of computed t
(Group_1_mean - Group_2_mean)
sqrt((Group_1_var/n1) + (Group_2_var/n2)) computed_t
-- computation of c for Group_1
(Group_1_var/n1)/((Group_1_var/n1)+(Group_2_var/n2)) Group_1_c
-- computation of c for Group_2
(1-(Group_1_var/n1)/((Group_1_var/n1)+(Group_2_var/n2))) Group_2_c
),0) df
count(*) n1
,avg(score) Group_1_mean
,var(score) Group_1_var
from ##temp_group_scores where group_id = 'Group_1'
) summary_Group_1
cross join
count(*) n2
,avg(score) Group_2_mean
,var(score) Group_2_var
from ##temp_group_scores where group_id = 'Group_2'
) summary_Group_2
inner join [dbo].[Critical_ts_for_mean_difference_test_1_tail_table] t
on c.df = t.df
if @tail_num = 2
when abs(computed_t) > t_001 then 'significant beyond .001 level for two-tailed test'
when abs(computed_t) > t_01 then 'significant beyond .01 level for two-tailed test'
when abs(computed_t) > t_05 then 'significant beyond .05 level for two-tailed test'
else 'not significant beyond .05 level'
end probability_of_significance
,@tail_num [test tail type]
-- detailed results
,sqrt((Group_1_var/n1) + (Group_2_var/n2)) t_div
,(Group_1_mean - Group_2_mean) mean_difference
-- computation of computed t
(Group_1_mean - Group_2_mean)
sqrt((Group_1_var/n1) + (Group_2_var/n2)) computed_t
-- computation of c for Group_1
(Group_1_var/n1)/((Group_1_var/n1)+(Group_2_var/n2)) Group_1_c
-- computation of c for Group_2
(1-(Group_1_var/n1)/((Group_1_var/n1)+(Group_2_var/n2))) Group_2_c
),0) df
count(*) n1
,avg(score) Group_1_mean
,var(score) Group_1_var
from ##temp_group_scores where group_id = 'Group_1'
) summary_Group_1
cross join
count(*) n2
,avg(score) Group_2_mean
,var(score) Group_2_var
from ##temp_group_scores where group_id = 'Group_2'
) summary_Group_2
inner join [dbo].[Critical_ts_for_mean_difference_test_2_tail_table] t
on c.df = t.df
The next two screen shots show the results from the one-tailed test and the two-tailed test. Both tests yield results that are statistically significant at beyond the .001 level. The data represent
the thickness in centimeters of two different decorative styles of ceramic sherds from an archaeological site. As you can see, the sample means are substantially different; this corresponds to the
outcome of the t tests. There are 25 ceramic sherd measurements for the first group and 40 sherd measurements for the second group. You cannot directly see a comparison of the variances of the two
groups, but that will be covered in the next section.
Are the variances from two samples unequal?
One of two t tests can apply to two-sample data when samples have unequal sizes.
• If the two samples also have unequal variances, then apply the t test from the preceding section.
• If two samples have unequal sample sizes, but their population variances cannot be verified to be significantly different, then apply the t test from the section just before the preceding
However, it may not be perfectly obvious just from examining data points from two samples when two samples come from populations with unequal variances. There are a couple of statistical tests for
comparing the variances between two samples. This tip illustrates the use of the F test for the ratio of the variances between two samples.
The NIST/SEMATECH e-Handbook of Statistical Methods presents a general discussion along with sample data for using the F test to assess if the variances of two populations from which two samples come
are equal. If the variances from two samples are unequal enough to suggest that they come from populations with different variances, the F values at df_num and df_dnom degrees of freedom for the
ratio of the two variances will be distributed in one of two critical F distribution regions. This concept is more fully explained in the "Computing and saving lookup tables containing critical
values" section and the next paragraph.
An F distribution has two degrees of freedom - one for the left tail and a second one for the right tail. When comparing the ratio of the variances for two samples, the ratios are distributed as an F
distribution. If sample size for the numerator of the ratio is n[1], then the degrees of freedom for the numerator is n[1] - 1. If sample size for the denominator of the ratio is n[2], then the
degrees of freedom for the denominator is n[2] - 1. You can lookup the F distribution for the F distribution's two tails from the Critical_Fs_for_variance_test_table object described in the
"Computing and saving lookup tables containing critical values" section.
The following script presents a create procedure statement for assessing if the ratio of the variances is statistically significant at beyond the .05 probability level, which is the level used for
the Critical_Fs_for_variance_test_table object with a two-tailed test. As you can see, a collection of nested queries resides within the procedure.
• The procedure's name is compute_F_between_2_groups_for_variances_with_lookup.
• The group_1_var and group_2_var subqueries compute the variances for the first and second samples, respectively. These subqueries also compute sample sizes and degrees of freedom for the first
and second samples.
• A cross join query between the two innermost subqueries combines the results from the first and second groups into a single row. Additionally, the subquery with the cross join computes the ratio
of the variances between the two groups.
• The outermost query joins the result set from the subquery with the cross join to the Critical_Fs_for_variance_test_table object. A case statement in the outermost query assigns a value of
'significant at beyond .05' or 'not significant at beyond .05' to the are_variances_different column based on the ratio of the variances.
create procedure compute_F_between_2_groups_for_variances_with_lookup
when (var_Group_1/var_Group_2) < f.[left-tail-f]
or (var_Group_1/var_Group_2) > f.[right-tail-f]
then 'significant at beyond .05'
else 'not significant at beyond .05'
end [are_variances_different]
,var_Group_1/var_Group_2 F
,var(score) var_Group_1
,count(*) n_Group_1
,count(*)-1 df_Group_1
from ##temp_group_scores
where group_id = 'Group_1'
group by group_id
) group_1_var
cross join
,var(score) var_Group_2
,count(*) n_Group_2
,count(*)-1 df_Group_2
from ##temp_group_scores
where group_id = 'Group_2'
group by group_id
) group_2_var
) c
inner join [dbo].[Critical_Fs_for_variance_test_table] f
on c.df_Group_1 = f.df_num
and c.df_Group_2 = f.df_dnom
The following script shows the application of the compute_F_between_2_groups_for_variances_with_lookup stored procedure to the data from the "Comparing two means from samples with unequal variances"
• The script initially creates an empty copy of the ##temp_group_scores table.
• Then, it uses a set of insert statements to pump the sample data points from the "Comparing two means from samples with unequal variances" section into the table.
• Finally, it invokes the compute_F_between_2_groups_for_variances_with_lookup stored procedure to compute the F value and assess if it is statistically significant.
-- create and populate ##temp_group_scores
begin try
drop table ##temp_group_scores
end try
begin catch
print '##temp_group_scores not available to drop'
end catch
create table ##temp_group_scores
group_id varchar(10)
,score float
-- from --https://www.unm.edu/~marcusj/2Sampletex2.pdf
-- Two-sample t-tests section with unequal sample variances
insert into ##temp_group_scores values ('Group_1', 19.7146)
insert into ##temp_group_scores values ('Group_1', 19.3516)
insert into ##temp_group_scores values ('Group_1', 20.8439)
insert into ##temp_group_scores values ('Group_1', 18.6316)
insert into ##temp_group_scores values ('Group_1', 23.7872)
insert into ##temp_group_scores values ('Group_1', 22.8245)
insert into ##temp_group_scores values ('Group_1', 29.1662)
insert into ##temp_group_scores values ('Group_1', 28.8265)
insert into ##temp_group_scores values ('Group_1', 22.4471)
insert into ##temp_group_scores values ('Group_1', 28.4952)
insert into ##temp_group_scores values ('Group_1', 26.3348)
insert into ##temp_group_scores values ('Group_1', 21.5908)
insert into ##temp_group_scores values ('Group_1', 23.8161)
insert into ##temp_group_scores values ('Group_1', 27.8443)
insert into ##temp_group_scores values ('Group_1', 27.9284)
insert into ##temp_group_scores values ('Group_1', 25.4338)
insert into ##temp_group_scores values ('Group_1', 25.0997)
insert into ##temp_group_scores values ('Group_1', 27.034)
insert into ##temp_group_scores values ('Group_1', 25.3329)
insert into ##temp_group_scores values ('Group_1', 22.2871)
insert into ##temp_group_scores values ('Group_1', 20.831)
insert into ##temp_group_scores values ('Group_1', 18.022)
insert into ##temp_group_scores values ('Group_1', 23.5834)
insert into ##temp_group_scores values ('Group_1', 26.679)
insert into ##temp_group_scores values ('Group_1', 13.2098)
insert into ##temp_group_scores values ('Group_2', 40.079)
insert into ##temp_group_scores values ('Group_2', 24.2808)
insert into ##temp_group_scores values ('Group_2', 34.6926)
insert into ##temp_group_scores values ('Group_2', 37.1757)
insert into ##temp_group_scores values ('Group_2', 26.5954)
insert into ##temp_group_scores values ('Group_2', 18.5252)
insert into ##temp_group_scores values ('Group_2', 23.5064)
insert into ##temp_group_scores values ('Group_2', 30.9565)
insert into ##temp_group_scores values ('Group_2', 29.3769)
insert into ##temp_group_scores values ('Group_2', 19.7374)
insert into ##temp_group_scores values ('Group_2', 35.8091)
insert into ##temp_group_scores values ('Group_2', 39.7922)
insert into ##temp_group_scores values ('Group_2', 29.9376)
insert into ##temp_group_scores values ('Group_2', 40.7894)
insert into ##temp_group_scores values ('Group_2', 33.9418)
insert into ##temp_group_scores values ('Group_2', 26.556)
insert into ##temp_group_scores values ('Group_2', 21.4682)
insert into ##temp_group_scores values ('Group_2', 23.9296)
insert into ##temp_group_scores values ('Group_2', 39.6987)
insert into ##temp_group_scores values ('Group_2', 30.6148)
insert into ##temp_group_scores values ('Group_2', 31.3332)
insert into ##temp_group_scores values ('Group_2', 13.1078)
insert into ##temp_group_scores values ('Group_2', 27.6245)
insert into ##temp_group_scores values ('Group_2', 27.1912)
insert into ##temp_group_scores values ('Group_2', 26.8967)
insert into ##temp_group_scores values ('Group_2', 39.6987)
insert into ##temp_group_scores values ('Group_2', 25.3269)
insert into ##temp_group_scores values ('Group_2', 37.2205)
insert into ##temp_group_scores values ('Group_2', 27.3089)
insert into ##temp_group_scores values ('Group_2', 28.4069)
insert into ##temp_group_scores values ('Group_2', 25.1476)
insert into ##temp_group_scores values ('Group_2', 30.2518)
insert into ##temp_group_scores values ('Group_2', 33.9531)
insert into ##temp_group_scores values ('Group_2', 36.1267)
insert into ##temp_group_scores values ('Group_2', 30.6148)
insert into ##temp_group_scores values ('Group_2', 29.6046)
insert into ##temp_group_scores values ('Group_2', 39.1803)
insert into ##temp_group_scores values ('Group_2', 32.0166)
insert into ##temp_group_scores values ('Group_2', 28.7846)
insert into ##temp_group_scores values ('Group_2', 33.8551)
exec compute_F_between_2_groups_for_variances_with_lookup
Here's a screen shot with output from the preceding script.
• Recall that the sample sizes for the first and second groups are, respectively, twenty-five and forty. Therefore, the corresponding degrees of freedom are twenty-four and thirty-nine.
• As you can see, the variances are significantly different, which is the a priori assumption for the data in the "Comparing two means from samples with unequal variances" section.
The next script shows the application of the compute_F_between_2_groups_for_variances_with_lookup stored procedure to the data in the "Comparing two means from samples with equal or unequal sample
sizes and equal variance" section. The data for this section had unequal sample sizes of twelve and seven for the first and second samples, respectively.
• The code starts by truncating the ##temp_group_scores table initially created and populated above.
• Next, a sequence of insert statements populates the table with data from the "Comparing two means from samples with equal or unequal sample sizes and equal variance" section.
• Finally, the compute_F_between_2_groups_for_variances_with_lookup stored procedure is invoked.
truncate table ##temp_group_scores
-- from --https://www.statsdirect.co.uk/help/parametric_methods/utt.htm
-- Two-sample t-tests section with unequal sample sizes
insert into ##temp_group_scores values ('Group_1', 134)
insert into ##temp_group_scores values ('Group_1', 146)
insert into ##temp_group_scores values ('Group_1', 104)
insert into ##temp_group_scores values ('Group_1', 119)
insert into ##temp_group_scores values ('Group_1', 124)
insert into ##temp_group_scores values ('Group_1', 161)
insert into ##temp_group_scores values ('Group_1', 107)
insert into ##temp_group_scores values ('Group_1', 83)
insert into ##temp_group_scores values ('Group_1', 113)
insert into ##temp_group_scores values ('Group_1', 129)
insert into ##temp_group_scores values ('Group_1', 97)
insert into ##temp_group_scores values ('Group_1', 123)
insert into ##temp_group_scores values ('Group_2', 70)
insert into ##temp_group_scores values ('Group_2', 118)
insert into ##temp_group_scores values ('Group_2', 101)
insert into ##temp_group_scores values ('Group_2', 85)
insert into ##temp_group_scores values ('Group_2', 107)
insert into ##temp_group_scores values ('Group_2', 132)
insert into ##temp_group_scores values ('Group_2', 94)
-- end of data prep for stored proc
exec compute_F_between_2_groups_for_variances_with_lookup
Here's a screen shot with the results from the preceding section. As the name of the section implies, the sample variances are not significantly different from each other at the .05 probability
Next Steps
Try the stored procedures and scripts for loading and running with your own data. If you encounter difficulties review the content in the remainder of this section.
All the sample scripts for this tip were run from the AllNasdaqTickerPricesfrom2014into2017 database with two new tables copied from two Excel worksheet files. You can download the file with the SQL
scripts and the worksheet files from here.
If you want to run code for the first two t tests demonstrated in this tip, then you will also need to download the initial version of the SQL statistics package because two stored procedures are
referenced in this tip from that release of the package.
While all code testing was performed for the AllNasdaqTickerPricesfrom2014into2017 database, you do not strictly need that database. You do, however, need the relevant script files and the SQL Server
tables based on the worksheet files. The worksheet files and the SQL scripts are available for download from this tip. If you want, you can download the AllNasdaqTickerPricesfrom2014into2017 database
from here.
About the author
Rick Dobson is an author and an individual trader. He is also a SQL Server professional with decades of T-SQL experience that includes authoring books, running a national seminar practice, working
for businesses on finance and healthcare development projects, and serving as a regular contributor to MSSQLTips.com. He has been growing his Python skills for more than the past half decade --
especially for data visualization and ETL tasks with JSON and CSV files. His most recent professional passions include financial time series data and analyses, AI models, and statistics. He believes
the proper application of these skills can help traders and investors to make more profitable decisions.
This author pledges the content of this article is based on professional experience and not AI generated.
View all my tips
Article Last Updated: 2018-09-25
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Euclidean Vector Spaces
Linear Vector Spaces: Euclidean Vector Spaces
In these pages, a Euclidean Vector Space is used to refer to an
Orthonormal Basis
An orthonormal basis set is a basis set whose vectors satisfy two conditions. The first condition is that the vectors in the basis set are orthogonal to each other and the second condition is that
each vector has a unit norm.
Orthogonal Projection
The dot product structure allows the definition of orthogonal projections. Given a vector
In the following tool, enter the components of the vectors
Cross Product in
The structure of the Euclidean vector space cross product operation. This is a unique map that gives the vector perpendicular to any two linearly independent vectors.
The cross product is the operation:
1. Denoting
2. The operation is skewsymmetric:
3. The operation is distributive over addition:
4. The operation is compatible with scalar multiplication:
The last property ensures that the norm of the resulting vector
The cross product operation is defined above using its algebraic properties. Equivalently, the cross product can be defined as follows, given
In the following, the algebraic properties of the cross product are used to show the traditional properties of the cross product given an orthonormal basis set
The cross product of the basis vectors
Where the positive sign is used to indicate a right-handed orientation.
This is straightforward from properties 1 and 5 above. From property 1, since
To find the value of
The cross product of linearly dependent vectors
Indicating that if
We first assume that either
Then, we assume that
which is a contradiction, therefore,
The explicit representation of the cross product
This is a direct consequence of property 3 above and the cross product of the basis vectors result.
The triple product
The triple product of any three vectors
This is a direct consequence of properties 1 and 2 above as follows:
The remaining equalities can be proven similarly
The triple product
The triple product of any three vectors
The one direction is straight forward, if
In the opposite direction we will argue by contradiction. Assuming that
The triple product
Similar to the proof for the cross product of the basis vectors.
Show that
The following tool calculates the cross product
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In How Many Ways Can You Solve the Thiagi Circles Jolt?
I was at a seminar yesterday with world famous (and awful fun) Thiagi. He keeps his participants on their toes with small activities called jolts. In this Thiagi GameLetter, jolts are defined at
interactive experiential activities and:
…force participants to re-examine their assumptions and revise their habitual practices.
One jolt we did yesterday got me to thinking about math. Okay, many of them had me thinking about math. But this one was about the multiplication principal of counting and sets.
The multiplication principal says that if you have 7 ways to do the first thing and 3 ways to do the 2nd thing, then you’ll have 3 * 7 = 21 ways to do them together. Assuming you pick one of each.
So if I want a sandwich (tuna, ham or turkey) and a side (chips, onion rings, fruit or fries) for lunch, I will have the option of 3 * 4 = 12 different lunches.
The Thiagi Circles Jolt
Here is the Circles Jolt that Thiagi offered: Draw this figure without ever lifting your pencil or retracing over any lines (or curves):
It isn’t hard, as you probably see. It takes anywhere from 10 to 45 seconds to figure it out. The question is, “Is there another way to do it than the one I thought of?”
So here are the ways I immediately thought of in the Thiagi Workshop:
The fancy pants teacher’s pet, Mark, was asked to demonstrate the answer. And he did it in an even different way! So I saw that there were 8 more ways to do it:
Trying to trump Mark and be the new teacher’s pet, I announced that there were indeed 16 ways to accomplish Thiagi’s goal.
And here I am. Still trying to win the favor of the teacher by producing the 16 ways via blog and video.
There are 16 ways because of the Multiplication Principal of Counting.
Notice there are four questions to answer when solving this:
1. Do you start from left or right side of the drawing?
2. Do you start by drawing the outer circle or go through and start at the inner circle?
3. Do you go up or down on the first circle you draw?
4. Do you go up or down on the second circle?
Each question has two different options. So there are
2 sides to start from * 2 circles * 2 directions to go in the 1st circle * 2 direction in the 2nd circle
2 * 2 * 2 * 2 = 16 different ways to draw the diagram!
Teach this with experiments.
Although you now know how many there are, don’t teach it this way. Instead, just give the jolt to the kids. Once they solve it, show them “your way,” which should be any way that is different than
Then tell them that for ever unique solution, you’ll give them $1. (Thiagi gave out $1 bills yesterday, too!)
Once they come up with all 16, offer the extra $4 for some way they can show the number 16 with numbers (logically based on their solutions). They might come up with 2 * 8 or 4 * 4, as long as they
can group their drawings in logical bundles.
What do you think?
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Static Rigid BodiesQuizwiz - Ace Your Homework & Exams, Now With ChatGPT AI
Static Rigid Bodies
Lakukan tugas rumah & ujian kamu dengan baik sekarang menggunakan
A non-uniform plank AB of mass 12kg and length 4 metres is hanging horizontally from two strings C and D. The distances AC and BC are 1 metres and 1.4 metres respectively. A particle of mass 5 kg is
placed on the plank at E, 2.4 m away from A. The tensions in the ropes have the same magnitude. What is the first thing that you would write for a question like this?
The reaction at C + the reaction at D = 12g + 5g
Explain how you would do the following question: A uniform plank AB of mass 20kg and length 6m is resting horizontally on two supports at A and C. The distance CB is 1.4 metres. A child of mass 25kg
is standing on the plank between C and B, x metres away from B. Find the minimum distance x so the plank will not tilt about C.
Do the question so that the reaction at A = 0, working out the moment about C and then minusing the distance found there from 1.4
If you have a rod and there is a point b which is just in the air, does it have a reaction?
Explain how you would solve the following question: 3: A uniform ladder of mass 20 kg and length 8 m rests against a smooth vertical wall with its lower end on rough horizontal ground. The
coefficient of friction between the ground and the ladder is 0.3. The ladder is inclined at an angle θ to the horizontal, where tan θ = 2. A boy of mass 30 kg climbs up the ladder. By modelling the
ladder as a uniform rod, the boy as a particle and the wall as smooth and vertical, (a) find how far up the ladder the boy can climb before the ladder slips.
P = Fr and R = 50g. Then do the moment around A, making P the subject then subbing in 0.3 x 50g, to work out the distance
A ladder AB, of weight W and length 2l, has one end A resting on rough horizontal ground. The other end B rests against a rough vertical wall. The coefficient of friction between the ladder and the
wall is 1/3. The coefficient of friction between the ladder and the ground is μ. Friction is limiting at both A and B. The ladder is at an angle θ to the ground, where tan θ = 5/3. The ladder is
modelled as a uniform rod which lies in a vertical plane perpendicular to the wall. Find the value of μ. Start the question (3) and write out the equation for the moment around A
R + Fr against wall = W and R + 1/3P = W and P = Fr of floor. Wcos feta x l = Psin feta x 2l + 1/3P x 2lcos feta
There is a ladder AB of mass 25kg and length 4m, resting in equilibrium with one end A on rough horizontal ground and the other end B against a smooth vertical wall. The ladder is in a vertical plan
perpendicular to the wall. The coefficient of friction between the ladder and the ground is 11/25. The ladder makes an angle of alpha with the ground. When Reece, who has made 75kg stands at the
point C on the ladder, where AC = 2.8m, the ladder is on the point of slipping. The ladder is modelled as a uniform rod and Reece is modelled as a particle. You answer a load of questions on this,
then you are asked to state how you have used the modelling assumption that Reece is a particle, what would you put?
This means that Reece's weight acts at a single point at C
Set pelajaran terkait
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Incidence graph of
Incidence graph of $(11,6,3)$-design
This is the incidence graph of the complementary design of the biplane on 11 points, and thus the distance-3 graph of its incidence graph.
Number of vertices: $22$
Diameter: $3$
Intersection array: $\{6,5,3;1,3,6\}$
Spectrum: $6^1 (\sqrt{3})^{10} (-\sqrt{3})^{10} (-6)^1$
Automorphism group: $ PGL(2,11)$
Distance-transitive: Yes
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How to Teach Basic Math Operations Using Flashcards
Teaching basic math can feel like juggling numbers and concepts while trying to keep kids entertained and engaged. But flashcards? Now that’s where the magic happens. Flashcards are like the cheat
codes for teaching math — simple, effective, and a ton of fun. How to teach basic math operations using flashcards can transform those numbers into an engaging game. Ready to make math lessons an
exciting adventure? Let’s break it down step by step!
Benefits of Using Flashcards for Teaching Math
Teaching basic math can feel like juggling numbers and concepts while trying to keep kids entertained and engaged. But flashcards? Now that’s where the magic happens. Flashcards are like the cheat
codes for teaching math simple, effective, and a ton of fun. Plus, fun math practise with Hit the Button adds an interactive twist that keeps kids on their toes! Ready to make math lessons an
exciting adventure? Let’s break it down step by step!
• Quick Recall Development: Flashcards train the brain to recognize math facts instantly. The more your kids practice, the faster they’ll get!
• Engaging Visual Aids: With bright colors and fun designs, flashcards can capture attention and keep kids engaged longer than traditional worksheets.
• Improving Memory Retention: The repetitive nature of flashcards boosts memory, helping kids lock in those math operations for the long haul.
Choosing the Right Flashcards
Not all flashcards are created equal. You don’t want to overwhelm young learners with too much information too soon. Keep it simple!
• Simple vs. Complex Designs: Avoid cards that look too “busy.” Stick to clean, easy-to-read numbers with just the right amount of color to keep it interesting.
• Colorful and Engaging Visuals: Bright visuals can keep attention spans in check. Look for designs that are colorful without being overwhelming.
Preparing Your Flashcards for Math Lessons
Ready to create your flashcards? You’ve got two options: make your own or grab a set from the store. Here’s how to decide:
• DIY vs. Store-bought Options: Making your own flashcards allows you to tailor them to your child’s current math level. Store-bought cards work just as well if you’re short on time.
• Tailoring Flashcards to Age and Skill Level: Keep the content age-appropriate. Start with simple addition and subtraction for younger learners, and gradually introduce multiplication and division
as they get more confident.
Introducing Addition Using Flashcards
Addition is usually the first math operation kids learn. Start with small numbers to build confidence.
• Start with Small Numbers: Use flashcards that introduce numbers like 1+1, 2+2, and 3+3. Kids can gradually work their way up to larger sums.
• Techniques to Reinforce Learning: Use everyday objects (like toys or fruit) alongside flashcards to make the math problems more relatable.
Making Subtraction Fun with Flashcards
Subtraction doesn’t have to be intimidating. With a little creativity, it can be just as fun as addition.
• Visualizing Subtraction Problems: Flashcards with pictures (e.g., “5 apples – 2 apples”) help kids visualize what’s happening in the problem.
• Engaging Activities to Promote Learning: Try using hands-on objects to help kids subtract in real-time. For example, have them physically remove items as they solve each subtraction problem.
Teaching Multiplication Through Flashcards
Multiplication can be tricky, but flashcards can make it feel like a walk in the park.
• Skip Counting and Patterns: Start by introducing the concept of skip counting (2, 4, 6, 8…) with flashcards. This helps kids see the patterns in multiplication.
• Grouping to Illustrate Multiplication: Use visuals to show how multiplying is simply adding groups of numbers. For example, “3 x 4” can be shown as three groups of four items.
Simplifying Division Using Flashcards
Division is where things can get a little tricky, but don’t worry — we’ve got it covered.
• Using Visuals to Break Down Division: Show division problems visually. For example, divide 12 objects into 3 equal groups to explain “12 ÷ 3.”
• Real-Life Examples to Make Division Relatable: Use real-life situations, like sharing snacks or toys among friends, to demonstrate the idea of division.
Incorporating Word Problems into Flashcards
Flashcards don’t have to be just numbers! Mixing words with numbers can create fun challenges.
• Mixing Words with Numbers: Create simple word problems on the back of your flashcards to challenge kids to apply their math skills in different contexts.
• Creating Real-Life Scenarios: Ask questions like, “If you have 5 apples and eat 2, how many are left?” Word problems are a great way to blend reading with math.
Flashcard Games to Teach Basic Math
Let’s turn math into a game! Flashcards are perfect for creating engaging math activities.
• Math Memory Match: Make pairs of matching flashcards (e.g., one card shows 5+3, and the other shows 8) and have kids match them up.
• Timed Flashcard Drills: Turn it into a race! See how many flashcards your child can solve in 1 minute, and try to beat their personal best next time.
Integrating Flashcards with Technology
Flashcards have gone digital! You can integrate tech to boost the learning experience.
• Flashcard Apps and Online Tools: There are tons of flashcard apps that let you create digital cards, or you can use online flashcards that offer interactive quizzes.
• How to Create Digital Flashcards: Tools like Quizlet allow you to create personalized flashcards that kids can use on their tablets or computers.
Tips for Keeping Kids Engaged
Keeping young learners engaged with flashcards can be challenging, but a few simple tricks can make a world of difference.
• Creating a Rewards System: Use small rewards, like stickers or extra playtime, to encourage consistent effort and improvement.
• Rotating Flashcards to Avoid Repetition: Keep things fresh by regularly switching out flashcards so the activities never feel stale.
Customizing Flashcards for Different Learning Styles
Every kid learns differently, and flashcards can be customized to fit those needs.
• Visual, Auditory, and Kinesthetic Learners: Some kids learn best by seeing, others by hearing, and some by doing. Flashcards can be adapted for each style by adding visual aids, sounds, or even
physical objects to help with learning.
• Adjusting Flashcards to Suit Each Style: Visual learners can benefit from colorful images, while auditory learners might enjoy accompanying the flashcards with sounds or songs.
Common Mistakes to Avoid
While flashcards are a great tool, there are some pitfalls to watch out for.
• Overloading with Too Much Information: Stick to one concept per flashcard to avoid confusion.
• Lack of Variety in Flashcard Exercises: Mix up the activities to keep it interesting. Use different games and approaches to challenge kids in new ways.
Teaching math with flashcards is an incredibly effective and engaging method. It’s all about using the right tools, keeping it fun, and adapting to each child’s unique learning style. With the right
approach, flashcards can turn math from a daunting task into an enjoyable learning experience.
How to Teach Basic Math Operations Using Flashcards FAQs
• How many flashcards should be used in a session?
It’s best to start with around 10-15 cards per session, then increase as your child’s confidence grows.
• Can flashcards be used for older children?
Absolutely! Flashcards can be adapted for more complex math problems, making them useful for older kids too.
• What’s the best way to organize flashcards?
Use categories like addition, subtraction, multiplication, and division, and rotate through them regularly.
• How can I make my own flashcards at home?
You can create your own by cutting out small pieces of paper or index cards and writing simple math problems on one side, with answers on the back.
• How long should each flashcard session last?
Aim for 10-20 minutes per session, depending on your child’s attention span and interest.
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Algebraic structures
Code: 31425
ECTS: 6.0
Lecturers in charge: prof. dr. sc. Marcela Hanzer
prof. dr. sc. Boris Širola
Lecturers: Bruno Predojević , mag. math. - Exercises
prof. dr. sc. Boris Širola - Exercises
English level: All teaching activities will be held in Croatian. However, foreign students in mixed groups will have the opportunity to attend additional office hours with the lecturer and
teaching assistants in English to help master the course materials. Additionally, the lecturer will refer foreign students to the corresponding literature in English, as well as
1,0,0 give them the possibility of taking the associated exams in English.
1. komponenta
Lecture type Total
Lectures 30
Exercises 30
* Load is given in academic hour
(1 academic hour = 45
COURSE AIMS AND OBJECTIVES:
The goal of the course is to
introduce the most important
notions and to learn
fundamental results of the
group theory, rings, modules,
fields and generally
algebras. Especially in
course will be described a
connections between main
objectives mentioned above
with another important math
theory like number theory and
representation theory.
COURSE DESCRIPTION AND
First section: Group theory
(7 weeks)
1. Definition of the
semigroup, group, subgroup
2. Normal subgroup and
quotient group
3. Examples of the groups
(cyclic groups, symmetric
group, dihedral group, etc.)
4. Homomorphisms of the
5. Direct products and sums
of the groups
6. The p-groups and the Sylow
Second section: Ring, Fields
and Algebras (6 weeks)
1. Definition of the rings,
subrings and ideals
2. Homomorphisms of the rings
3. Factorizations in
commutative rings and
principal ideal domain
4. Rings of polynomials and
the other important examples
5. Definition of the fields,
subfield and the field
6. Simple and finitely
7. Finite fields
8. Examples of algebras
(associative and
Third section: Modules (2
Definition of the modules,
submodules and quotient
1. The main theorems and
examples of simple
(semisimple) modules
Prerequisit for:
Enrollment :
Passed : Elementary
mathematics 1
Passed : Linear algebra 2
1. komponenta Lecture typeTotal Lectures 30 Exercises 30 * Load is given in academic hour (1 academic hour = 45 minutes)
COURSE AIMS AND OBJECTIVES: The goal of the course is to introduce the most important notions and to learn fundamental results of the group theory, rings, modules, fields and generally algebras.
Especially in course will be described a connections between main objectives mentioned above with another important math theory like number theory and representation theory. COURSE DESCRIPTION AND
SYLLABUS: First section: Group theory (7 weeks) 1. Definition of the semigroup, group, subgroup 2. Normal subgroup and quotient group 3. Examples of the groups (cyclic groups, symmetric group,
dihedral group, etc.) 4. Homomorphisms of the groups 5. Direct products and sums of the groups 6. The p-groups and the Sylow theorems Second section: Ring, Fields and Algebras (6 weeks) 1. Definition
of the rings, subrings and ideals 2. Homomorphisms of the rings 3. Factorizations in commutative rings and principal ideal domain 4. Rings of polynomials and the other important examples 5.
Definition of the fields, subfield and the field extensions 6. Simple and finitely extensions 7. Finite fields 8. Examples of algebras (associative and nonassociative) Third section: Modules (2
weeks) Definition of the modules, submodules and quotient modules 1. The main theorems and examples of simple (semisimple) modules
Enrollment : Passed : Elementary mathematics 1 Passed : Linear algebra 2
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