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## Calculus 2
### Unit 2: Lesson 3
Integrating using trigonometric identities
# Integral of sin^2(x) cos^3(x)
Another example where u substitution combined with certain trigonometric identities can be used.
## Want to join the conversation?
• I think I'm much more comfortable with the u-substitution as compared to the REVERSE CHAIN RULE. Will that be a problem or is it alright?
• Reverse chain rule is basically doing u substitution in your head, so it would be a bit faster than u-sub but using u-sub won't be a problem, it might even increase accuracy as you right it down
• I got a different answer after substituting the trig identity for sin(x)^2 = 1 - cos(x)^2
So what I have now is ∫(1 - cos(x)^2) * cos(x)^3 * dx
Then after distributing the cos(x)^3 I have, ∫cos(x)^3 - ∫cos(x)^5 * dx
Evaluating this give me: (cos(x)^4)/4 - (cos(x)^6)/6
Is this equivalent to the answer in the video, or where have I gone wrong?
(1 vote)
• Basically you can't integrate the cos(x)^3 and the cos(x)^5. The reason for this is because they represent cos(x)*cos(x)*cos(x) and you cant integrate that without the u substitution. So in the end your only choice is to sub the trig identity for the cos(x)^2
(1 vote)
• I will assume you intend the integrand to be interpreted as `[ (2 - sin 2x) / (1 - cos 2x) ]eᵡ`. To solve the integral, we will first rewrite the sine and cosine terms as follows:
` I) sin(2x) = 2sin(x)cos(x);`
`II) cos(2x) = 2cos²(x) - 1.`
Rewriting yields
`2 - sin(2x)`
`= 2 - 2sin(x)cos(x)`
`= 2[1 - sin(x)cos(x)],`
and
`1 - cos(2x)`
`= 1 - [2cos²(x) - 1]`
`= 2 - 2cos²(x)`
`= 2[1 - cos²(x)]`
`= 2sin²(x).`
Hence
`[ (2 - sin 2x) / (1 - cos 2x) ]eᵡ`
`= ( 2[1 - sin(x)cos(x)] / [ 2sin²(x) ] )eᵡ`
`= [ 1/sin²(x)`` - cos(x)/sin(x) ]eᵡ`
`= eᵡ / sin²(x) - eᵡcot(x).`
Thus `∫ [ (2 - sin 2x) / (1 - cos 2x) ]eᵡ dx = ∫ [ eᵡ / sin²(x) - eᵡcot(x) ] dx`. This may be split up into two integrals as `∫ eᵡ / sin²(x) dx - ∫ eᵡcot(x) dx`. We will first focus on the first of these integrals.
Recall that `d/dx cot(x) = -1 / sin²(x)`. Using integration by parts on the expression `∫ eᵡ / sin²(x) dx` yields
`∫ eᵡ / sin²(x) dx`
`= -eᵡcot(x) + ∫ eᵡcot(x) dx.`
When we plug this into the expression `∫ eᵡ / sin²(x) dx - ∫ eᵡcot(x) dx`, we get
`∫ eᵡ / sin²(x) dx - ∫ eᵡcot(x) dx`
`= -eᵡcot(x) + ∫ eᵡcot(x) dx - ∫ eᵡcot(x) dx`
`= -eᵡcot(x) + C.`
Therefore, the answer is `-eᵡcot(x) + C`, where `C` is an arbitrary real number.
• Can anyone help me with this: ∫ ln(x)/sin(x) dx
• I'm pretty sure there is no standard way to write the antiderivative of that function. The best you can probably do is use a calculator if you needed a definite integral.
• Where does the du go after we solve for the anti derivative of U?
• If you were just solving the integral of x^2 dx, what would happen with the dx? It's just part of the operation and goes away when it is used. Let me know if that doesn't quite make sense.
• why can't ∫sin^2(x)*cos^2(x)*cos(x) be written as ∫1*cos(x)?
after all trig identity says sin^2(x)*cos^2(x)=1
• Is it possible to use U substitution without distributing cosx to (1- sinx^2)? I didn't distribute cos(x) and therefore my solution was different: x - 1/3(sin^3(x)) + C instead of sin(x) - 1/3(sin^3(x)) + C. I feel that when I do distribute my professor does not and ends up finding a different solution than what I would have done and then when I dont distribute, he does. Is there a rule I am missing here?
• In the beginning of learning a concept (like U-sub), it is a good idea to exaggerate the amount of steps it takes to form a solution.
Some little things might be lost if we jump too far between the different steps.
It might be a good idea to control the solutions by deriving the finished antiderivative.
(x - 1/3(sin^3(x)) + C)'=cos^3(x)-cos(x)+1
(sin(x) - 1/3(sin^3(x)) + C)'=cos^3(x)
What could we do to make these derivatives equal eachother?
I hope this was a little helpful!
(1 vote)
• How do we know not to do sin^2(x)-sin^4(x)=-sin^2(x)?
(1 vote)
• Hello, thanks a lot for the video!
I would like to know why does Sal distributes this way ():
`sin² (1 - sin²)`
`(1 - sin²) cos`
Is it an intuitive thing to do or is there any kind of hint for doing this?
Cheers :)
(1 vote)
• He did that to end up with an integrable amount inside the bracket multiplied by its derivatives outside of it so that he could apply u-substitution or the reverse chain rule...
(1 vote)
• let y= (cosx)^4
dy/dx = 4 (cos x) ^3 * (sin x)^2
so my answer is 1/4 { cosx) ^4 } + C, is that correct?
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# Thinking like a theorist: Are complex numbers real?
UPDATE: My answer to this problem. Which, at the end of the day, isn't really an answer at all.
As part of our explorations of "why this math" for aspects of physics, I pose an obvious and seemingly simple question: which type of numbers is required to describe the world around us?
We use different types of numbers in mathematics. For example, we have the integers, the rational numbers, the real numbers, and the complex numbers. Furthermore, there is a distinct hierarchy among number types - the hierarchy for the four types above is:
$$integers \subset rational ~numbers \subset real ~numbers \subset complex~ numbers$$.
There are additional number types such as the hypercomplex numbers (which are fascinating if you've never run across them). Physics, however, should only need to use a certain type of number to describe the universe, and which type should be dictated by nature. Which number type is required for our current physical theories, and why?
A couple comments for this question:
• Arguing that one number type is simply more convenient to describe phenomena is not a valid argument, as we are looking for what is required.
• On its surface this seems like a very straightforward question, but I warn everyone that it is actually rather subtle.
• There are different answers to this question, depending on what you believe about measurement and/or the fundamental structure of our world. Hence I hope everyone will provide a number of interesting viewpoints.
Finally, as always - only if enough interest and discussion is shown by the members of our community will I post my own answer. I and your peers on Brilliant want to hear everyone's thoughts!
Note by David Mattingly
3 years, 11 months ago
Sort by:
You asked us what number type is required for our current physical theories. Well, I can name certain theories where complex numbers are a necessity. Quantum Mechanics is such an example. It seems that you can't even approach Quantum Mechanics without introducing complex numbers.
In QM the probability of something happening is the square of the magnitude (absolute value) of the “probability amplitude”. This probability amplitude can be complex valued. For example if the probability amplitude is$$\frac{i}{\sqrt2}$$, then the probability is $$|{\frac{i}{\sqrt2}}|^2=\frac{1}{2}$$.In fact, the Schrödinger equation, which is arguably the backbone of Quantum Mechanics, has an '$$i$$' right at the beginning. You can't escape it! The equation of a single particle looks something like this:
$$i\hbar\frac{\partial}{\partial t}\Psi(\mathbf{r},\,t)=-\frac{\hbar^2}{2m}\nabla^2\Psi(\mathbf{r},\,t)+V(\mathbf{r})\Psi(\mathbf{r},\,t)$$ [It took me a really long time to render this in LaTeX!].
I don't know much about Quantum Mechanics and most of what I wrote here is copied and pasted from things I found on the internet. So, I will stop talking about Quantum Mechanics now.
A classmate of mine once asked me, "Why are we studying about complex numbers anyway? They don't exist after all! So why make a fuss about things that don't even exist?"
In reply, I asked him another question, "What is the physical significance of multiplying something by $$-1$$? Multiplication by $$-1$$ rotates something by $$180$$ degrees. If something is heading towards north at a velocity of $$1 ms^{-1}$$. This means that thing is heading towards south at a velocity of $$-1ms^{-1}$$.
What if we could find a number that would rotate something by $$90$$ degrees? Assume that we have such a number. What would happen if we multiplied something by this number twice? It would rotate that thing by $$90+90=180$$ degrees. This is remarkable! Because that's very thing multiplying by $$-1$$ does. In other words,
$$something\times new$$ $$number \times new$$ $$number = something \times (-1)$$
Or in other words, $$new$$ $$number^2=-1$$.
And this is how the imaginary unit $$i$$ is defined. Just because you can't count $$i$$ chickens doesn't mean complex numbers are any less real. By this definition, even negative numbers don't exist (you can't actually count $$-5$$ chickens)."
My answer was able to convince my classmate.
Another point: we know that the distance between two points in Euclidean space is $$\sqrt{\bigtriangleup x^2+\bigtriangleup y^2+\bigtriangleup z^2}$$ [this is just the Pythagorean theorem].
When relativity came along, we realized that space and time were very closely related. And the distance [this is also known as the space-time interval] between two points (events) in space-time is:
$$\sqrt{\bigtriangleup x^2+\bigtriangleup y^2+\bigtriangleup z^2 -c^2t^2}$$.
What does that have to do with complex numbers? Watch again: the formula is
$$\sqrt{\bigtriangleup x^2+\bigtriangleup y^2+\bigtriangleup z^2 +(ict)^2}$$.
So $$i$$ also creeps up here! So physical theories need numbers and sometimes those numbers happen to be complex. But as you have said, there are different answer to this question and a lot of people will come up with different viewpoints. According to me, in order to understand and describe everything [I'm putting a lot of stress on this] in this universe, complex numbers are necessary.
Look at the size of this comment! I think I'll stop now.
EDIT: there have been a couple of comments recently that say that complex numbers are not actually necessary for describing the universe. They are absolutely right!
But this raises another question:
Do we even need numbers to describe the universe around us?
Any physical theory is a model of the universe. Numbers are a tool to describe a theory. They are not a property of the theory itself. Numbers have properties and we use those properties as a tool to try to describe a theory that in turn describes the universe to some extent. We can have theories that don't use numbers at all!
There are certain things that we observe in the universe and we try to capture those things with numbers and end up using those numbers in theories. Complex numbers exist in nature in the same way other numbers exist. They have certain properties that we can experience, perceive and observe. I tried to demonstrate that with the example of rotation. I tried to illustrate current physical theories that use the properties of complex numbers to describe natural phenomenon without getting too philosophical about it. The universe doesn't care if we use numbers to understand it. Numbers are merely a tool for us.
I understand that this has gotten a little bit more philosophical than I would have wanted. So, I'm stopping here. · 3 years, 11 months ago
Although I agree with Mursalin and have nothing to add about QM, I do object to his observation that $$i$$ also creeps up in special relativity when considering the interval ("distance") between two points in spacetime. Unfortunately, properly explaining why this interpretation is not suitable, requires quite a lengthy explanation.
First off, vectors with complex components are used in QM for instance. The norm squared of such a vector is (considered to be) equal to the inproduct with its conjugate rather than itself, so even with complex components you still get something strictly positive.
The four-vector as used in special (and general) relativity is not a proper vector in the mathematical sense, or at least not in the normal four-dimensional vector space. The scalar product of two four-vectors is defined as
$$\mathbf{A} \cdot \mathbf{B} = -A_0 B_0 + A_1 B_1 + A_2 B_2 + A_3 B_3 = \sum_{\mu,\nu=0}^{3} \eta_{\mu\nu} A^\mu B^\nu$$,
where $$\eta$$ is the Minkowski (or flat-space) metric tensor with $$\eta_{00} = -1, \eta_{11} = \eta_{22} = \eta_{33} = 1$$ and all other components zero. It should be noted that sometimes the scalar product (and hence $$\eta$$) is defined with the exact opposite sign; there is no ironclad convention for this. Also, typically the summation sign is omitted; summation is implicit whenever an index is repeated (that is ironclad).
At this point I cannot resist but demonstrate what makes four-vectors and their special scalar product so useful. The fundamental principle (or postulate, if you like) behind (special) relativity is that it does not matter what (inertial) frame of reference you use to describe something. This is nicely reflected in this scalar product: the scalar product of two four-vectors is invariant with respect to a change of reference frame. That goes for the norm of the spacetime four-vector $$(ct, x, y, z)$$,
$$\eta_{\mu\nu} x^\mu x^\nu = -c^2 t^2 + x^2 + y^2 + z^2$$,
which gives you the spacetime interval. But it also works for other four-vectors such as the energy-momentum four-vector $$(E/c, p_x, p_y, p_z)$$, for which we have
$$\eta_{\mu\nu} p^\mu p^\nu = -E^2/c^2 + p_x^2 + p_y^2 + p_z^2 = -m^2 c^2$$,
where $$m$$ is the invariant restmass. You might recognize the above formula for the special case $$\vec{p} = 0$$, but this is the more general form.
Going back to the definition of a scalar product, it may seem that using $$\eta$$ is just overly complicated; why introduce a 16-component tensor for a sum of four products? And why not use imaginary components to deal with the minus sign? I suppose the most convincing answer (which is what I have been working towards) is that, when you go to general relativity, the scalar product generalizes to
$$\mathbf{A} \cdot \mathbf{B} = g_{\mu\nu} A^\mu B^\nu$$,
where $$g$$ is still real-valued but may now have nonzero off-diagonal components.
Finally! Sorry for the long post. · 3 years, 11 months ago
I agree with you. It's like algebra, that one use to describe a system, and same system can be described by different algebras, but it doesn't mean that algebraic model exists physically. As we have different tools like matrices system, similarly complex numbers and it's properties are used to describe systems, it doesn't have to do with physical existence. · 3 years, 11 months ago
Hey all. Sorry I haven't been able to monitor this discussion as closely as I'd like. We got hit by lightning over the weekend and it took out our internet connection. However, it's a great discussion. I'm not going to reply to everyone's comments as I'll post my overall take on this question on Wednesday (after I consult Mariam B's tarot cards). The comments on QM are right in a certain way of looking at it, and the comments about deriving complex numbers from simpler number types are also right. I'll try and distill these viewpoints into my own response.
There is a physics question besides QM involved too, that no one has incorporated yet. I'll toss this question into the ring as well:
Is there a fundamental limit as to how much information one can squeeze into a certain region of space and time? If yes, then our universe, which is finite in extent, could be built out of a large but finite number of these small regions. How would this change the required number system?
Note, this goes a bit beyond the limitation "current physical theories", which was a deliberate choice. Staff · 3 years, 11 months ago
I like this question!
Some top answers have focused on the fact that complex numbers are required because basic equations describing quantum mechanics rely on complex numbers. Others have countered that only natural numbers are required because from them we can construct a system that mimics the behavior of, e.g., complex numbers using only the naturals.
As others have pointed out the Natural numbers purists are kind of cheating - if you construct a system that behaves like complex numbers it doesn't matter that you have avoided using the symbol 'i', you are still using complex numbers. Why not go even further and claim that we don't need numbers at all, just the concept of and axioms of a set (from which we can reconstruct all math)... On the other hand, I don't think that those on the complex numbers side of the debate go far enough.
Let's examine the meaning of the phrase "a number system". A number system is a collection of items (which we will call numbers) and relations and operations (different ways to compare or combine these numbers).
Numbers are useful because we can talk about them in the abstract, replace an actual number with an unknown variable, and we know how the operations will be evaluated when we assign any specific value to the variable. The problem with this abstraction is that sometimes you can write down unsolvable questions. For example if your number system is the natural numbers, you can write down 4+x=3, but now there is no number that makes the equation true unless we introduce integers. Similarly if we want to solve all types of equations we must expand our number set to include rations, reals, and complex numbers to solve polynomial equations. The thing is, as soon as we have introduced 0, 1, the operations of addition and multiplication, and, crucially, the idea that we want to be able to 'undo' those operations. That is, we want to be able to 'do algebra'. Then we have magically already created all the complex numbers.
What about vectors? surely in a multi-dimensional world we need vectors to describe things. Vectors appear to just be a list of numbers, but they have different operations. For example, Maxwell's laws require the use of cross product. We need matrices to deal with vectors. And while the matrix is just an array of numbers, and the matrix operations can be described as a sequence of steps using basic number operations, the overall algebra of matrices has a different structure to that of 'numbers'. So I believe that we have to include matrices on our list of required number systems for the same reason that we have to include complex numbers.
It is a false cop-out to claim that because we can describe a more complex system using the symbols of a simpler system, we don't really need the complex system at all.
Are there other number systems needed? Surely. I don't pretend to know any details about cutting edge particle physics theory, but I know that the ways that these particles interact can be quite complicated, and cannot be modeled within the algebra of basic numbers.
On the very cutting edge of what might not even be real physics after all, string theorists make predictions about reality using even more exotic algebraic structures.
At the root, numbers are an abstraction that allow us to makes predictions about the universe. For any complicated way that objects in the universe can interact, we need a number system that incorporates and models that kind of interaction. As our knowledge of physics in incomplete, I'm sure that there is no final answer to the types of number systems that we need to describe the world around us. · 3 years, 11 months ago
I find that many of the people here that argue for the coarser left side of David M.'s inclusion chain above argue something like this: "You don't need complex numbers because you can construct those from real numbers by algebraic closure" or like this: "You don't need real numbers because you can construct those from the rationals using Dedekind cuts or Cauchy sequences", If that is the case, then you're still using those numbers, you've just created them anew. Just call a spade a spade and admit it. Also, I think a few interesting options are left out: the algebraic numbers, the Gaussian integers and $$\mathbb Q(i)$$
For the current quantum theory, I believe we need a continuum of values (or, at least, something dense in the continiuum) that multiply without changing modulus. Thus we need values from all over the complex unit circle. Also, all rational numbers should be present, since ratios are a thing. So, at least all complex numbers with rational polar coordinates (with argument a rational multiple of $$\pi$$), extended to a field (Again, since ratios are a thing, I think we need to have a field). That would be $$\mathbb Q(i)$$ extended with all possible values of $$\sin (q\,\pi),\; q\in \mathbb Q$$. Some of these might be trancendental. I can't tell this late at night.
Do we really need $$\pi$$ or $$e$$? How about all the other trancendental numbers? I don't know. I am not certain enough to give an answer, but I have told you what I believe to be a minimum of numbers needed. · 3 years, 11 months ago
I was quite impressed by Mursalin's answer to the question. I know some basics about quantum mechanics, so I would try to make the reason here:- The Schrodinger's equation gives the probability of finding an electron in a given amount of space. It is the square of the wavefunction, and the wavefunction itself consists of a complex number. In fact, even the Newtonian mechanics can be derived from the Schrodinger's equation after assigning some values and parameters to the equation. So this clearly shows that this equation is a much more basic way of describing the universe than the newton's laws itself. So the answer comes here, everything in this universe is described by its wavefunction, and this wavefuction is an imaginary quantity. Hence complex numbers are the more basic and essential number type required for describing the universe around us. I don't know anything about hypercomplex numbers, but as per my knowledge complex numbers may satisfy the requirements. · 3 years, 11 months ago
I'd like to make another pitch for an answer that seems mostly neglected on this thread, which is the answer of INTEGERS!
One of the first issues with this answer is obviously the irrational numbers. However, I think that there is an easy way around this. Consider that no measurement is exact, that any measurement has a finite amount of significant digits in any measurement. Then, any measurement we could ever take has a finite number of decimal points, and can then be expressed as a fraction of two integers. In other words, it should never be necessary to use every infinite digit of pi to describe the circumference of a circle, or any other physical quantity. Only in pure math would you require these irrational numbers.
I am not so far in physics to be able to speak confidently about the issues of these claims in quantum mechanics, nor the role or necessity of imaginary numbers in quantum mechanics, so I will leave that alone. But perhaps there is a similar argument to be made there. · 3 years, 11 months ago
I like it! Any applied science needs only enough digits to have the calculations turn out accurately enough. · 3 years, 11 months ago
Mursalin gave an answer which I think everybody is liking. But honestly speaking I don't think that it is right. If you ask what type of numbers is required to describe universe around us then my answer would be Natural numbers! Yes.. It looks very stupid but let me explain. First of let me clear to you that it is not true that it is impossible to imagine quantum mechanics without complex numbers. This is a very famous misconception. Let us ask.. if nobody would have thought about complex numbers before the advent of quantum mechanics, wouldn't scientists have solved puzzles of blackbody radiation or photoelectric effect? Actually quantum mechanics can be constructed without using complex numbers, we just need to write equations for 2 variables instead of 1 with proper constraints for them. Complex number has two such quantities contained in it (magnitude and phase) and so if we use complex number then we need to write only single equation. Thus mathematically quantum mechanics or LCR circuit equations become easier if we use complex numbers.
Now what about 0,negative integers,rational numbers and irrational numbers? It is well known that all these quantities can be constructed mathematically. I think everybody knows about constructions of 0 and negative integers: they are constructed as solutions to certain equations. Last of all irrational numbers are constructed using well known procedures of Dedekind cuts or Nested intervals (you may read wikipedia articles about these).
Now the next question: suppose we don't construct any of these and only use natural numbers. Then is it possible to write say theory of general relativity or quantum mechanics? And answer is YES! Only problem would be that it would be too complicated since each variable would need lot of description! May be Kroneker had realized this long back and said "God made the natural numbers; all else is the work of man."
People who don't like my answer may also want to read about "Preintuitionism" on wikipedia. · 3 years, 11 months ago
Though members of this thread have successfully acknowledged that the entities that represent rationals and irrationals are completely built from natural numbers (e.g. equivalence classes of ordered pairs for fractions and sequences for irrationals), one significant component of the construction of real numbers has been ignored: the operations on these new structures, addition and multiplication.
At each stage of construction, one must redefine addition and multiplication for ordered pairs, sequences, etc. For example, if the ordered pairs (a,b) and (c,d) are fractions, we have the rule, which we must define their sum by the rule (a,b) + (c,d) = (ad + bc, bd). So even if we only use ordered pairs and sequences of ordered pairs of natural numbers, we have to provide new axioms that define the structures of the real numbers. So even employing this construction, one is still using a system isomorphic to the real numbers, equipped with the entire structure of the continuum. Thus if physics uses any idea of the continuum, it uses real numbers, no matter how they are disguised · 3 years, 11 months ago
I disagree on one point,can all irrational numbers be constructed from natural numbers?How would you construct pi or e? · 3 years, 11 months ago
Well, one possible way is this:
$$\pi=4(1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}- \cdots$$
$$e=1+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+ \cdots$$ · 3 years, 11 months ago
How do you know that L.H.S. is equal to R.H.S. here? You first need to construct an object which we call irrational number and only then you can think about representing them using infinite series. This is not the correct way. You can either use nested intervals or Dedekind cut. · 3 years, 11 months ago
mursalin's way is obviously not a feasible construction-It will require an infinite number of terms to be calculated.Moreover,there are irrational numbers which can't be represented as such explicit series. If I've Understood the method of Dedekind cuts correctly,it requires you to construct a set of rational numbers where each element is less than the number to be constructed. My point is,in both of the above methods,we are talking about infinite sequences,and thus,not applicable in actual physics calculations,no matter how complicated a calculation or analysis we are ready to undertake.
But it has got really interesting.I read that there are many irrational numbers which can't be explicitly represented(by some root,ratio,or series).Do these numbers have any significance in physics? · 3 years, 11 months ago
The first one comes from the obvious fact that $$tan\frac{\pi}{4}=1$$ and the Taylor Expansion of $${tan}^{-1}x$$.
The second one is a direct implementation of the Taylor Expansion of $$e^x$$.
I'm trying to express them just from their definition and without using any idea of where they might be on the number line. But I think we're deviating away from the topic a little bit.... · 3 years, 11 months ago
Dear Mursalin, I think you are not getting my point. May be this is because yet you didn't study analysis. Read first what I said and then you will understand. · 3 years, 11 months ago
Read about Dedekind cut on wikipedia or in any book on analysis. · 3 years, 11 months ago
What you've said is absolutely right! But that wasn't what I was going for. See the EDIT part of my comment. · 3 years, 11 months ago
I think that complex numbers are necessary, because they make a complete system. This means all equations can be described. This is important to mathematics (I don't know about QM and that stuff). And for all those people who say "everything can be constructed by integers", that doesn't matter. For example, some people said that fractions are simply integers over integers. Yes, but they aren't INTEGERS, they're RATIONAL NUMBERS. If you need to construct something to use it, face it, you need it. Now real numbers. I'm sure we all love calculating out infinite fraction sequences (not really), but for all purposes, it's better to use the irrational numbers. Complex numbers are similarly useful, even if they can't describe actual amounts. If you only count amounts, then sure, natural numbers are fine. But the universe is obviously more complex than that. And like shown dozens of times, physical equations use complex numbers. So I think that complex numbers are necessary, if not in a too obvious way, to our physical system. · 3 years, 10 months ago
Interesting, i find this kind of discussion fascinating, and would really like to see what Davids own answer is... One method of reasoning is that all rational numbers can be expressed using integers - simply use fractions.... real numbers also tend to be able to be expressed using integers - taylor expansions etc, although i do not know if it would be possible to express all real numbers in this way... maybe not.... but the ones necessary for current physical theories, quite possibly. Complex numbers such as 'i' can just be expressed as 'root(-1)', using integers and common notation.
The possible flaw in this method is that although you can express these using integers, overall what you have expressed isnt actually an integer - if you replace i with 'root(-1)' surely you're still using complex numbers - I guess it just depends on how you like to think about things.
If you ignore this flaw and continue using this reductionist method, you can actually dispose of "numbers" altogether, and instead use symbols and notation - sort of like 'typographical number theory', it is probably possible to express current physical theories using pure logic. It would be horrific - but I think its possible.
On the topic of beyond current physical theories, if we assume that the universe is finite, and that in a certain region there can only be a certain finite amount of information, I think that this information can be expressed/approximated in lots of different ways, to varying degrees of accuracy - different constructs are required for different approximations. I think if there was an ultimate fundamental "theory of the universe", it is likely to be simple in nature, in that it shouldn't require as a necessity certain man-made constructs (which I believe numbers essentially are), but would have complex implications - think fractals and chaos theory, however I also feel that such a theory would also be impossible... because surely any such system must be either incomplete or inconsistent - according to Godel. If this is applicable to 'the universe' or what implication it would have if it was is beyond me.
Whichever which-way you look at it, the "fundamental structure of our world" seems to be so far beyond human understanding that all we can do is approximate using crude models based on artificial constructs which seem to agree with our observations, which are also crude. · 3 years, 10 months ago
Fascinating view. · 3 years, 10 months ago
I have another question.... Are quaternions real? · 3 years, 11 months ago
All number systems are essentially a mathematical construct to explain the space, e.g. 1-D space can be described by a real number, 2-D space by a complex number or an ordered pair of two real numbers, 3-D space by vectors, and 4-D by quaternions, etc. · 3 years, 11 months ago
Really, all we ever need is $$1$$, $$-1$$, addition, and multiplication. Because with these few tools and a little algebra, you can construct all other numbers. However, I did realize that transcendental numbers aren't the easiest to construct with this system, so we should also include helpful numbers like $$\pi$$, $$e$$, or anything else like that. (Here is a more detailed explanation of this construction system and transcendental numbers) Whereas real and complex numbers are convenient for most physical applications, they are not necessary. Furthermore, if you consider $$1$$ and $$-1$$ as "real," then undoubtably, all these other numbers are real. · 3 years, 11 months ago
I think it should be complex numbers. Though I have not seen them(purely complex numbers) being used so often, but I know one field where it is used. My teacher had showed me using it in solving parallel LCR circuits. I don't know whether I am going as per the requirements of the discussion. · 3 years, 11 months ago
Aah, so this is a good example of ease of description. Is it necessary to have complex numbers for LCR circuits, or is it merely a convenient description because it makes the math easier? Staff · 3 years, 11 months ago
From what I can tell, in classical wave analysis, complex numbers are introduced as a way to simplify representations of wave equations, using Euler's formula. In the example of LCR circuits, one sets up the differential equation describing the system; an oversimplified method of seeing how complex numbers relate to this is by showing that the solution of the differential equation may take the form of $$e^{i(wt-c)}$$, where w is angular frequency and c is phase shift. This is convenient, as it is represents the wave equations involving sine and cosine, so we know intuitively that we're on the right track like this. When you plug it all back into the original differential equation and solve with initial conditions and such, we find that, indeed, all the imaginary terms from the $$e^{ix}$$ term disappear by the time we arrive at the final solution. Thus, it becomes apparent that the complex numbers were only introduced to help solve the equation, and are devoid of physical meaning. Quantum mechanics is a different story, however... · 3 years, 11 months ago
Well I think most of the things can be simplified out using one or more approaches. It depends on the effectiveness( I mean accuracy) of the technique being employed. · 3 years, 11 months ago
It makes the math easier I guess. I remember that while learning LCR circuits, I would end up with a second order differential equation and solving them isn't included in High School mathematics. Instead, we were introduced to phasors and use vectors(?) to solve the problems. · 3 years, 11 months ago
I like the book of Paul J. Nahin, "An Imaginary Tale: The Story of i" (you can see on Amazon) :) · 3 years, 10 months ago
First of all focussiong on what numbers are? guyz they are nothing except something used as a magnitude to describe teh physical quantities if the universe.ANd according to Model delpendent realism it occurs in our brain.Numbers are only in our brain and are create dfrom logic and i dont know the numbers which will decribe the universe but i know is the last and frist number of this number system- the numbers are -0 and Infinity...and guyzz thay are same · 3 years, 10 months ago
Seems like this answer has not come up: real numbers. Complex numbers, can actually be represented as an ordered pair, and just make the operations just like the ones in complex numbers. I do not have much time to build this up but I believe it is possible. In my opinion, complex numbers is just a tool for convenience. Other stuff can just replace it. It is obvious that rational numbers are needed, otherwise how are you going to evaluate $$P=\frac{F}{A}$$ (just an example)? So we have left why we need irrational numbers. Again I think this is obvious, as many constants are irrational and we need them. $$\pi$$ is significant enough. Well I think this is not the answer, just my opinion, so I hope someone out there can tell me my misconceptions, mistakes, gaps or whatsoever. · 3 years, 11 months ago
I think complex numbers should be used , though we can only use the reals and the tasks where complex numbers were used should be assciated with pairs of real numbers. · 3 years, 11 months ago
I think COMPLEX NUMBERS is required to describe the whole world around us.
Reason being , Complex numbers exists in Conjugate pairs . In the same way , In physics Forces exists in conjugate pairs (Fundamental theory of classical mechanics : newtons 3rd law). We can't move a inch without forces .SO, forces are there in every fundamental structure of the world and exists in pairs (analogous to complex numbers) .
Secondly , In physics we use vectors everywhere which is analogous to complex numbers.
Thirdly , We can't compare complex numbers. We can't say whether 3 + i or 4 - i greater. Can we? In the same way in physics , We can't compare different measuring units in physics. For example , we can't compare ' Kilo grams and pascals ' etc. etc. ( somewhat analogous )
(I think my 3rd reason makes no sense , But I am just trying to prove it )
Fourthly , Maths in the mother of all sciences. It is required in every walk of life and physics. In physics we use mathematical equations instantly. There is always a tendency to get a purely complex number as a root of the equation. And there are infinitely many pure complex numbers and infinitely many real numbers (or integers etc) .SO the probability of getting a pure complex number is equal to the probability of getting a real number. So , we can say , complex are used in almost every equations of maths (as integers⊂rational numbers⊂real numbers⊂complex numbers. )
Combining above four conditions , I can say that COMPLEX NUMBERS is required to describe the whole world around us. · 3 years, 11 months ago
I think that the whole process of classifying number types has begun from simple systems such as integers to intricate ones like complex numbers or hypercomplex numbers (quaternions, tessarines, coquaternions, etc. ). Now one can question what would be the need for newer (or more complex) number systems and the answer to that lies in the fact that these new number systems provide the appropriate mathematical formalism to realize ideas and theories. Now why are they appropriate and the answer to that would be if any other number system be used to formalize a theory it's be insufficient to describe the nuances of that theory. This brings us to question as to why advanced mathematics like lie algebra, or group theory, etc. is needed to do advanced physics. One would not be able to get far along if one continues to abandon complex numbers to do Quantum Mechanics ( as someone rightly pointed out). Coming back to number systems, the process with which new number systems were developed tells us an important fact as we made progress with the advent of complex numbers and hypercomplex numbers, our understanding of the universe improved. Does this not hint towards the fact that there is a possibility of an all encompassing number system which will qualify as the singular type of system needed for all physical theories. Definitely this number system can't be integers, real numbers, or even complex numbers because these prove to be parts of a bigger better system for theories. Taking a crude example, we can always use complex numbers instead of real numbers or integers, but let's say does it serve any purpose to use complex numbers in simple arithmetic. The point here is these are all subsets of what is required. Striving for a better, bigger, "all encompassing" and universal number system is what would be required for our current physical theories. And as complex numbers is in a sense a subset of hypercomplex numbers. So "currently" my answer would be hypercomplex numbers. And yeah, i could be totally wrong but this is what I feel should be. Hope I have conveyed my idea properly. · 3 years, 11 months ago
I think complex numbers must be the numbers used for describing the universe around us. · 3 years, 11 months ago
Can you give a reason? Staff · 3 years, 11 months ago
yes sir ,in real life situations like free fall under all the effects of nature like drag, viscosity, etc..the energy dissipated by the falling object is in the form of complex number magnitudes.. · 3 years, 11 months ago
I think all we need is real numbers.other things are all structures constructed from real numbers.A complex number is only an ordered pair of real numbers. An n-tuple of real numbers constitute an n-dimensional vector,a matrix of real numbers is an n cross n tensor,and so on.If necessary,we can also add another dimension and get a 3 dimensional analogue of a tensor,which will have n cross n cross n elements.So I think faith in 'reality' should be restored :) · 3 years, 11 months ago
hmmm · 3 years, 11 months ago
[Deleted for irrelevance- Peter] · 3 years, 11 months ago
Internet preachers are everywhere! Unbelievable! · 3 years, 11 months ago
What this has to do with the original question?? LOL · 3 years, 11 months ago
!!!!??? · 3 years, 11 months ago
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# Principle of Consistency
Financial Mathematics → Principle of Consistency
According to the principle of consistency
For example,
Example 1.
For all $t_1\leq t_2$ prove that the principle of consistency holds for the accumulation factor
Solution:
Let $t_1\leq t_2$, then for the principle of consistency we have to show that
We have
So,
Hence,
So, the principle of consistency is proved.
Posted
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# Manuals/calci/PERMUTATION
MATRIX("PERMUTATION",order)
• is the size of the Permutation matrix.
## Description
• This function returns the matrix Permutation matrix of order 3.
• A permutation matrix is a square binary matrix obtained by permuting the rows of an nxn identity matrix according to some permutation of the numbers 1 to n.
• This matrix has exactly one entry 1 in each row and each column and 0's elsewhere.
• A permutation matrix is nonsingular, and its determiant + or -.
• Also permutation matrix A having the following properties , where is a transpose and I is the identity matrix.
• Permutation matrices are orthogonal .Hence, their inverse is their transpose: .
• A permutation matrix allows to exchange rows or columns of another via the matrix-matrix product.
• In calci MATRIX("permutation",4) gives the permutation matrix of order 4.
## Examples
• 1.MATRIX("permutation",5,200..210)
0 0 0 200 0 0 201 0 0 0 202 0 0 0 0 0 0 203 0 0 0 0 0 0 204
• 2.MATRIX("permutation",18)._(SUM) = 18
• 3.MATRIX("permutation",5).(SUM)=
1 1 1 1 1
• 4.MATRIX("permutation",5).(SUM) =
1 1 1 1 1
## Related Videos
Permutation Matrix
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# How can photons destructively interfere?
This is a concept I don't fully understand. If I have two photons each with frequency $$\nu$$, then they each have an energy of $$E = h\nu$$. If they get matched with an inverted phase, then the summed wave will be null due to destructive interference. Then where does the energy go? It cannot radiate, since that would produce an extra E-M wave, right?
• Are you wanting to consider exactly 2 photons and nothing else? Or are you wanting to consider full EM waves and look at two photons within that? – BioPhysicist Dec 17 '20 at 22:00
• Electromagnetic waves have frequency and wavelength. Photons are point particles with energy and momentum. For your question it is enough to discuss electromagnetic waves. – my2cts Dec 17 '20 at 22:17
• This question is not a duplicate of what it was closed as. This question specifically seems interested in photons, not just waves in general. – BioPhysicist Dec 18 '20 at 17:37
• Maybe this exposures your question a bit physics.stackexchange.com/a/601091 and the link therein – HolgerFiedler Dec 18 '20 at 22:43
• Photons are particles. Waves interfere. – ProfRob Dec 19 '20 at 10:16
You cannot have complete destructive interference everywhere unless the photons have exactly the wave vector (that is, are propagating in the same direction with the same frequency*). At some places there is constructive interference, while in some places there is destructive interference. The total energy, including regions of constructive and destructive interference, is just the sum of the energies of the constituent waves.
Consider, as a simple example, two waves (of equal amplitude and polarization) traveling in opposite directions (and, for now, only worry about their electric fields), $$\vec{E}_{1}(\vec{r},t)=E_{0}\hat{\epsilon}\cos(kz-\omega t)\\ \vec{E}_{2}(\vec{r},t)=E_{0}\hat{\epsilon}\cos(-kz-\omega t)$$ When you add these together, you get a standing wave $$\vec{E}=\vec{E}_{1}+\vec{E}_{2}=2E_{0}{\hat\epsilon}\cos(kz)\cos(\omega t),$$ for which the time-average of the electric energy density** $$u_{E}=\frac{\varepsilon_{0}}{2}\vec{E}^{2}$$ is $$\langle u_{E}\rangle=\varepsilon_{0}E_{0}^{2}[1+\cos(2kz)].$$ There are places [nodes, where $$2kz=n\pi$$ for $$n$$ odd], where the electric energy density is zero because of destructive interference; and there are places [antinodes, with $$2kz=n\pi$$ for $$n$$ even] where it is four times that of each original wave, because of constructive interferences. Averaged over all space, the total energy is twice that of a single propagating wave—exactly what we expect for a system with two waves.
*If you really want to consider two waves with identical wave vectors, then you cannot emit a second photon that is $$180^{\circ}$$ with the first. Generating an electric field with that phase is actually absorbing the first photon, not emitting another one.
**The magnetic energy has as similar structure, but it is situated slightly differently spatially, because there is an additional relative minus sign between the magnetic fields $$\vec{B}_{1}$$ and $$\vec{B}_{2}$$ of the two waves.
• I think the key here is in your first "*". A complication is that you can't really create a wave with a single wave vector nevermind two with identical wave vectors. But apply the same question to point sources. Again, you can't put the two sources in exactly the same place. But in an imaginary world where you can (dangerous: your results are always meaningless if physics is applied to unreal situations) I think you have to conclude that each source does such work on the other that the energy remains in the system to be dissipated in whatever mechanism is driving the sources. – garyp Dec 17 '20 at 23:05
• I think the question is about photons, and this answer is about classical EM waves – anna v Dec 18 '20 at 10:20
• Related question on classical EM waves: physics.stackexchange.com/q/23930 (although I did not flag the question as a duplicate of this, there is significant overlap between Buzz's answer and those answering the linked question) – Nihar Karve Dec 20 '20 at 9:07
In the present day physics standard model photons are elementary particles , on par with the other particles in the table. This means they are point particles, of fixed (in this case zero) mass with spin 1 and $$E=hν$$ . The $$ν$$ is the frequency that the classical light will have, as it is composed of zillions of photons This can can be seen experimentally, how classical interference appears because the beam is composed out of a large number of same energy photons.
camera recording of photons from a double slit illuminated by very weak laser light. Left to right: single frame, superposition of 200, 1’000, and 500’000 frames.
Single photons leave a point consistent with the particle nature. It is the accumulation of photons that shows the classical interference pattern.
So two photons will not interfere in any way, except if one is studying photon photon scattering, which is very improbable for low energy photons. For high energy photons, gamma rays, a lot of particle antiparticle pairs can be created and there are plans of gamma colliders.
So there is no problem with the individual photons, they do not interfere. It is the wavefunction of the set up ( in the case above "photon scattering through two given slits given distance apart") that carries the frequency information of the photon, and can thus appear in the probability distribution. This should not be surprising as it is a quantized maxwell equation that gives the photon wavefunctions.
Your question is correct, photons do not really interfere. The DSE taught at the high school level is a convenient theory and it also works well mathematically but 2 photons cancelling is a violation of conservation of energy. In university in quantum optics courses deeper explanations are provided.
Think of 2 tsunamis one from Japan and the other from USA, starting with opposite phase .... when they meet (at say Hawaii) they cancel and Hawaii is saved ... but a second later the waves emerge again and continue on their way to Japan and USA, the energy was only stored temporarily in the elasticity of the water! The energy will only be absorbed when the wave crashes on the land. For photons we can never really observe the field directly ... we can only see a photon when our eye or camera absorbs it. We assume the photons are interfering in the EM field .... it makes sense .... but every photon is created by an atom and eventually absorbed by an atom.
• Dear PhysicsDave. It is usually frown upon to directly copy-paste identical answers. (The problem is if everybody start to copy-paste identical answers en mass.) – Qmechanic Dec 22 '20 at 20:37
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# Get Answers to all your Questions
#### State whether the given statement is true (T) or false (F).4 : 7 = 20 : 35
True
As $20: 35= \frac{20}{35}=\frac{4}{7}=4: 7$
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# Windows library inclusion in CMake Project
I have a growing project in CMake. It is time to link to a library, which at this time exists only in Windows, Linux functionality will have to wait. I am attempting to do this with preprocessor directives as recommended in an answer to this question:
// MyLibHeader.hpp
#ifdef WIN32
#include <windows.h>
#define ProcHandle HINSTANCE
#define LoadLib LoadLibraryA
#define LoadSym GetProcAddress
#else
// ... I'll fill these in with dlopen etc. when necessary
This is the first platform specific inclusion I have had to put in my code, and It seems there is more to it than this. This is generating this error:
C:\Program Files (x86)\Microsoft Visual Studio 8\VC\include\intrin.h(944) : error C2733: second C linkage of overloaded function '_interlockedbittestandset' not allowed
The error is repeated four times, twice in intrin.h, and twice in winnt.h. So here's my question. Are there other includes or preprocessor steps I need to take to get this to work inside windows (up to now it has been a basic console application) and is there something in CMake I can leverage to make this easier.
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## 2 Answers
From what I've been able to scrape up with some help and some google, one solution is indeed to comment out the duplicate definitions of _interlockedbittestandset in initrin.h
This may have been fixed in later versions of Visual Studio.
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I must say, I find myself averse to such hackery. – 2NinerRomeo Mar 13 '12 at 17:25
That hackery was indeed successful. – 2NinerRomeo Mar 16 '12 at 15:26
One such discussion is cataloged here: social.msdn.microsoft.com/Forums/en-US/vcprerelease/thread/… – 2NinerRomeo Mar 16 '12 at 15:27
You could look at the source code of CMake, there is a C++ class that does cross platform library loading. It is a BSD style license so you could just copy the code. Works on lots of platforms. The code is here:
http://cmake.org/gitweb?p=cmake.git;a=blob;f=Source/kwsys/DynamicLoader.cxx;h=c4ee095519fe27742a0a9a9e392be4ce095af423;hb=HEAD
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Service interruption on Monday 11 July from 12:30 to 13:00: all the sites of the CCSD (HAL, EpiSciences, SciencesConf, AureHAL) will be inaccessible (network hardware connection).
# Real Time Detection of Soil Moisture in Winter Jujube Orchard Based on NIR Spectroscopy
Abstract : The measurement and control of soil moisture are the key technologies of precision agriculture. In order to real-time detect soil moisture content faster and more accurately, a portable soil moisture sensor based on NIR spectroscopy was developed. With the sixty soil samples collected from a winter jujube orchard, a linear regression model was established. The determination coefficients of the calibration ($R^2_c$) and validation ($R^2_v$) reached 0.88 and 0.92, respectively. The model passed F-test and t-test and showed robust. Subsequently, two spatial distribution maps of soil moisture were generated based on the data obtained by the portable soil moisture detector and the data obtained by oven drying method, respectively. Finally, the correlation between these two maps was investigated by using the software of Surfer 8.0. The zones of dry and wet soil could be distinguished easily in both maps. The results of the study showed that the developed detector was practical.
Keywords :
Document type :
Conference papers
Domain :
Cited literature [19 references]
https://hal.inria.fr/hal-01348262
Contributor : Hal Ifip Connect in order to contact the contributor
Submitted on : Friday, July 22, 2016 - 3:57:59 PM
Last modification on : Monday, November 12, 2018 - 10:22:02 AM
Long-term archiving on: : Sunday, October 23, 2016 - 1:17:32 PM
### File
978-3-642-36137-1_52_Chapter.p...
Files produced by the author(s)
### Citation
Xiaofei An, Minzan Li, Lihua Zheng, yumeng Liu, yajing Zhang. Real Time Detection of Soil Moisture in Winter Jujube Orchard Based on NIR Spectroscopy. 6th Computer and Computing Technologies in Agriculture (CCTA), Oct 2012, Zhangjiajie, China. pp.447-455, ⟨10.1007/978-3-642-36137-1_52⟩. ⟨hal-01348262⟩
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[SOLVED] Bizarre nil issue
Questions about the LÖVE API, installing LÖVE and other support related questions go here.
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milon
Party member
Posts: 210
Joined: Thu Jan 18, 2018 9:14 pm
[SOLVED] Bizarre nil issue
SUMMARY
A variable which I've just successfully concatenated in a string gets copied to another variable, and decides to be nil instead
DETAILS
I have absolutely no idea why this is happening. I can't figure out a reproducer subset of code, so I'm attaching the whole .love file. When the maze builder hits a dead end it will backtrack to the previous node and look for other possible routes. When all routes are exhausted, it will backtrack to the very beginning and the maze building is done. Note that you need to look at the console output to actually see the error. In file maze.lua, line 74 is where y is assigned to the vertical backtrack co-ord. I can use the stored y0 (at notes[x][y].y0) in a string function before copying it into y, so it's fine there. But after setting y, it has a seemingly random chance of being nil instead. This seems to only happen on the very last backtrack step of a completed maze. (Yes, I could code around this, but it's a bug and I hate bugs! )
There's a second bug which seems to be unrelated, but occasionally the backtrack goes back 2 steps instead of 1. This rarely means part of the maze doesn't get dug out. I'd like to solve this issue too, but I'm primarily focused on the strange nil issue above.
Can anyone else reproduce this? Any idea why it's happening?
Attachments
maze.love
(2.57 KiB) Downloaded 59 times
Last edited by milon on Wed Sep 22, 2021 2:27 pm, edited 1 time in total.
BrotSagtMist
Citizen
Posts: 62
Joined: Fri Aug 06, 2021 10:30 pm
Re: Bizarre nil issue
Firstoff, for debug reasons you should use math.random instead love.math.random.
That makes debugging predictable.
Ok look at your code:
x = notes[x][y].x0
y = notes[x][y].y0
You are changing x then try to access the table with the CHANGED x. So your y is something completely different than in the debug backtrack.
obey
grump
Party member
Posts: 813
Joined: Sat Jul 22, 2017 7:43 pm
Re: Bizarre nil issue
BrotSagtMist wrote: Thu Sep 16, 2021 7:30 pm Firstoff, for debug reasons you should use math.random instead love.math.random.
That makes debugging predictable.
Why? Elaborate please?
monolifed
Party member
Posts: 192
Joined: Sat Feb 06, 2016 9:42 pm
Re: Bizarre nil issue
Code: Select all
x = notes[x][y].x0 -- x is never affected by this bug...
y = notes[x][y].y0 -- somehow this can set y to nil even though I've just shown a non-nil value for y0...
first statement sets x to notes[x][y].x0
but the second sets y to notes[notes[x][y].x0][y].y0 Not notes[x][y].y0
You can do
Code: Select all
x, y = notes[x][y].x0, notes[x][y].y0
Last edited by monolifed on Thu Sep 16, 2021 7:48 pm, edited 1 time in total.
milon
Party member
Posts: 210
Joined: Thu Jan 18, 2018 9:14 pm
Re: Bizarre nil issue
BrotSagtMist wrote: Thu Sep 16, 2021 7:30 pm Firstoff, for debug reasons you should use math.random instead love.math.random.
That makes debugging predictable.
Ok look at your code:
x = notes[x][y].x0
y = notes[x][y].y0
You are changing x then try to access the table with the CHANGED x. So your y is something completely different than in the debug backtrack.
Bwa ha ha! Yup, I did that. Good catch! How in the world did it ever work at all? o_O
For anyone who cares, my fix is to combine the two lines into one:
Code: Select all
x, y = notes[x][y].x0, notes[x][y].y0
Both x0 and y0 get evaluated together and assigned together, so no index-monkeying occurs.
EDIT - This also fixes the second bug.
grump wrote: Thu Sep 16, 2021 7:41 pm
BrotSagtMist wrote: Thu Sep 16, 2021 7:30 pm Firstoff, for debug reasons you should use math.random instead love.math.random.
That makes debugging predictable.
Why? Elaborate please?
I think because love.math.random is seeded with the OS timer to make it more random, and lua's basic math.random always has the same seed unless you give it a different one.
Last edited by milon on Thu Sep 16, 2021 8:05 pm, edited 1 time in total.
grump
Party member
Posts: 813
Joined: Sat Jul 22, 2017 7:43 pm
Re: Bizarre nil issue
milon wrote: Thu Sep 16, 2021 7:47 pm I think because love.math.random is seeded with the OS timer to make it more random, and lua's basic math.random always has the same seed unless you give it a different one.
math.random is platform-dependent and can/will produce different numbers on different platforms, even with the same seed.
If you want a reproducible, platform-independent sequence of random numbers, use LÖVE's RNG with a fixed seed.
https://love2d.org/wiki/love.math.setRandomSeed or https://love2d.org/wiki/love.math.newRandomGenerator
milon
Party member
Posts: 210
Joined: Thu Jan 18, 2018 9:14 pm
Re: Bizarre nil issue
That's true, I'd forgotten that. I knew there was a reason I was using Love's version.
BrotSagtMist
Citizen
Posts: 62
Joined: Fri Aug 06, 2021 10:30 pm
Re: Bizarre nil issue
It is yet not clear if there are actually differences between platforms, i have not seen them in the wild so far. It is only that there is no guarantee that it is the same. And on further downsides, loves version is a slight tick slower then luajits function.
In terms of map creation if we write math.randomseed(1) before a generation this will always create the same level and we can therefore say: There is a bug in level 1 instead of "sometimes it bugs".
Personally i use math.randomseed(love.math.random()) which gives the best of booth.
obey
grump
Party member
Posts: 813
Joined: Sat Jul 22, 2017 7:43 pm
Re: Bizarre nil issue
BrotSagtMist wrote: Thu Sep 16, 2021 8:18 pm It is yet not clear if there are actually differences between platforms, i have not seen them in the wild so far.
It is clear though, because different implementations of the C runtime library use different implementations of PRNGs.
It is only that there is no guarantee that it is the same.
love.math.random guarantees it.
And on further downsides, loves version is a slight tick slower then luajits function.
True, but also almost irrelevant unless you have to generate hundreds of millions of numbers per second.
In terms of map creation if we write math.randomseed(1) before a generation this will always create the same level
Only on the same platform and with the same runtime version, while love.math.random always produces the same sequence on all platforms.
Edit: I just remembered that LuaJIT has a different PRNG than vanilla Lua with more guarantees, so the point about platform independence might be moot
Last edited by grump on Thu Sep 16, 2021 8:43 pm, edited 1 time in total.
BrotSagtMist
Citizen
Posts: 62
Joined: Fri Aug 06, 2021 10:30 pm
Re: Bizarre nil issue
I challenge you to find me a platform where the results differ then.
obey
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# Luna Tech
Tutorials For Dummies.
# 0. 前言
Reference: 8.18
You might conclude from this that there is some underlying structure to the web that clusters together web sites that are similar on some level.
One graph algorithm that can help find clusters of highly interconnected vertices in a graph is called the strongly connected components algorithm (SCC).
# 1. Strongly Connected Components (SCC)
## 定义
strongly connected component - C
C 属于一个 Graph - G
C 需要满足这个条件：在 C 里面的每一对 node（如 v, w），都有 v 到 w 的 edge 以及 w 到 v 的 edge（也就是双向关联）。
# 2. 概念：Transpose Graph
The transposition of a graph 𝐺 is defined as the graph $𝐺^𝑇$ where all the edges in the graph have been reversed.
SCC: ABD
SCC: ABD
# 3. SCC 步骤
SCC 算法用来计算 graph 里面的 SCC，它基于我们之前讲过的 DFS。
## 步骤
1. 在 G 里面调用 dfs function，计算每个 node/vertex 的 finishing time；
2. 计算 $G^T$
3. 在 $G^T$ 里面调用 dfs function，但是 DFS 最主要的那个 loop 是以 desceding order of finish time 来遍历 node/vertex 的；
4. Step 3 的结果是一个 forest，这个 forest 里面的每棵树都是 SCC，我们最终输出在 forest 里面的每棵树的每个 node 的 id（一个 list），用来标识 SCC。
# 思考
## 为什么要转置？
https://en.wikipedia.org/wiki/Transpose_graph
Although there is little difference mathematically between a graph and its transpose, the difference may be larger in computer science, depending on how a given graph is represented. For instance, for the web graph, it is easy to determine the outgoing links of a vertex, but hard to determine the incoming links, while in the reversal of this graph the opposite is true. In graph algorithms, therefore, it may sometimes be useful to construct the reversal of a graph, in order to put the graph into a form which is more suitable for the operations being performed on it. An example of this is Kosaraju’s algorithm for strongly connected components, which applies depth first search twice, once to the given graph and a second time to its reversal.
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Together these charts cover the majority of control chart needs of healthcare quality improvement and control. The Xbar & R Control Chart An Xbar & R Control Chart is one that shows both the mean value ( X ), ... Each of these values then becomes a point on the control chart that then represents the characteristics of that given day. In healthcare, which, you may have guessed, is my domain, most quality data are count data. There is a subtle but important distinction between counting defects, e.g. number of pressure ulcers, and counting defectives, e.g. number of patient with one or more pressure ulcers. Find if the element is outside control limit using the ucl calculator. 6. San Francisco: John Wiley & Sons Inc. David B. Laney (2002). Douglas C. Montgomery (2009). Joseph Berk, Susan Berk, in Quality Management for the Technology Sector, 2000. Defects are expected to reflect the poisson distribution, while defectives reflect the binomial distribution. However, you are more interested in what your average score is on a given night. Figure 2: I chart, special cause variation. When defects or defectives are rare and the subgroups are small, C, U, and P charts become useless as most subgroups will have no defects. One can do them all at the same time. Second calculate sigma.The formula for sigma varies depending on the type of data you have. R-chart example using qcc R package. Just remember, it is three sigma limits of what is being plotted. Finally, we see two red lines labeled lower control limit (LCL) and upper control limit (UCL). I highly recommend Montgomeryâs Introduction to Statistical Process Control (Montgomery 2009). The presence of special cause variation makes the process unpredictable. Figure 3 shows that the average weekly number of hospital acquired pressure ulcers is 66 and that anything between 41 and 90 would be within the expected range. Control chart constants are the engine behind charts such as XmR, XbarR, and XbarS. In particular, the sections on rare events T and G control charts and the detailed explanation of prime charts are most helpful. A Control Chart is also known as the Shewhart chart since it was introduced by Walter A Shewhart. So another idea is to plot the average of the three ga⦠On average, 8% of discharged patients have 1.5 hospital acquired pressure ulcers. Additionally, two lines representing the upper and lower control ⦠Also, The Healthcare Data Guide (Provost 2011) is very useful and contains a wealth of information on the specific use of control charts in healthcare settings. Figure 4 displays the number of pressure ulcers per 1000 patient days. First calculate the Center Line. The larger the numerator, the narrower the control limits. Control chart Selection. Individuals Chart Limits The lower and upper control limits for the individuals chart are calculated using the formulas ... the R chart center line is given by values ... goes above the upper control limit, the chart gives no indication that a change has taken place in the process. However, I suggest that you avoid the chapter on run charts in this book, since it promotes the use of certain run chart rules that have been proven ineffective and even misleading (Anhoej 2015). It was not my intention to go deep into the theoretical basis of run and control charts. It is a beginnerâs mistake to simply calculate the standard deviation of all the data points, which would include both the common and special cause variation. Introduction to Statistical Process Control, Sixth Edition, John Wiley & Sons. To demonstrate the use of C, U and P charts for count data we will create a data frame mimicking the weekly number of hospital acquired pressure ulcers at a hospital that, on average, has 300 patients with an average length of stay of four days. Improved control charts for attributes. Calculate the upper control limit for the X-bar Chart b. Finally, as mentioned, the diagnostic value of run charts is independent of the number of data points, which is not the case with control charts unless one adjusts the control limits in accordance with the number of data points. Chart demonstrating basis of control chart Why control charts "work" The control limits as pictured in the graph might be 0.001 probability limits. UCL (R) = R-bar x D4 Plot the Upper Control Limit on the R chart. Jacob Anhoej, Anne Vingaard Olesen (2014). In a recent study, using simulated data series, I found that run charts (using appropriate rules) are more sensitive to moderate, persistent shifts in data (< 2 SD) than control charts, while keeping a low rate of false positive signals that is independent of the number of data points (Anhoej 2014). Luckily, one does not have to choose between C, U and P charts. R Chart Results. Figure 15: Example of R Chart. A control chart is a chart used to monitor the quality of a process. 490 0 obj <>/Filter/FlateDecode/ID[<7D1A6AE204E4E141B614C09A67857257>]/Index[472 31]/Info 471 0 R/Length 87/Prev 1062849/Root 473 0 R/Size 503/Type/XRef/W[1 2 1]>>stream The P chart is probably the most common control chart in healthcare. For example, the rate of pressure ulcers may be expressed as the number of pressure ulcers per 1000 patient days. The purpose of the MR chart is to identify sudden changes in the (estimated) within subgroup variation. Figure 11: T chart displaying time between events. Laney proposed a solution to this problem that incorporates the between subgroup variation (Laney 2002). Together with my vignettes on run charts it forms a reference on the typical day-to-day use of the package. We can also call it as process behavior chart. On the other hand, one looses the original units of data, which may make the chart harder to interpret. Note that the first patient with pressure ulcer is missing from the chart since, we do not know how many discharges there had been since the previous patient with pressure ulcer. Lets review the 6 tasks below and how to solve them a. is caused by phenomena that are not normally present in the system. Figure 6 displays a G chart mimicking 24 discharged patient with pressure ulcers. The U chart plots the rate of defects. I do not use any other sensitising control chart rules. Each of the data frameâs 24 rows contains information for one week on the number of discharges, patient days, pressure ulcers, and number of discharged patients with one or more pressure ulcers. For that, seek out the references listed below. Calculate the lower control limit for the X-bar Chart It is important to note that neither common nor special cause variation is in itself good or bad. There is one exception to this practice: When dealing with rare events data, it often pays to do the G or T control chart up front, as it may otherwise take very long time to detect improvement using run chart rules alone. The purpose of this vignette is to demonstrate the use of qicharts for creating control charts. Walther A. Shewhart, who invented the control chart, described two types of variation, chance cause variation and assignable cause variation. # Lock random number generator to reproduce the charts from this vignette, # Introduce an outlier at data point number 18, 'Hospital acquired pressure ulcers (C chart)', 'Hospital acquired pressure ulcers (U chart)', 'Hospital acquired pressure ulcers (P chart)', # Create vector of random values from a geometric distribution, 'Patients between pressure ulcers (G chart)', # Plot I chart of individual birth weights, 'Pairwise differences in birth weights (MR chart)', # Vector of 24 subgroup sizes (average = 12), # Plot Xbar chart of average birth weights by date of birth, # Plot S chart of within subgroup standard deviation, 'Standard deviation of birth weight (S chart)', 'Patients with hospital acquired pressure ulcers (Standardised P chart)', 'Prime P chart of patients with pressure ulcer', P chart for proportion of defective units, G chart for units produced between defective units, I and MR charts for individual measurements, Xbar and S charts for average measurements, Laney says that there is no reason not always to use prime charts, Diagnostic Value of Run Chart Analysis: Using Likelihood Ratios to Compare Run Chart Rules on Simulated Data Series, Run Charts Revisited: A Simulation Study of Run Chart Rules for Detection of Non-Random Variation in Health Care Processes. is also called random variation or noise. h�bf�aa�4gb@ !�+P�������������B��@'���p� %~5����Tp��d�ޖ��;�FX�5��wP��5��2ӹ�o����/��R}��A�f�� �zF\�\$W&��q�8�0L����c����lY!�JLu�biK۲"���l8��� ��� ��������Ѡ�����l�� dht4wt0e-:�r��P�H CK�*Mg���H �8����|�T��w����%������s̆F��Lf�>�N�y6��@98J�Āt���b �Dr �� is also called non-random variation or signal. The R-chart generated by R also provides significant information for its interpretation, just as the x-bar chart generated above. X-bar control limits are based on either range or sigma, depending on which chart it is paired with. Figure 9: Xbar chart of average measurements, Figure 10: S chart of within subgroup standard deviations. The standardised chart has fixed control limits at $$\pm3$$ and a centre line at 0. In an interview Laney says that there is no reason not always to use prime charts. The control limits, also called sigma limits, are usually placed at $$\pm3$$ standard deviations from the centre line. To illustrate the control chartâs anatomy and physiology, we will use a simple vector of random numbers. Like the G chart, the T chart is a rare event chart. h�bbdb��� �H�����]@�i�uDT��� �� �HpG��)�� SJ�@,�)q�Ӂ,F҈�b� n 2 Most of the hard work has been done. Control charts monitor the quality of the elements. In this vignette I have demonstrated the use of the qicharts package to create control charts for measure and count data. Thirty-five samples of size 7 each were taken from a fertilizer-bag-filling machine at Panos Kouvelis Lifelong Lawn Ltd. The standardised chart shows the same information as its not-standardised peer, but the straight control lines may appear less confusing. To explain further, ... Upper Control Limit (UCL) = D4 * R bar. Figure 13 is a prime P chart of the same data as in figure 5. Jacob Anhoej (2015). Figure 4: U chart displaying the rate of defects. If data points fall outside of these lines, it indicates that it is statistically likely there is a problem with the process. In short it is the intended result on the metric that is measured. Traditionally, the term âdefectâ has been used to name whatever it is one is counting with control charts. And since time is a continuous variable it belongs with the other charts for measure data. The statistical process control has the highest level of quality for a product in the ucl lcl calculator. By this, we can see how is the process behaving over the period of time. The Center Line equals either the average or median of your data. Instead we could plot the number of discharges between each discharge of a patient with one or more pressure ulcers. However, a control chart is being used at the initial stage to see the process behavior or to see the Voice of Process (VoP). I recommend that you read the vignette on run charts first for a detailed introduction to the most important arguments of the qic() function. Data point no. The results were: Overall mean = 57.75 lb. The X-Bar chart and Individuals chart both use A2 and E2 constants to compute their upper and lower control limits. The upper control limit for the range (or upper range limit) is calculated by multiplying the average of the moving range by 3.267: U C L r = 3.267 M R ¯ {\displaystyle UCL_{r⦠It is used to plot the proportion (or percent) of defective units, e.g. the proportion of patients with one or more pressure ulcers. The standard deviation is the estimated standard deviation of the common cause variation in the process of interest, which depends on the theoretical distribution of data. In theory, the P chart is less sensitive to special cause variation than the U chart because it discards information by dichotomising inspection units (patients) in defectives and non-defectives ignoring the fact that a unit may have more than one defect (pressure ulcers). This happens when the numerator (area of opportunity) differs between subgroups. So, what does that mean? %%EOF Also note that the G chart rarely has a lower control limit. Figure 6: G chart displaying the number of units produced between defectives. The center line in the control chart is the mean, the two horizontal line is the ucl and lcl. Although in Six Sigma study, we usually read Control chart in the Control phase. Figure 7: I chart for individual measurements. X-bar control limits are based on either range or sigma, depending on which chart it is paired with. Explanation: The Upper Limit, Lower Limit, and Central/Control Line are the parameters of the control chart. Additionally, two lines representing the upper and lower control limits are shown. Similar to the run chart, the control charts is a line graph showing a measure (y axis) over time (x axis). ... Knowing which control chart to use in a given situation will assure accurate monitoring of process stability. Without it we cannot estimate the control limits using equation (4). When an X-Bar/R chart is in statistical control, the average The number of units between defectives is modelled by the geometric distribution. Third, calculate the sigma lines.These are simply ± 1 sigma, ± 2 sigma and ± 3 sigma from the center line. Figure 2 is an example of special cause variation. If your process is in statistical control, ~99% of the nails produced will measure within these control limits. On the other hand, the P chart often communicates better. 0 However, Provost and Murray (Provost 2011) suggest to use prime charts only for very large subgroups (N > 2000) when all other explanations for special cause variation have been examined. We calculate these terms because we have a theory base for that. Especially, the set of rules promoted by Provost and Murray (Provost 2011), have very poor diagnostic properties (Anhoej 2015). In practice I always do the run chart analysis first. Usually there is no relationship whatsoever. A process that is in statistical control is predictable, and characterized by points that fall between the lower and upper control limits. In contrast to the run chart, the centre line of the control chart represents the (weighted) mean rather than the median. Select all the data in those four columns and create a line chart based on that data. However, from time to time I stumble across measure data, often in the form of physiological parameters or waiting times. 18, lies above the upper control limit, which indicates that special causes are present in the process. Also they are easier to construct (by pen and paper) and understand than are control charts. Having gathered preliminary data and used it to define the grand average, the average range, and the upper and lower control limits for each, we can now begin plotting data and using the chart to control the process. The Range (R) chart shows the variation within each variable (called "subgroups"). One idea is that you could plot the score from each game. 502 0 obj <>stream 18 is under influence of forces that are not normally present in the system. X-bar Chart Limits The lower and upper control limits for the X-bar chart are calculated using the formulas = â n LCL x m ÏË = + n UCL x m ÏË where m is a multiplier (usually set to 3) chosen to control the likelihood of false alarms (out -of-control signals when the process is in control). The R chart is the control chart for the subgroup ranges. Figure 7 is an I chart of birth weights from 24 babies. If our process i⦠In the same way, engineers must take a special look to points beyond the control limits and to violating runs in order to identify and assign causes attributed to changes on the system that led the process to be out-of-control. endstream endobj startxref The qicharts package employs a handful of the classic Shewhart charts for measure and count data plus a couple of rare events charts. Before we start, we will load the qicharts package and lock the random number generator in order to make reproducible data sets for this vignette. is caused by phenomena that are always present within the system. When the X-bar chart is paired with a range chart, the most common (and recommended) method of computing control limits based on 3 standard deviations is: X-bar UCL - Upper Control Limit UCL, (Upper Control Limit), as it applies to X Bar, (mean), and R Bar, (range), charts, is a formula that will calculate an upper most limit for samples to evaluate to.There is usually a LCL, (Lower Control Limit), that is also calculated and used in process control charts.. You can also use Pre-Control to establish control limits on control charts. As mentioned, defectives are modelled by the binomial distribution. I charts are often accompanied by moving range (MR) charts, which show the absolute difference between neighbouring data points. Q.No 2: what are the values for Central Line, Upper control limit and lower control limit, also show the entire calculations for the response Can you please let me know if I should employ an Xbar and R chart t- OR â Xbar and S chart. It is a common misunderstanding that control charts are superior to run charts. You are interested in determining if you are improving your bowling game. There are many more arguments available for the qic() function than I have demonstrated here. If the subgroup size is between 7 and 10, select the appropriate constant, called D3, and multiply by R-bar to determine the Lower Control Limit for the Range Chart. You bowl three games a night once a week in a bowling league. The correct control chart on the number of pressure ulcers is the C chart, which is based on the poisson distribution. If not, I suggest that you buy a good, old fashioned book on the subject. Lloyd P. Provost, Sandra K. Murray (2011). There is no Lower Control Limit for the Range Chart if the subgroup size is 6 or less. The Upper Control Limit (UCL) = 3 sigma above the center line = 23.769. Please study the documentation (?qic) for that. I assume that you are already familiar with basic control chart theory. %PDF-1.6 %���� Suppose you are a member of a bowling team. This may be an artefact caused by the fact that the âtrueâ common cause variation in data is greater than that predicted by the poisson or binomial distribution. What are some different approaches you could use? If, for example, 8% of discharged patients have a hospitals acquired pressure ulcer and the average weekly number of discharges in a small department is 10, we would, on average, expect to have less than one pressure ulcer per week. Control limit equations are based on three sigma limits. R-bar (mean of Ranges) = 6.4. How do you calculate control limits? You can specify a lower bound and an upper bound for the control limits. These were later renamed to common cause and special cause variation. D3 = 0. Control Chart Constants Depend on d2. Similar to the run chart, the control charts is a line graph showing a measure (y axis) over time (x axis). In theory, overdispersion will often be present in real life data but only detectable with large subgroups where point estimates become very precise. In statistical process monitoring (SPM), the ¯ and R chart is a type of scheme, popularly known as control chart, used to monitor the mean and range of a normally distributed variables simultaneously, when samples are collected at regular intervals from a business or industrial process.. In contrast to the run chart, the centre line of the control chart represents the (weighted) mean rather than the median. UCL = D4 (RÌ
) LCL = D3 (RÌ
) Grand mean (for mean of Xbars) = 15.11. PLoS ONE 9(11): e113825. The centre line of the G chart is the theoretical median of the distribution ($$mean \times 0.693$$). The formulas for calculation of control limits can be found in Montgomery 2009 and Provost 2011. The indicator is the number of discharges between each of these. Figure 3: C chart displaying the number of defects. Specification limits are the targets set for the process/product by customer or market performance or internal target. Figure 1: I chart showing common cause variation. The U chart is different from the C chart in that it accounts for variation in the area of opportunity, e.g. the number of patients or the number of patient days, over time or between units one wishes to compare. Since the calculations of control limits depend on the type of data many types of control charts have been developed for specific purposes. Control charts, on the other hand, are quicker to pick up large (transient) shifts in data. X-bar and range chart formulas. In both cases we need the d2 constant. Figure 5: P chart displaying the percent of defectives. But the end goal of improvement is always a stable process functioning at a satisfactory level. LCL(R) = R-bar x D3 PLoS ONE 10(3): e0121349. Control limits for the R-chart. Note that the control limits vary slightly. A rate differs from a proportion in that the numerator and the denominator need not be of the same kind and that the numerator may exceed the denominator. The Health Care Data Guide: Learning from Data for Improvement. Quality Control Grid Calculator; Control Limit Calculator; Reportable Range Calculator: Quantifying Errors; Reportable Range Calculator: Recording Results; Dispersion Calculator and Critical Number of Test Samples The upper and lower control limits are two horizontal lines drawn on the chart. The control limits are slightly wider. This chart must exhibit control in order to make conclusions on the Xbar chart. endstream endobj 473 0 obj <. If, and only if, the run chart shows random variation and I need to further investigate data for outliers or to know the limits of common cause variation, I would do a control chart analysis combining the run chart rules with Shewhartâs original 3 sigma rule (one or more data point outside control limits). D4 =2.114. But instead of displaying the number of cases between events (defectives) it displays the time between events. Run Charts Revisited: A Simulation Study of Run Chart Rules for Detection of Non-Random Variation in Health Care Processes. What is the relationship between control limits and specification limits? A stable process may function at an unsatisfactory level, and an unstable process may be moving in the right direction. The Lower Control Limit (LCL) = 3 sigma below the center line = 22.131. We often hear control limits and specification limits discussed as if they are interchangeable. Diagnostic Value of Run Chart Analysis: Using Likelihood Ratios to Compare Run Chart Rules on Simulated Data Series. For most people, not to mention the press, the percent of harmed patients is easier to grasp than the the rate of pressure ulcers expressed in counts per 1000 patient days. This is because the geometric distribution is highly skewed, thus the median is a better representation of the process centre to be used with the runs analysis. Select Largest Contributor to identify the variable that contributes If one does not like the wavy control lines in U, P, Xbar and S charts, one can do a standardised chart, which turns the indicator into a Z score by subtracting the mean from each value and dividing by the standard deviation. Maintaining an x bar:r Chart. If there is more than one measurement in each subgroup the Xbar and S charts will display the average and the within subgroup standard deviation respectively. And, if you've made a control chart by hand or sat in a class, you'll likely have memories of bizarre constants like d2, A2, etc. Minitab labels the lower bound as LB and the upper bound as UB. One big advantage of run charts is that they are oblivious to assumptions on the theoretical distribution of data. One data point, no. If there are many more patients in the hospital in the winter than in the summer, the C chart may falsely detect special cause variation in the raw number of pressure ulcers. If any data point in the MR is above the upper control limit, one should interpret the I chart very cautiously. Pane Options Point Symbols: select Beyond Limits to draw special point symbols only for points falling above the control limit. a) For the given sample size, the control limits for 3-sigma x chart are: Upper Control Limit (UCLZ) = Ib. The confusion may stem from the fact that different sets of rules for identifying non-random variation in run charts are available, and that these sets differ significantly in their diagnostic properties. Estimating the R Chart Center Line Your output should be a zigzag line in the middle with your actual observations, crossing and re-crossing the straight center line showing the process mean, with the upper control limit as a horizontal line above it and the lower control limit as a horizontal line below it. If the calculated control limit is farther from the center line than the value that you specify, Minitab displays the bound instead of the control limit. upper control limits is a signal of a potential out-of-control condition. makes the process predictable (within limits). 472 0 obj <> endobj An alternative to the G chart is the T chart for time between defects, which we will come back to later. ; Average range R = 1.78 lb. The control limits represent the boundaries of the so called common cause variation inherent in the process. The R chart must be in control to draw the Xbar chart. For the sample data, 3 out-of-control signals are given by the chart. Quality Engineering, 14(4), 531-537. ref : AIAG manual for SPC Chart for Averages Chart for Averages Control Limits Factor Divisors to Estimate ÏÏÏÏ x Control Limits Factor Divisors to estimate ÏÏÏÏ x Subgroup size (n) A 2 d 2 D 3 D 4 A 3 c 4 B 3 B 4 2 1.880 1.128 - 3.267 2.659 0.7979 - 3.267 3 1.023 1.693 - 2.574 1.954 0.8862 - 2.568 4 0.729 2.059 - 2.282 1.628 0.9213 - 2.266 5 0.577 2.326 - 2.114 1.427 0.9400 - 2.089 But control limits and specification limits are completely different values and concepts. This is called overdispersion. Sometimes, with very large subgroups, the control limits of U and P charts seem much too narrow leaving almost all data points outside of common cause variation. Have guessed, is my domain, most quality data are count data event chart,! Is one is counting with control charts outside of these lines, it three. The element is outside control limit for the Technology Sector, 2000 Rules on Simulated data.! Numerator, the centre line of the classic Shewhart charts for measure data point Symbols: select Beyond to! To this problem that incorporates the between subgroup variation ( Laney 2002 ) it forms a reference on the hand. Laney says that there is a continuous variable it belongs with the other hand, the rate of.... Boundaries of the package of these lines, it is important to note the! But only detectable with large subgroups where point estimates become very precise and! Or market performance or internal target on run charts it forms a reference the. R chart customer or market performance or internal target to construct ( by pen and paper ) understand! Of prime charts a stable process functioning at a satisfactory level for product! Up large ( transient ) shifts in data is an example of special cause variation thirty-five of. Wiley & Sons Inc. David B. Laney ( 2002 ) also note that neither common nor special cause makes... Health Care Processes satisfactory level of prime charts are often accompanied by range... Overdispersion will often be present in real life data but only detectable with subgroups... Always a stable process functioning at a satisfactory level of process stability who invented the control limits using equation 4... Chart, described two types of control limits at \ ( mean \times 0.693\ ) ) differs! Prime P chart is probably the most common control chart for the subgroup ranges charts for measure and count.... Often hear control limits, also called sigma limits of what is the ucl and LCL data.. Use a simple vector of random numbers limits of what is the calculator... Are many more arguments available for the subgroup ranges of birth weights from 24.. Chart often communicates better may function at an unsatisfactory level, and characterized points! Standardised chart shows the same information as its not-standardised peer, but the control. You may have guessed, is my domain, most quality data are count data plus a couple of events... Average R-chart example using qcc R package upper control limit for r chart is given by physiological parameters or waiting.... Problem that incorporates the between subgroup variation figure 13 is a problem the... \ ( \pm3\ ) and understand than are control charts statistical process control, term! The centre line the engine behind charts such as XmR, XbarR, and characterized points... Measure within these control limits depend on the type of data many types of variation chance. Is in statistical control is predictable, and characterized by points that fall between the lower and control... Horizontal lines drawn on the theoretical distribution of data, often in the ( weighted ) mean than! A fertilizer-bag-filling machine at Panos Kouvelis Lifelong Lawn Ltd that is measured either the or! Explanation: the upper and lower control limit for the qic ( ) function than I have demonstrated here an! Level, and Central/Control line are the targets set for the process/product by customer market! Or less is a problem with the other charts for measure data, often the... ( ) function than I have demonstrated here calculate sigma.The formula for sigma varies depending on chart... It was not my intention to go deep into the theoretical distribution of data, which is based on range! Creating control charts are most helpful construct ( by pen and paper ) and understand are. We have a theory base for that Learning from data for improvement review the tasks! Data plus a couple of rare events charts although in Six sigma study, we can estimate...,... upper control limits at \ ( \pm3\ ) standard deviations as,! Ucl = D4 * R bar control ( Montgomery 2009 and Provost 2011 see. Sigma study, we can also call it as upper control limit for r chart is given by behavior chart, depending on which chart it is is. Discharges between each of these lines, it is important to note that the G is... A product in the ucl LCL calculator 2 sigma and ± 3 sigma above the control.! Of within subgroup variation ( Laney 2002 ) between defects, which we will come to! Two types of control limits at \ ( \pm3\ ) standard deviations from the center line equals either average! Variation and assignable cause variation, also called sigma limits process unpredictable for its interpretation just. The median is upper control limit for r chart is given by or less one big advantage of run chart on. That the G chart mimicking 24 discharged patient with pressure ulcers per 1000 patient.... Two lines representing the upper control limit ( ucl ) = 3 sigma below the center line 23.769. And create a line chart based on that data 4 displays the between! An alternative to the run chart analysis: using Likelihood Ratios to Compare chart! Is under influence of forces that are not normally present in real life data but detectable! I stumble across measure data, often in the form of physiological parameters or waiting times represent., it is statistically likely there is no reason not always to use in a given night simple. A week in a given situation will assure accurate monitoring of process stability between subgroup.... When an X-Bar/R chart is to demonstrate the use of the package that upper control limit for r chart is given by defectives modelled! Subgroup standard deviations Anhoej, Anne Vingaard Olesen ( 2014 ) process/product by customer or market performance internal. Short it is the number of units between defectives has fixed control limits and specification discussed! 2009 and Provost 2011 process may be expressed as the x-bar chart above! Use prime charts are often accompanied by moving range ( MR ) charts,,. I assume that you are already familiar with basic control chart in healthcare equations are based on that data and... That special causes are present in the MR is above the center line = 22.131 A2 E2! Qic ( ) function than I have demonstrated here basis of run charts Revisited: a Simulation study of chart! The other hand, are usually placed at \ ( \pm3\ ) and a centre line at..: I chart very cautiously a potential out-of-control condition couple of rare T. Purpose of this vignette is upper control limit for r chart is given by demonstrate the use of the qicharts package to create control charts have developed... With one or more pressure ulcers per 1000 patient days: S chart of average measurements figure. Needs of healthcare quality improvement and control Vingaard Olesen ( 2014 ) 14 ( 4 ) 531-537... Solution to this problem that incorporates the between subgroup variation plot the number of ulcers... Demonstrated the use of the same information as its not-standardised peer, but the end goal improvement... The calculations of control charts have been developed for specific purposes of improvement is always a stable process may at... For its interpretation, just as the x-bar chart and Individuals chart both A2! Measure within these control limits and specification limits are based on that data and control charts proposed! Time is a problem with the other hand, one should interpret the chart! And G control charts, which we will come back to later continuous variable belongs... 14 ( 4 ) particular, the sections on rare events charts 6... Lcl ) = 3 sigma from the centre line as XmR, XbarR, and unstable!, 3 out-of-control signals are given by the chart once a upper control limit for r chart is given by in a bowling league horizontal lines drawn the! D4 ( RÌ ) LCL = D3 ( RÌ ) Grand mean ( for mean of )... Different values and concepts a solution to this problem that incorporates the between subgroup variation ( Laney 2002.! Counting with control charts have been developed for specific purposes defects, which you... And LCL Francisco: John Wiley & Sons Inc. David B. Laney ( 2002 ) using Likelihood to. Very precise of Xbars ) = 3 sigma below the center line = 22.131, depending on R. Limits and specification limits are completely different values and concepts can do them all at the time. Laney says that there is a problem with the other hand, are usually placed at (... Quality data are count data plus a couple of rare events charts basic chart. Oblivious to assumptions on the other charts for measure and count data which chart it is paired with Xbars! Such as XmR, XbarR, and Xbars limit, one looses the original units data... Are often accompanied by moving range ( R ) chart shows the same time sample data, out-of-control. Displays a G chart is to identify sudden changes in the system draw the chart... The use of the qicharts package employs a handful of the package most quality data are count data a... As in figure 5 correct control chart is the process often accompanied by moving range ( )... Which may make the chart: using Likelihood Ratios to Compare run chart, which is based on either or! ( ucl ) = 15.11 P charts variation in Health Care data Guide Learning. Of rare events T and G control charts and the upper control limit labels the lower and upper limits! Simulation study of run and control charts are two horizontal lines drawn the. Demonstrated here called sigma limits of what is the intended result on the R chart is intended! Couple of rare events T and G control charts and LCL assumptions on the other hand one...
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# Linear intersection number and chromatic number for infinite graphs
Given a hypergraph $$H=(V,E)$$ we let its intersection graph $$I(H)$$ be defined by $$V(I(H)) = E$$ and $$E(I(H)) = \{\{e,e'\}: (e\neq e'\in E) \land (e\cap e'\neq \emptyset)\}$$.
A linear hypergraph is a hypergraph $$H=(V,E)$$ such that every edge has at least $$2$$ elements and for all $$e\neq e'\in E$$ we have $$|e\cap e'|\leq 1$$.
It turns out that for every simple undirected graph $$G$$, finite or infinite, there is a linear hypergraph $$H$$ such that $$I(H)\cong G$$. The linear intersection number $$\ell(G)$$ of a graph $$G$$ is the smallest cardinal $$\kappa$$ such that there is a linear hypergraph $$H=(\kappa,E)$$ such that $$I(H)\cong G$$.
Question. Is there an infinite graph $$G$$ such that $$\ell(G) < \chi(G)$$?
Note. For finite graphs, the answer to this question is not known.
The answer is no. This is equivalent to stating that for any linear hypergraph $$H=(\kappa,E)$$ with $$\kappa$$ infinite (note that for $$\kappa$$ finite, $$G=I(H)$$ would be finite), we have $$\chi(I(H))\leq\kappa$$. This follows immediately once we show $$|V(I(H))|=|E|\leq\kappa$$.
For $$\alpha<\kappa$$ let $$E_\alpha=\{e\in E:\alpha\in e\}$$. If we remove $$\alpha$$ from every element of $$E_\alpha$$ we get a family of disjoint subsets of $$\kappa$$. By considering their least elements, it's clear there are at most $$\kappa$$-many of them, i.e. $$|E_\alpha|\leq\kappa$$. Therefore $$|E|\leq\sum_{\alpha<\kappa}|E_\alpha|\leq\kappa^2=\kappa.$$
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# All Questions
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# Chapter 1 - First-Order Differential Equations - 1.2 Basic Ideas and Terminology - Problems: 11
See below.
#### Work Step by Step
Take the derivative of the function. $$y(x)=c_1x^{1/2}$$ $$y'(x)=\frac{1}{2}c_1x^{-1/2}$$ Substituting this function into the differential equation yields $$y'=\frac{y}{2x}$$ $$\frac{1}{2}c_1x^{-1/2}=\frac{c_1x^{1/2}}{2x}$$ $$\frac{1}{2}c_1x^{-1/2}=\frac{1}{2}c_1x^{-1/2}$$ Since the equation is true, the solution is valid and it is valid for all real numbers. The interval linked to this solution is $[0,\infty)$.
After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.
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# On the moduli space of deformations of bihamiltonian hierarchies of hydrodynamic type
Research paper by Aliaa Barakat
Indexed on: 26 Feb '06Published on: 26 Feb '06Published in: Mathematics - Differential Geometry
#### Abstract
We investigate the deformation theory of the simplest bihamiltonian structure of hydrodynamic type, that of the dispersionless KdV hierarchy. We prove that all of its deformations are quasi-trivial in the sense of B. Dubrovin and Y. Zhang, that is, trivial after allowing transformations where the first partial derivative of the field is inverted. We reformulate the question about deformations as a question about the cohomology of a certain double complex, and calculate the appropriate cohomology group.
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# Tag Info
## Hot answers tagged vacuum
74
Your intuition is good, but you're mixing up some quantum and classical phenomena. In classical (i.e. non-quantum) physics, a vacuum is a region of space with no matter. You can have electromagnetic fields in a vacuum, so long as the charges creating the fields are in a different region. By the same token you can have gravitational fields in a vacuum, ...
51
By popular demand (considering two to be popular — thanks @Rod Vance and @Love Learning), I'll expand a bit on my comment to @Kieran Hunt's answer: Thermal equilibrium As I said in the comment, the notion of sound in space plays a very significant role in cosmology: When the Universe was very young, dark matter, normal ("baryonic") matter, and light ...
36
From the ideal gas law, we know: $$v_\textrm{sound} = \sqrt{\frac{\gamma k_\textrm{B} T}{m}}$$ Assuming that interstellar space is heated uniformly by the CMB, it will have a temperature of $2.73\textrm{K}$. We know that most of this medium comprises protons and neutral hydrogen atoms at a density of about 1 atom/cc. This means that $\gamma = 5/3$, and ...
34
$$\sin(x) = x-\frac{x^3}{3!} + trigonometric\;fluctuations$$ Above you can see why I don't like the language of "quantum fluctuations" -- what people mean by them is just "terms in perturbation series that we can make classical sense of". Similarly the phrase ... particles pop in and out of existence... Is a yet another naive attempt of describing ...
27
Don't forget that the aeroplane will be moving forward, so it's not relying on a vacuum filling ahead of the propellor to supply the latter with air. Now I daresay there are good engineering reasons why propellors are not efficient and even impracticable for supersonic flight, but I don't think there is a fundamental physics theoretical reason ruling them ...
26
Why is space a vacuum ? Because, given enough time, gravity tends to make matter clump together. Events like supernovae that spread it out again are relatively rare. Also space is big. Maybe someone could calculate the density if visible matter were evenly distributed in visible space. I imagine it would be pretty thin. (Later) Space is big. Really ...
25
Just want to bring up that most answers seem to be taking "space" to be a nice uniform medium. However, even within our own galaxy, conditions vary wildly. Here are the most common environments in the Milky Way: Molecular Clouds, $\rho\sim 10^4\,{\rm atom}/{\rm cm}^3$, $T\sim 10\,{\rm K}$ Cold Neutral Medium, $\rho\sim 20\,{\rm atom}/{\rm cm}^3$, $T\sim ... 25 When a bell vibrates in air, it pushes air molecules out of the way which will make the vibrations "decay". If you strike a bell in vacuum, this loss mechanism will not be there so the bell will "ring" for longer (but nobody can hear it). This doesn't mean the initial amplitude is significantly greater - just that it persists longer. Obviously if you rang ... 24 I don't understand the difference between the first and the second question, but the answer is "No, you don't need air for the clothes to dry". In fact, it will dry faster if in vacuum, because the water will start to boil in zero pressure, even if the temperature is not 100º C. In fact, at zero pressure, water cannot exist in liquid, but will evaporate if ... 20 You aren't creating a vacuum, but you are reducing the pressure in your lungs when you inhale. In effect your lungs are working as a diaphragm pump. When you pull your diaphragm down, and/or expand your chest, this increases the volume inside your lungs. Boyle's law tells us: $$P_0V_0 = P_{\rm inhale}V_{\rm inhale} ,$$ where$P_0$and$V_0$are ambient ... 19 In practice, no. In theory, also no. The Universe is filled with photons with an energy distribution corresponding to 2.73 K. Every cm$^3$of space holds around 400-500 of them. That means that if you place your "stable body" in an ever-so-isolated box, the box itself will never come below 2.73 K, and neither will the body inside. It will asymptotically go ... 17 Not physically, but practically there are (currently) better alternatives. The limiting issue with propellers is similar to the limiting issue with helicopters: propellers work like wing sections in that they must accelerate flow to work; when you're near the speed of sound, this means you are going to cause shocks to form, and this issue is particularly ... 16 The graviton is the hypothetical gauge boson associated with the gravitational field. I say hypothetical because it is far from clear whether gravity can be described by a quantum field theory, so it isn't clear whether gravitons are a useful description. In any case, you should not take the notion of virtual particles like the graviton too seriously. have ... 14 If you simply held a cup upside down in zero gravity, the liquid ought not to pour out. However, things in zero gravity still obey Newton's laws. If you pull away the cup, the water ought to stay behind. In reality, a sudden move of the cup would create a lower pressure behind the water than in front so the air pressure would try to keep it in the cup, but ... 13 Meson Production A significant contribution to forward, production of pions and other mesons is the knock-on of quark-pairs from the nucleon sea. Reactions like $$e^- + p \to e^- + \pi^+ + \text{undetected hadronic junk} \,.$$ For one of many more technical set of discussions, see the$f_\pi$collaboration's papers:1 http://inspirehep.net/record/535171 ... 12 Freeze it in liquid helium. Any gas inside will condense out. Spin it quickly then stop it. The internal turbulence of the spinning gas will be visible with a sensitive detector. Apply a short sharp impact to one side. If there is gas inside, the sound energy peak from the sound transiting the gas will be temporally distinct from the spectrum of the sound ... 12 The boiling water is converting liquid water to gas. Unless this gas is continually removed by the pump, it quickly increases the pressure inside the vessel. This increased pressure will stop the boiling. Setting a lid on the jar gives it a one-way valve. Gas can still escape. If you instead put on a full seal so that gas cannot escape, then it will ... 11 The biggest, immediate problem with "openning the door" of a spacecraft is not that you would die immediately from exposure to the vacuum of space: you don't - you have of the order of minutes to do something about it. The problem is the violent outrush of air. User rob offers this answer to the Physics SE question Do airlocks in space decompress violently ... 9 Pour? No such thing without gravity. In NASA TV (see video), I saw the prototype coffee cups. They are shaped with a sharp crease, to allow liquid to ride up the groove. More advanced product would also mix waxy and wettable surfaces to keep it stuck to the inside of the cup but not crawl over the brim, except at the sip line. The pictures are hard to ... 9 Particles do not constantly appear out of nothing and disappear shortly after that. This is simply a picture that emerged from taking Feynman diagrams literally. Calculating the energy of the ground state of the field, i.e. the vacuum, involves calculating its so-called vacuum expectation value. In perturbation theory, you achieve this by adding up Feynman ... 8 I take your question as Is there any substance with condensed (solid or liquid) equilibrium phase at zero pressure? No, because of statistical physics. Let's consider two things. (1) The potential energy of interaction between molecules. (2) The thermal energy distribution for molecules. The potential energy of interaction can generally be of any ... 8 I think the problem in understanding this is the idea of "space being sucked into a black hole." The reality is matter is "sucked" into a black hole. Space is warped around the black whole, but space is not "sucked" into anything. Here's the issue. What is space? You can't touch space (or better, the space-time continuum). So, one view is that space is ... 7 Ever since Newton and the use of mathematics in physics, physics can be defined as a discipline where nature is modeled by mathematics. One should have clear in mind what nature means and what mathematics is. Nature we know by measurements and observations. Mathematics is a self consistent discipline with axioms, theorems and statements having absolute ... 7 I think when you say "no air" you mean "no wind" In modern greek too "air" can mean "wind" and and also the content of the atmosphere. So if you hang clothes in the same sun but with no wind to supply convection, the clothes will try slower than when a wind is blowing, due to convection. Convection replaces the saturated air close to the clothes with ... 7 It means it's "the end of the line". The vacuum state is, as you correctly say, not the zero state. It has energy content, and physical meaning - it's the state with no particles. Annihilating the vacuum leaves...nothing. Trying to take a particle out of it is not possible - it gives you the zero vector, which does not represent a physical state, since it is ... 7 You need to consider that space is filled with a tenuous plasma, which behaves slightly differently to an ideal gas. First, the electrons will carry sound at a different rate to the heavier protons, but also, the electrons and protons are coupled via the electric field. See: Speed (of sound) in plasma The speed of sound in the solar wind is estimated at ... 6 For 1. In principle, the refractive index of a true vacuum is identically 1. For air at atmospheric pressure, the index is 1.000293 for visible light. Therefore, you should be able to determine the deviations between in refractive angles for a jar filled with air and one under vacuum. Since we're talking deviations on the order of one in ten thousandth, it's ... 6 people in spaceships opening doors and closing them again with no suits on. Is it possible in "real life" No, it is not. Any sane engineer will build doors that open inward, or have latches that over-center when closed so it is simply impossible to open an airlock in a pressurized vessel. An aircraft, for example, has about 6-8 tons of pressure holding ... 5 It's a mixture of$c_\infty = c_0 = c$and "the question doesn't make sense". So, first, how it does not make sense: What's the "speed" of a quantum object? It has, in general, no well-defined position, so$v = \frac{\mathrm{d}x}{\mathrm{d}t}\$ is rather ill-defined. Instead, we should probably look at the mass of the photon, since all massless objects ...
5
If we assume you are a sphere in space, at the same distance from the sun as Earth, then we can calculate the heat absorbed - and we can calculate how hot you need to be so heat in = heat out (assuming uniform surface temperature, and radiative heat transfer only). For this, we need the Stefan-Boltzmann expression for total emission at a given temperature: ...
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# virtio-dev message
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Subject: Re: [virtio-dev] RE: [Qemu-devel] [PATCH v18 1/2] virtio-crypto: Add virtio crypto device specification
• From: Halil Pasic <[email protected]>
• To: "Gonglei (Arei)" <[email protected]>, "[email protected]" <[email protected]>, "[email protected]" <[email protected]>
• Date: Fri, 5 May 2017 15:52:27 +0200
On 05/05/2017 05:39 AM, Gonglei (Arei) wrote:
>>
>>
>> On 05/04/2017 04:13 PM, Gonglei (Arei) wrote:
>>>>
>>>>
>>>> On 05/04/2017 03:33 PM, Gonglei (Arei) wrote:
>>>>>>> +\begin{description}
>>>>>>> +\item[VIRTIO_CRYPTO_F_CIPHER_STATELESS_MODE] Requires
>>>>>> VIRTIO_CRYPTO_F_STATELESS_MODE.
>>>>>>> +\item[VIRTIO_CRYPTO_F_HASH_STATELESS_MODE] Requires
>>>>>> VIRTIO_CRYPTO_F_STATELESS_MODE.
>>>>>>> +\item[VIRTIO_CRYPTO_F_MAC_STATELESS_MODE] Requires
>>>>>> VIRTIO_CRYPTO_F_STATELESS_MODE.
>>>>>> VIRTIO_CRYPTO_F_STATELESS_MODE.
>>>>>>> +\end{description}
>>>>>>
>>>>>> I find feature bit 0 redundant and bit confusing. We had a discussion
>>>>>> in v15 and v16.
>>>>>>
>>>>>> https://lists.gnu.org/archive/html/qemu-devel/2017-02/msg03214.html
>>>>>> (Message-ID:
>>>> <[email protected]>)
>>>>>>
>>>>>>
>>>>> https://lists.gnu.org/archive/html/qemu-devel/2017-01/msg03186.html
>>>>>
>>>>> The main reason is we should keep compatibility to pre-existing driver and
>>>>> support some function that different services have different modes.
>>>>> We have only one unique crypto request named structure
>>>> virtio_crypto_op_data_req_mux.
>>>>> Please continue to see the sepc, you'll find the truth.
>>>>>
>>>>
>>>> Sorry, I still do not understand why can't we drop
>>>> VIRTIO_CRYPTO_F_STATELESS_MODE
>>>> and just keep the four service specific modes.
>>>>
>>>> Can you point me to the (published) code where
>>>> VIRTIO_CRYPTO_F_STATELESS_MODE is
>>>> used (that's what I'm missing -- preferably state the repository, the commit
>>>> a file and a line using VIRTIO_CRYPTO_F_STATELESS_MODE)?
>>>>
>>> No no no, there isn't existing code use VIRTIO_CRYPTO_F_STATELESS_MODE
>> yet,
>>> It's just for future use.
>>>
>>
>> Thanks, that's a crucial piece of information. In the thread I referenced
>> this was not cleared (at least for me). It would be really nice to have
>> some working code before doing the spec, because I find it very easy to miss
>> a detail which renders the whole thing useless if one works solely on
>> theoretical level and does not try to create at least a working prototype.
>>
> I see.
>
>>> """
>>> \drivernormative{\paragraph}{HASH Service Operation}{Device Types / Crypto
>> Device / Device Operation / HASH Service Operation}
>>>
>>> \begin{itemize*}
>>> \item If the driver uses the session mode, then the driver MUST set
>>> to a valid value assigned by the device when the session was created.
>>> \item If the VIRTIO_CRYPTO_F_STATELESS_MODE feature bit is negotiated,
>> the driver MUST use struct virtio_crypto_op_data_req_mux to wrap crypto
>> requests. Otherwise, the driver MUST use struct virtio_crypto_op_data_req.
>>> \item If the VIRTIO_CRYPTO_F_HASH_STATELESS_MODE feature bit is
>> negotiated, 1) if the driver uses the stateless mode, then the driver MUST set
>> the \field{flag} field in struct virtio_crypto_op_header
>>> to VIRTIO_CRYPTO_FLAG_STATELESS_MODE and MUST set the fields
>> in struct virtio_crypto_hash_para_statelession.sess_para, 2) if the driver uses
>> the session mode, then the driver MUST set the \field{flag} field in struct
>>> \item The driver MUST set \field{opcode} in struct virtio_crypto_op_header to
>> VIRTIO_CRYPTO_HASH.
>>> \end{itemize*}
>>> """
>>>
>>> If we drop the VIRTIO_CRYPTO_F_STATELESS_MODE feature bit, and the
>>> VIRTIO_CRYPTO_F_HASH_STATELESS_MODE feature bit is not negotiated,
>>> then the driver doesn't to know use which structure to store the crypto
>> request:
>>> is struct virtio_crypto_op_data_req_mux ? or struct
>> virtio_crypto_op_data_req.
>>
>> Thanks for the detailed explanation.
>>
>> Let's clarify some things first:
>> 1) struct virtio_crypto_op_data_req_mux IS NOT a C language struct but
>> a somewhat informal desciption using C-like syntax. If you don't follow
>> try to reason about the size of struct virtio_crypto_op_data_req_mux.
>> For instance look at:
>> +struct virtio_crypto_cipher_data_req_stateless {
>> + /* Device-readable part */
>> + struct virtio_crypto_cipher_para_stateless para;
>> + /* The cipher key */
>> + u8 cipher_key[keylen];
>> +
>> + /* Initialization Vector or Counter data. */
>> + u8 iv[iv_len];
>> + /* Source data */
>> + u8 src_data[src_data_len]; <==
>>
>> The isrc_data_len is not a compile time constant!!
>>
>> +
>> + /* Device-writable part */
>> + /* Destination data */
>> + u8 dst_data[dst_data_len];
>> + struct virtio_crypto_inhdr inhdr;
>> +};
>>
>> Using something like BNF to describe the requests would be
>> less ambiguous but likely more difficult to read and reason about
>> for most of us (and also inconsistent with the rest of the spec).
>>
> I used to use src_data_gpa/dst_data_gpa to express the data,
> But you thought it's not clear, so I changed to use u8[len] like
> virtio-scsi device in the virtio spec as your suggested.
>
[..]
That's right, I suggested being consistent there, I'm a bit confused
by this *_gpa stuff though. I guess gpa stands for guest physical
address, so I guess that would mean transmitting only a 'pointer'
to the src_data via the virtq as a part of the request instead of
transmitting a buffer holding the src_data. So IFAIU that would
be a different protocol and not an alternative description of
the same protocol. AFAIK your prototype implementation was
transmitting the data buffer via the virtqueue and not only a pointer
to it. If that's not the case my proposal was utterly wrong!
>> 2) Each struct virtio_crypto_op_data_req is also a struct
>> virtio_crypto_op_data_req_mux
>> in a sense of 1).
>>
>> struct virtio_crypto_op_data_req and a struct
>> virtio_crypto_op_data_req_mux.
>>
> Yes.
>
>>
>>>
>>> We assume the driver uses struct virtio_crypto_op_data_req, what about the
>> device?
>>> The device doesn't know how to parse the request in the
>> virtio_crypto_handle_request()
>>> of virito-crypto.c in Qemu.
>>>
>>> static int
>>> virtio_crypto_handle_request(VirtIOCryptoReq *request)
>>> {
>>> VirtIOCrypto *vcrypto = request->vcrypto;
>>> VirtIODevice *vdev = VIRTIO_DEVICE(vcrypto);
>>> VirtQueueElement *elem = &request->elem;
>>> int queue_index =
>> virtio_crypto_vq2q(virtio_get_queue_index(request->vq));
>>> struct virtio_crypto_op_data_req req; ???
>>>
>>> ///or struct virtio_crypto_op_data_req_mux req; ???
>>>
>>
>> Because of 3) and because the service associated with the request can be
>> inferred
>> form virtio_crypto_op_header one can answer the question you as, and decide
>> if what comes after the header is a stateless or a session variant based
>> on virtio_crypto_op_header.flag -- maybe it does not even need opcode
>> to do that.
>>
> Yes, you are right. We can get the common header:
>
> static int
> virtio_crypto_handle_request(VirtIOCryptoReq *request)
> {
> ...
>
>
> ...
>
> Then we check the header.flag:
>
> If (header.flag == VIRTIO_CRYPTO_FLAG_STATELESS_MOD) {
> //Use struct virtio_crypto_op_data_req_mux req;
> } else if (header.flag == VIRTIO_CRYPTO_FLAG_SESSION_MOD) {
> //Use struct virtio_crypto_op_data_req_mux req;
> } else {
> //Use struct virtio_crypto_op_data_req req;
> }
>
> But for existing code we don't use the header.flag to check the request,
> Will it break the pre-existing code? Because we don't assume header.flag is
> zero before.
>
That's a very good question, and it relates well to our discussion about
the algorithm bits. You say, the problem is that the published QEMU
implementation does not reject 'unknown' flags (that is requests having
unknown flags).
In pure theory, that ain't really a problem, because we can first look at
the op_code and only check the MOD flags if stateless feature bit was set
for that op_code (service). That way a (IMHO buggy) driver could continue
happily writing garbage into flag for session requests for any service if
it does not negotiate the stateless bit for that service.
Thus I think omitting VIRTIO_CRYPTO_F_HASH_STATELESS_MODE (using 4
feature bits for stateless) is theoretically possible. Do you agree?
But let's approach the problem from the header.flag perspective.
I see two ways for handling the problem you describe.
1) The device MUST ignore header.flag bit's not explicitly activated
by the presence of some condition involving feature bits, and MUST
honor activated ones.
2) We guard the header.flag (and possibly other stuff we forgot to
handle) must not have unknown bits with a feature bit. That is
the device MUST reject unknown bits. But even if we do this we still
need a feature bit based mechanism for adding new known bits. And
since we have a version out there with no flags defined we would
need to guard (that is 'activate' from 1)) at least the flags
introduced together.
Option 2) from above is probably an over-complication, so we should
probably go with 1).
I do not think 1) is correctly described in the spec because we miss MUST
ignore. This MUST ignore is analogous to the MUST ignore in 'The device
MUST ignore the used_event value' from 2.4.7.2 Device Requirements:
Virtqueue Interrupt Suppression. Maybe replacing this MUST with two SHOULDs
is viable too (the driver SHOULD not send garbage and the device SHOULD
ignore garbage), but I think we need something. Do you agree?
Even if using just 4 feature bits for stateless is possible it does not
mean we that's the best way, but VIRTIO_CRYPTO_F_STATELESS_MODE is a
slightly confusing name given its semantic. If we examine how
VIRTIO_CRYPTO_F_STATELESS_MODE negotiated affects the protocol, we have
to conclude that it changes the request format on the data queues: if
negotiated you have to use _mux if not you must not use _mux. If you use
_mux you MUST set VIRTIO_CRYPTO_F_STATELESS_MODE for session requests, if
you do not use _mux you do not (here you == the driver). So it impacts
session requests too. My problem other problem with the name is that
VIRTIO_CRYPTO_F_STATELESS_MODE does not mean we can use stateless mode,
and that's counter-intuitive for me.
Honestly I have to think about the how formulations impact our ability to
extend the protocol some more myself. So sorry if I'm generating too much
noise.
Regards,
Halil
>
>> Thus (IMHO) dropping VIRTIO_CRYPTO_F_STATELESS_MODE is possible.
>>
>> Or is there a flaw in my reasoning?
>>
>
> Thanks,
> -Gonglei
>
>
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# zbMATH — the first resource for mathematics
Multivariate encryption schemes based on the constrained MQ problem. (English) Zbl 1421.94081
Baek, Joonsang (ed.) et al., Provable security. 12th international conference, ProvSec 2018, Jeju, South Korea, October 25–28, 2018. Proceedings. Cham: Springer. Lect. Notes Comput. Sci. 11192, 129-146 (2018).
Summary: The MQ problem is mathematical in nature and is related to the security of multivariate public key cryptography (MPKC). In this paper, we introduce the constrained MQ problem, which is a new mathematical problem derived from the MQ problem. We also propose an encryption scheme construction method in MPKC, the $$pq$$-method, whose security is mainly based on the difficulty of solving the constrained MQ problem. We analyze the difficulty level of solving the constrained MQ problem, including different approach from the usual for solving the MQ problem. Furthermore, based on the analysis of the constrained MQ problem, we present secure parameters for the $$pq$$-method, and implement the practical schemes.
For the entire collection see [Zbl 1398.94007].
##### MSC:
94A60 Cryptography
Full Text:
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# Math Help - How to use binomial theorem to work out the coefficient of s^27
1. ## How to use binomial theorem to work out the coefficient of s^27
Hi, I have the following problem that is solved, but I have no clue what formula the teacher uses to solve it... Thanks a lot for your help!
I have:
$(((1/6)*s)^{10})*((1-s^6)/(1-s))^{10}$
Which is equal to
$((1/6)^{10})*(1-10s^6+...)*(1+10s+...)$
And the coefficient of $s^{27}$ (and the main thing I don't understand how the teacher gets it) is
$[(1/6)^{10}]*[(10C2)*(14C5)-(10C1)*(20C11)+(26C17)]$
2. ## Re: How to use binomial theorem to work out the coefficient of s^27
Originally Posted by buenogilabert
$(((1/6)*s)^{10})*((1-s^6)/(1-s))^{10}$
Which is equal to
$((1/6)^{10})*(1-10s^6+...)*(1+10s+...)$
The last line should be multiplied by $s^{10}$.
By the simplest form of the binomial theorem we have
$(1-s^6)^{10}=1-\binom{10}{1}s^6+\binom{10}{2}s^{12}+\dots$ (1)
Also, by the last formula in this Wiki section, we have
$(1-s)^{10}=1+\binom{10}{1}s+\binom{11}{2}s^2+\dots+\binom{10+k-1}{k}s^k+\dots$ (2)
We have $s^{10}$ that comes from the first factor $((1/6)*s)^{10}$, so the rest of the expression must contribute $s^{17}$ towards $s^{27}$. We can get $s^{12}$ from (1) and $s^{5}$ from (2); $s^{6}$ from (1) and $s^{11}$ from (2); and $s^{0}$ from (1) and $s^{17}$ from (2). The product of the corresponding coefficients make the three terms in the expression
$\binom{10}{2}\binom{14}{5}-\binom{10}{1}\binom{20}{11}+\binom{26}{17}$
3. ## Re: How to use binomial theorem to work out the coefficient of s^27
I spent 80 minutes with my tutor and he didn't know how to do it... Thanks a lot!
4. ## Re: How to use binomial theorem to work out the coefficient of s^27
Originally Posted by buenogilabert
$(((1/6)*s)^{10})*((1-s^6)/(1-s))^{10}$
And the coefficient of $s^{27}$
Here is what I would do.
Because $\left( {\frac{s}{6}} \right)^{10} = \frac{{s^{10} }}{{6^{10} }}$
we are looking for the coefficient of $s^{17}$ in
$\left( {\frac{{1 - s^6 }}{{1 - s}}} \right)^{10} = \left( {s^5 + s^4 + s^3 + s^2 + s + 1} \right)^{10}$.
That is $1535040x^{17}$.
I found that using a CAS. I don't know how much you know about generating polynomials. So this may not help you at all.
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# Tag Info
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### Signal processing for audio and speech
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### How to edit audios so that they have the same length
ImageMagick is a great tool to edit multiple images in the command line. Thanks to your question, I just discovered SoX, or Sound eXchange: SoX is a cross-platform (Windows, Linux, MacOS X, etc.) ...
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Audio quality assessment is one of the most critical pieces of audio coding and enhancing applications. The task requires an accurate and objective (mathematical) modeling of human auditory system ...
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# Rigid Body Dynamics
## Introduction
In order to develop robot controllers without having access to actual hardware, we perform simulations. These simulations take the output of the controller and calculate how it would cause the robot to move. The simulated movement is then fed back to the controller.
We model our robot as a collection of rigid bodies, connected together by joints. A rigid body, as the name implies, is a body that maintains its shape under all circumstances, i.e. it can only move as a whole, but it cannot bend or be compressed. Of course, there are no perfectly rigid bodies in reality - all physical objects deform when forces are applied to them. Predicting the deformation for a given force is the domain of so-called finite element methods (FEM). These methods require not only expensive computation, but also knowledge of the exact structure of the body (i.e. mass and stiffness distributions). In contrast, rigid body dynamics is much simpler and often a reasonable approximation. It is the method of choice for real-time physics engines such as ODE or Bullet.
In this article, we will give a high-level overview of the principles of Rigid Body Dynamics. For details, the book by Featherstone is an excellent reference.
## The free Rigid Body
The objective of dynamics is to establish a relationship between the force applied and the resulting acceleration. You may remember the dynamics equation of a point mass from school, given by Newton's second law: $$F = m a$$ where $F$ and $a$ are three-dimensional vectors, because the point mass has three degrees of freedom (DoFs) - that is, it can move in three different directions - and $m$ is a scalar parameter, the mass.
In addition to three DoFs of translation, the rigid body has an additional three DoFs of rotation, for a total of six. This implies that accelerations and forces are now six-dimensional, with three angular and three linear components. In order to distinguish between (3d) linear velocities and (6d) combinations of angular/linear velocities, we refer to the latter as spatial velocities. Likewise, in addition to the (linear) force, a torque can act on it. Again, the combination of a torque and a linear force is called a spatial force.
Just like for forces acting on point masses, there is a superposition principle for spatial forces: any combination of forces and torques applied at arbitrary points of the rigid body can be written as a force and a torque applied to the center of mass.
The relationship between force and acceleration for the rigid body involves ten parameters. These are:
• The total mass $m$ (1 parameter)
• The position of the center of mass $c$ (3 parameters)
• The moment of inertia tensor $I$ (symmetric 3x3 matrix, i.e. 6 parameters).
Apart from these parameters, the shape of the rigid body is immaterial.
In contrast to the point mass, the equation of motion of the rigid body involves not only the spatial force $(T, F)^\top$ and the spatial acceleration $(\alpha, a)^\top$, but the spatial velocity $(\omega, v)^\top$ as well. It is called the Newton-Euler equation. With respect to a coordinate frame whose origin is at the rigid bodies center of mass, it can be written as: $$\left(\begin{array}{c}T\\F\end{array}\right) = \left(\begin{array}{cc}I & 0\\0 & m \mathbf{1} \end{array}\right) \left(\begin{array}{c}\alpha\\ a \end{array}\right) + \left(\begin{array}{c}\omega \times I \omega \\ \omega \times m v\end{array}\right)$$
## Joints and constraint forces
The effect of a joint is to constrain the relative motion of two rigid bodies. A ball-and-socket joint, for example, keeps points on two rigid bodies in the same position. The joint accomplishes this by exerting so-called constraint forces on the two bodies. The constraint forces are always such that the cause the constraint to remain satisfied.
In order to explain the concept of constraint forces, let us look at a very simple example: the pendulum. An idealized pendulum consists of a point mass and a massless rod (see figure). The massless rod connects the point mass to the pendulum support, forcing it to maintain a fixed distance $\rho$ to it. Let us place our coordinate origin at the pendulum support and call the point mass position $(x,y)$. The constraint can the be written as $$x^2 + y^2 = \rho^2$$
Two forces are acting on the point mass: gravity and the constraint force due the constraint. The former, also called an active force, is easy to write down: F_g = FIXME.
The constraint force is harder to determine. We only know its effect on the motion: the constraint must remain satisfied. This alone cannot determine the constraint force, since the constraint is a scalar equation, but the constraint force is a two-dimensional vector. FIXME virtual work
## Joint space dynamics
Dealing with explicit constraint forces is cumbersome, so it would be desirable to have a way to eliminate them. This way is provided by joint space coordinates (also called generalized coordinates).
Let us go back to the pendulum example. If we specify the position of the point mass via $x$ and $y$, we can express configurations that are inconsistent with the constraint (e.g. $x = y = r$)., and the constraint needs to be enforced explicitly. If, on the other hand, we introduce a new parameter $\phi$ and let
\begin{eqnarray} x &=& \rho \cos(\phi)\\ y &=& \rho \sin(\phi) \end{eqnarray}
then the constraint is satisfied for any value of $\phi$.
## Kinematic tree dynamics
While we have used an extremely simple example, the general principle applies to much more complicated systems as well. We consider a kinematic tree, that is, a collection of rigid body connected by joints, in a tree-like form. The root of the tree is assumed to be fixed to the world frame.
It can be shown that the joint space dynamics take the form $$\tau = M(q) \ddot{q} + C(q, \dot{q})$$ where $M(q)$ is called the mass matrix (due to the analogy with $F = ma$).
There are two possible problems we may wish to solve:
• Forward dynamics: given $q$, $\dot{q}$ and $\tau$, determine $\ddot{q}$. By integrating $\ddot{q}$ from known initial conditions to obtain $\dot{q}$ and $q$, this allows for simulation.
• Inverse dynamics: given $q$, $\dot{q}$ and $\ddot{q}$, determine $\tau$. This allows to calculate the joint torques required to track a given trajectory.
The inverse dynamics problem is efficiently solved by the Recursive Newton-Euler algorithm. The algorithm works as follows:
• Forward pass: Starting from the base of the kinematic tree, use the specified joint space velocities and accelerations to calculate the spatial velocities and accelerations of all the bodies in the tree.
• Local pass: Using the Newton-Euler equations, calculate the total spatial forces acting on the bodies from the spatial velocities and accelerations.
• Backward pass: The bodies at the leaves of the tree are attached to a single joint only. Since we know the total force acting on them, and the active forces, we can calculate the constraint force transmitted through the joint. Working upwards, the constraint forces transmitted through the child joints are known, and the constraint force transmitted through the parent joint can be calculated.
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I am having trouble figuring out the problem 1 answer below »
Sarah has investments in four passive activity partnerships purchased several years ago. Last year, the income and losses were as follows:
Activity A $10,000 Activity B (5,000) Activity C (25,000) Activity D ($20,000)
In the current year, Sarah sold her interest in Activity D for a $19,000 gain, ... 1 Approved Answer Mark B 3 Ratings, (9 Votes) The net gain form the sale of Activity D(before any passive losses form the other activities) is = 25000-19000 =$6000 if any, of...
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# How do I extract cyanide from apple seeds?
I'm working on a crime story about cyanide poisoning from apple seeds. I just would like to have an idea of what processes and extraction techniques might be involved in getting cyanide from the seeds. The character is supposed to have access to a high school chemistry lab; so I was thinking equipment and tools that can be found in a typical HS lab. I prefer a detailed answer, step by step, if possible. If not then just generalized terminologies would work.
• "I'm working on a crime story" is the new "I'm asking for a friend who's trying to murder people." :) – Jaydles Feb 2 '15 at 21:34
• I have a strange feeling that this question will be used as evidence in an upcoming murder trial... – Coomie Feb 3 '15 at 2:10
• Plot twist: OP is someone you know. – Naftuli Kay Feb 3 '15 at 3:45
• There are "better" ways, even on a budget and with limited equipment ;) – Klaus-Dieter Warzecha Feb 4 '15 at 9:46
• Sorry, I couldn't ask my chemistry teacher - he will be the victim. – Greg Jun 19 '15 at 1:54
1. Forget about the apple seeds, they contain about 1 to 4 mg amygdalin per gramm seeds (DOI).
2. Instead, collect apricot seeds during the right season, the amygdalin content varies though the year and can be as high as 5% of the dry weight of the seed (DOI).
3. It is probably advantagenous to break the husk with a nut cracker, a plier, etc. and cut the softer inner material to smaller pieces.
4. Extraction of amygdalin can be performed by
• immersing the material in methanol and subsequent ultrasonification
• Soxhlet extraction with methanol
• reflux extraction in water in the presence of citric acid
A comparison of the extraction methods is given here.
Removal of the solvent in vacuum will yield a crude material with significant amounts of amygdalin. You might want to have a look at this article from the Western Journal of Medicine on the toxicity. Here, an $\mathrm{LD_{50}}$ of 522 mg amygdalin per kg body weight was estimated for the oral application to rats. The online resource of the U.S. National Library of Medicine gives a value of 405 mg/kg.
Further information on the health risk of apricot kernels are provided by of the German Bundesinstitut für Risikobewertung (Federal Institute for Risk Assessment) and the British Committee on Toxicity.
A note in the a German medical journal, Deutsche Ärzteblatt, (PDF) describes a case where boy of four years (110 cm, 18 kg body weight) was given apricot kernels during an alternative cancer treatment. Upon additional treatment with a single dose of 500 mg amygdalin, the kid showed agitation, spasms and the eyes started to roll.
I'll leave it up to your fantasy as a writer on how to apply the poison, but spicing some marzipan with it might help ;)
• Weird that the NFPA health rating is 1 if amygdalin is a potential murder toxic. How much do you need to ingest to become severely ill? – Jori Feb 2 '15 at 17:09
• @Jori Unless I'm completely off, the blue NFPA ratings are for acute intoxications and health risks upon direct exposure. Release of cyanide from amygdalin takes place in the smaller intestine (Dünndarm). Anyway, I've added some more information to my answer. – Klaus-Dieter Warzecha Feb 2 '15 at 18:27
• There is bitter almonds and "normal" almonds, the same seems to be the case for apricot. Of course only the bitter variants contain amygdalin. – Georg Feb 2 '15 at 18:34
• @Georg To my knowledge, this is true for the sweet and bitter variants of the almonds, but not for the apricots (Prunus armeniaca). – Klaus-Dieter Warzecha Feb 2 '15 at 18:49
• The leaves of many Prunus species contain high concentrations of amygdalin and prunasin, both of which are cyanogenic glucosides capable of doing the "job". There are better common garden plants for poisoning people, but I could see that there is something poetic about apples. – George of all trades Feb 7 '17 at 9:10
Honestly, you probably wouldn't be able to get this done with just a highschool chem lab, at least not to a high purity (whether or not this is needed is debatable; in fact it could be interesting in the story because it would give a clue as to how it was produced, whereas pure cyanide wouldn't really tell the detective much).
Having said that, apples don't contain the harmful version of cyanide (hydrogen cyanide) in and of themselves. They do however contain amygdalin, which can be metabolized to hydrogen cyanide. A quick search ends up with wikipedia giving a brief overview of how it is isolated:
Amygdalin is extracted from almonds or apricot kernels by boiling in ethanol; on evaporation of the solution and the addition of diethyl ether, amygdalin is precipitated as white minute crystals.
In case your orgo isn't all that great (mine is a little rusty too), what you should read from the above quote is that ethanol is used to oxydize the primary alcohol group to an aldehyde or a carboxylic acid (depending on how long you let it boil for) making it polar and so non-soluble the diethyl ether (an organic solvent) which leads it to precipitating out as a white crystal. Once it precipitates out the easiest way to isolate it is just to use a paper filter, since it should be the only solid present. Suction filtration would be optimal but that's generally not available in high schools. Lastly you could use sublimation to get a really pure final product, but again this isn't normally available in high school labs.
If you're writing a novel I recommend you take the time to read up on it and how it is metabolized within the body. The more you understand the process the more authentic it'll feel, imo. Best of luck.
EDIT: You may want to consider switching to apricot kernels as opposed to apple seeds. The process is effectively the same but it's more realistic, since you'd need a LOT of apple seeds to get a decent amount of cyanide. Also, like I said my orgo is a bit rusty so if someone else could look over this that would be nice.
• Ethanol can oxidize another alcohol? – Dissenter Dec 27 '15 at 8:31
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# What is the perplexity of a mini-language of numbers [0-9] where 0 has prob 10 times the other numbers?
I'm reading Speech and Language Processing, Jurafsky and Martin, in particular chapter 4 where they introduce perplexity see https://web.stanford.edu/~jurafsky/slp3/4.pdf (page 8-9)
Here a brief excerpt:
There is another way to think about perplexity: as the weighted average branching factor of a language. The branching factor of a language is the number of possible next words that can follow any word. Consider the task of recognizing the digits in English (zero, one, two,..., nine), given that each of the 10 digits occur with equal probability $P = 1/10$ . The perplexity of this mini-language is in fact 10. To see that, imagine a string of digits of length N. By Equation (4.17), the perplexity will be:
$PP(W) = ({\frac{1}{10}}^{N})^{-{\frac {1}{N}}} = ({\frac{1}{10}})^{-1} = 10$
But now suppose that the number zero is really frequent and occurs 10 times more often than other numbers. Now we should expect the perplexity to be lower, since most of the time the next number will be zero. Thus although the branching factor is still 10, the perplexity or weighted branching factor is smaller. We leave this calculation as an exercise to the reader.
Now this should be fairly simple, I did the calculation but instead of lower perplexity instead I get a higher one.
My calculations were:
Probability of number zero is 10 times the other probabilities.
$P(0) = 10 * P(n =\{1,2,..,9\})$
The sum of the probabilities of all numbers have to add up to 1
$10 * P(n =\{1,2,..,9\}) + 9 * P(n =\{1,2,..,9\}) = 1$
Which implies:
$P(n =\{1,2,..,9\}) = {\frac {1}{19}}$
$P(0) = {\frac {10}{19}}$
So plugging it into the perplexity formula $PP(W) = P(w_1,w_2,..,w_N)^{-{\frac{1}{N}}}$ the numbers I get:
$PP(0,1,..,9) = ({\frac {10}{19}} * {\frac {1}{19}}^9)^{-{\frac{1}{10}}} = 15.09224$
which is more that the perplexity calculated earlier where all numbers had equal ${\frac{1}{10}}$ probability.
The book anticipates a lower perplexity instead, what am I doing wrong?
• Re the last equation: Is that really what your book's formula says? I suspect you might have misread it. – whuber Apr 6 '16 at 21:22
• I have edited the question and linked to the book, as far as I can tell I'm reporting the formula correctly but again the calculation result is not the one expected so I might as well have got it wrong, I cannot work out what is wrong. – Giuseppe Romagnuolo Apr 6 '16 at 21:27
• That is indeed a limited formula. Use (4.39) instead. Alternatively, as an approximation, construct a long random sequence of independent occurrences of these ten digits and compute its perplexity. – whuber Apr 6 '16 at 21:40
• @whuber isn't the formula at (4.39) about discounting, I don't think it is about perplexity. I'm not sure how constructing a long random sequence of independent occurrences will help me understand if I got the correct understanding of the formula to calculate the perplexity. – Giuseppe Romagnuolo Apr 6 '16 at 22:06
• I believe all of section 4.6 is devoted to explaining this connection. – whuber Apr 6 '16 at 22:28
The reason why you get the wrong answer is because the way the calculation has been described in the book is a little confusing. The book says "imagine a string of digits of length $N$". This means a long string of digits, not just a string of $10$ digits.
Imagine a long string of digits from the new language. On average, in $19$ digits from this sequence, you have $10$ zeros and $1$ of each of the other numbers, because $0$ occurs ten times as often as each of the other digits. Imagine dividing your long sequence into blocks of length $19$. Say there are $M$ of these blocks. Then the perplexity is
$$\left(\left(\frac{10}{19}\right)^{10M}\left(\frac{1}{19}\right)^{9M}\right)^{-1/19M}$$
and just like in the example in the book, the $M$'s cancel, so the perplexity is:
$$\left(\left(\frac{10}{19}\right)^{10}\left(\frac{1}{19}\right)^{9}\right)^{-1/19}$$
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The Word 2007/2010 Equation Editor
Contents
The Word 2007/2010 Equation Editor
When the Equation Editor Should Be Used
Why the Equation Editor Should Be Used
How to Enter the Equation Editor Quickly
Equation Display Modes
Equation Editor Options
Math Autocorrect: A Useful Look-up Tool
Insertion of Single Symbols
Insertion of Spaces
Grouping and Brackets
Superscripts and Subscripts
Division and Stacking
Parentheses size control with \phantom and \smash
Square Roots and Higher Order Roots
Integrals, Products and Sums
Function Names
Other Font Changes
Accent Marks, Overbars, Underbars, Above and Below
Greek Alphabet
Hebrew Characters
Equation Numbers
Symbol that Lack Keywords
References
# When the Equation Editor Should Be Used
The equation editor should be used to format your equations. In some cases you can use simple Word commands, such as superscript (<control>+) and subscript (<control>=) to format simple variables, as when you wish to say, “L1 is the length of the beam,” but in doing so, you should pay attention to the font in which the variable is displayed. For example, variables should be formatted in italic font, while function names and units of measure should be in regular font. (It is often easiest to use a shortcut key, as described below, to jump into the equation editor, even if you are simply typing a variable name).
# Why the Equation Editor Should Be Used
Some equations will be nearly impossible to represent without this editor. Others will simply look unprofessional. Compare the following:
dy ax2 + bx + c
--- = --------------- sin(q)
dx (x – a)2
The second form looks better, requires about a third of the time to create with the equation editor, and is far easier to modify. You can save substantial time if you become familiar with the shortcut commands within the equation editor. This document describes the use of the editor available in Word 2007. This environment differs from the keystroke-based editor that was available in previous versions of Word or in Mathtype. Its syntax is similar to that of TeC a typesetting program that pre-dates Microsoft Word.
# How to Enter the Equation Editor Quickly
The quickest way to enter the equation editor is the shortcut key <alt>= (hold down the <alt> key while you type “=”). You can also click on “Equation” under the “Insert” tab, but this sequence can become cumbersome when you are setting a large number of equations or defining multiple variables within text. The need to move your hands from the keyboard to the mouse (or mouse pad) slows your typing.
You now have no excuse not to use the equation editor on a casual basis. It is only one keystroke away.
While in the equation editor, you can use various symbols and keywords instead of the more cumbersome menu bar. A more complete description of the codes used by the equation editor and the syntax and philosophy behind it is given by Sargent [1].
Single characters, such as _, ^ and / that have special meanings.
Keywords such as \alpha that will be translated to symbols (in this case, ).
Keywords such as \sqrt and \overbrace will modify expressions that are correctly grouped.
Note: Spaces that you type are important to the equation editor because they tell the editor when it is time to translate a part of the equation you are typing. Where it is necessary for clarity in this document, a space will be represented by the sequence <sp>.
# Equation Display Modes
If you click on an equation, it will become highlighted, as shown in Figure 1. When you then click on the blue downarrow at the lower right, five options appear. “Save as new equation…” allows you to keep the equation as a building block, which makes it available from the “Insert” ribbon. “Professional” means that the equation should be displayed as a formatted equation. “Linear” means to show the equation in its raw form, similar to the way that the equation was typed, but with some of the typed codes translated into special characters. “Display” means that the equation will be formatted in a way that is appropriate for an equation between paragraphs. “Inline” means that the equation will be formatted in a way that is appropriate for an equation within a paragraph. Inline equations tend to be more compressed than displayed equations.
Figure 1: The appearance of an equation.
# Equation Editor Options[1]
If you find that the equation editor does not format some aspects of equations according to your taste, you may be able to change those aspects with the Equation Options menu. To find this menu, enter the equation editor (<alt>=), and when the “Design” ribbon appears, click on the arrow (circled in red in Figure 2) at the lower-right corner of the “Tools” group. This menu allows you to change, for example, the default font for equations, the placement of integral limits, and margins. It also is the gateway to the Math Autocorrect window discussed in the next section.
Figure 2: The tools group of the Design ribbon.
Figure 3: The Equation Options menu.
# Math Autocorrect: A Useful Look-up Tool
Math Autocorrect is an easily remembered tool that will allow you to look up the commands for certain symbols. Click “Math Autocorrect” on the next menu to obtain the display shown in Figure 3. This display shows a large number of the commands required to obtain specific symbols. It also allows you to define your own shortcuts. For example, if you frequently needed the character (Gha), you could enter the equation editor, go to the Insert ribbon, and use the Symbol menu to insert the character (selected from the Latin Extended-B font). Then highlight that character and go to the Math Autocorrect menu. The menu will appear as shown in Figure 4. Now type \Gha in the window under “Replace,” click on Add, and return to word. Thereafter, the equation editor will replace \Gha with .
Figure 3: Math Autcorrect Window
Figure 4: When you call up the autocorrect window with a character highlighted, it will prompt you for a replacement sequence for that character.
# Insertion of Single Symbols
Keywords can be used to quickly insert a limited number of frequently used symbols. Keywords are case-sensitive (e.g., \rightarrow is different from \Rightarrow). If you need a single symbol that does not appear on this list, see the section Symbol that Lack Keywords.
To Insert Type Or To Insert Type Or \infty \hbar \rightarrow \-> \leftarrow \uparrow \downarrow \nearrow \nwarrow \searrow \swarrow \leftrightarrow \updownarrow \Rightarrow \Leftarrow \Uparrow \Downarrow \partial \nabla \le <= \ge >= \ll << \gg a\times b f(t)\otimes g(t) a\cdot b a\odot b x\oplus y a\ominus y a\mapsto b \hookrightarrow a\dots b a\cdots b a\bot b a\top b A\bigcap B A\bigcup B A\bigsqcup B A\biguplus B a\star b \forall \in \exists \bigwedge \bigvee \ne \approx \equiv \cong \sim \simeq \bowtie \box \subset \emptyset ~= \therefore \because \pm +- \mp -+ \angle \propto 22 \degree ”C”
# Insertion of Spaces
Because spaces have special meaning in the equation editor, and because the equation editor usually handles spacing appropriately, the spacebar cannot usually be used to add spaces within equations. However, spaces can be inserted with the following keywords: (Keywords \medsp, \thicksp and vthicksp also exist, but \medsp is a not-so-useful synonym for no space at all, and \thicksp and \vthicksp are synonyms for \thinsp).
\hairsp a small space e.g.
\thinsp a wider space e.g.
# Grouping and Brackets
The equation editor causes brackets (such as [], {} and ( )) to grow to the size of the expression within them. However, parentheses are the grouping character and will not display when used as such. To force parentheses to display, you must double them. To prevent brackets from being reformatted, precede them by the “\” character. Some examples follow.
To Display Type Comments [a/b] The “/” command for fractions is described in a later section. {a/b} (a/b) Parentheses display. a/(b+1) Parentheses used for grouping do not display. a/((b+1)) Double parentheses display. { a\atop b \close y The \close keyword completes the opened brace. The \atop command is described in a later section. |(a|b|f)/(c+d)| The parentheses are, again, used for grouping. |a|b|f/(c+d)| y=$a/b$ Backslashes prevent [ and ] from growing. \norm a\norm
# Superscripts and Subscripts
The _ and ^ keys are used to insert superscripts and subscripts. Grouping is important because it distinguishes between and . Terms can be grouped by enclosing them in parentheses, where the parentheses themselves do not print. Some simple examples follow:
To Display Type Notes x_i\timesy^n The spacebar is needed. x^(i+1) The parentheses do not show. See “grouping.” x_i^n F_n^(k+1) F_(n^(k+1)) All of the parentheses are needed. (_0^9)H
# Division and Stacking
Use the “/” character for division. The equation editor will reformat the expression to place the numerator above the denominator. To prevent vertical buildup of the fraction, use “\/” instead of “/” alone. As with superscripts and subscripts, you can use parentheses to group expressions into a numerator and denominator. Examples follow:
To Display Type Comments a/b (a+b)/(c+d) Parentheses do not print. ((a+b))/(c+d) The double parentheses force the single parentheses to print in the numerator. ((a+b)/(c+d) + n)/(f(x)+e^(1\/2)) The “/” is preceded by “\” in the exponential to provide a horizontal fraction ( instead of ).
If you need to stack expressions without the horizontal division line, use \atop or \matrix. The vertical bar “|” can also be used in place of \atop.
To Display Type Comments a\atop b or a|b Do not add spaces between the expression and the vertical bar. (a+b)\atop(c+d) Parentheses do not print. f(x)={\matrix (\infty x=0@0 x\ne 0) \close The @ character ends a row of the matrix. A=[\matrix(x_11&x_12&x_13@ x_21&x_22&x23@ x_31&x_32&x_33)] The matrix must be enclosed in ()’s. The & character separates columns of the matrix. The @ separates rows.
The syntax for matrices tends to lead to expressions that are difficult to read. An easier approach is to enter the equation editor with <alt>+ and then use the Matrix dropdown menu in the structures group to insert the closest approximation to the matrix that you need. To add extra columns and rows, click on the equation and then click on the small blue down-arrow and scroll down to “Linear” to change the display to linear, then insert “&” and “@” symbols as appropriate, and return to “Professional.”
# Parentheses size control with \phantom and \smash
The keywords \phantom and \smash can be used to force brackets and parentheses to have a specific size.
To Display Type Comments [\phantom (a\atop b)] The \phantom command creates an object as large as the expression in parentheses, but does not print it, so you can create, for example, large empty brackets. [\smash(a\atop b)\close \smash creates the object, but makes its size zero so that the enclosing bracket does not grow. [\hphantom((a+b)/c)] The \hphantom command creates an object with the width of the expression in parentheses, but zero height. [\vphantom((a+b)/c)] The \vphantom command creates an object with the height of the expression in parentheses, but zero width.
This example shows a Routh matrix that was constructed with the aid of “\vphantom.”
Without the \vphantom, the vertical spacing of the stack of symbols on the left would not match with that of the two columns on the right.
The syntax is
\open\matrix(s^3@s^2@(1/4) s^1@s^0 )| \matrix(1&8\vphantom(s^3 )@4&48\vphantom(s^2 )@-1/4 (48-32)&0@b_1&0\vphantom(s^0 ))
Yes. I do realize that this example is far more than you want to worry about.
# Square Roots and Higher Order Roots
The square root keyword \sqrt operates on the argument that follows it. The equation editor also has keywords for higher order roots. Examples are:
To Display Type Comments \sqrt x \sqrt(x+1) \cbrt(x+1) \qdrt(x+1) (-b\pm\sqrt(b^2-4ac))/2a \sqrt(n&x) The & separates the root order from the argument
# Integrals, Products and Sums
Integrals, products and sums are inserted with the keywords \sum, \int and \prod. Use subscripts and superscripts to insert the limits. Examples are:
To Display Type \sum_(n=0)^N x^n \prod_(n=0)^N x^n \int_-\infty^\infty f(t) e^-i\omega tdt \iint f(x) dx \iiint f(x) dx \oint f(x,y) dl
# Function Names
The equation editor switches between “variable style” or “function style,” depending on whether it interprets part of an equation as a variable or a function (compare the two styles in the equation , which would not look right if it were displayed as ). You must type a space after the function name to allow the editor to interpret it as a function. Some of the functions that are recognized are:
If a function keyword is not recognized, you can force the editor to treat it as a function if you follow it with the \funcapply keyword. For example, is not recognized as a function, but the sequence sinc\funcapply<sp><sp> will produce (as opposed to the less attractive .
# Other Font Changes
Many of the utilities on the Font group of the Home ribbon can be used to modify text within an equation. The \scriptX command (where X is any single letter) can also be used to quickly create a script letter. For example \scriptL produces . Similarly, \doubleL produces , and \frakturL produces . The size of an equation can be increased with the shortcut <control>] and decreased with <control>[. The variable style can be overridden with several commands. See Gardner [2] for more information.
To Insert Use
a script character (e.g. ) \scriptF (Notice that there is no space between \script and F)
regular text Enclose in quotes. E.g. “a”= “b” produces instead of .
italic text toggle italic on and off with <control>i
bold text toggle bold on and off with <control>b
hello there <control>i hello <control>I there
hello there <control>i hello <control>b there
\scriptL {f(x)}
x=\Re(x+iy)
y=\Im(x+iy)
a large character Select the character. Then, on the Design ribbon, under the “Tools” group, click on “Normal Text.” Now go to the “Home” ribbon and change the font size. Go back to the Design ribbon and click on “Normal Text” again to get back to the correct font design (e.g. italic if it is a variable).*
* Sorry about this level of complication. There should just be a command like \size24 to change the point size, but there is not. Also, if you try to change font size without first clicking on “Normal Text,” the font size of the entire equation changes, rather than just the characters you selected.
# Accent Marks, Overbars, Underbars, Above and Below
Certain keywords can be used to place accents, overbars, overbraces and other modifiers on characters and expressions. Examples are:
To Display Type Comments \overbraceF^”force” = \overbrace(ma)^”mass times acceleration” Overbrace text is introduced with ^, as if it were a superscript. \underbraceF^”force” = \underbrace(ma)^”mass times acceleration” Underbrace text is introduced with _, as if it were a subscript. \overbar(a+b) \overparen(a+b) \underbar(a+b) lim_(x\rightarrow 0)f(x) You can replace “_” with “\below”. lim\below(x\rightarrow 0)f(x) Because “lim” is a keyword it is not displayed in italics. lim\above(x\rightarrow 0)f(x)
Some accents are designed to fit over a single character. The keyword must be typed after the modified character and followed by two spaces.
For Type For Type For Type x\bar x\ddot x\hvec x\Bar x\hat x\tvec x\check x\tilde x\dot x\vec
Prime marks also follow the expression that they modify, but are followed by only a single space:
To Display Type x\prime x\pprime x\ppprime
# Greek Alphabet
To include a Greek letter in an equation, spell the name of the letter, preceded by the backslash character (\). If the name begins with a lower case letter, a lower case Greek letter is inserted. If the name begins with an upper case letter, an upper case Greek letter is inserted. The table below provides the names for each of the lower case Greek letters (with some variations).
For Type For Type For Type For Type \alpha \eta \omicron* \upsilon \beta \iota \pi \varpi \chi \varphi \theta \omega \delta \kappa \vartheta \xi \epsilon \lambda \rho \psi \phi \mu \sigma \zeta \gamma \nu \tau
*\omicron should work, but did not when I tried it in the Word 2007 editor. If you have trouble with this letter, you can use the Symbols group of the Insert ribbon.
# Hebrew Characters
The equation editor’s collection of Hebrew characters is limited to the first four.
\aleph
\beth
\daleth
\gimel
# Equation Numbers
The easiest way to number equations is to put the equation in a table. If you wish to have the equation centered, use a table with three columns so that the left column balances the right column. Here is an example. The table borders are included for clarity, but in most cases, it is best to remove them.
Eq. 1
You can make the equations automatically numbered if you click inside the table, go to the “References” ribbon, and click on “Insert Caption.” Choose “Equation,” and use the label that fits your taste (in this case, I have used “Eq.”). Finally, change the “borders and shading” for the table to “None.”
To generate additional numbered equations, you can copy and paste the table that you just generated into different locations and simply change the equation. Highlight the equation number, right click, and select “Update Field” to have Word automatically change the number. To simplify matters further, you can just copy and paste the above table into your document and use it as a template.
Eq. 2
# Symbol that Lack Keywords
The equation editor lacks keywords for some symbols. Some missing keywords are surprising, such as one for , which one would expect to be simply \Angstrom.[2] If the keyword cannot be found, the symbol is probably still available. First, check the Symbols group on the Design ribbon under Equation Tools (Figure 5), which should appear when you type <control>+ to enter the equation editor.
Figure 5: The Symbols group under the Design tab of “Equation Tools.”
Most of the default symbols in this group have already been described. However, if you click on the down-arrow (circled in Figure 5) and then click on the down-arrow at the top of the resulting menu, the following options will appear:
Basic Math
Greek Letters
Letter-Like Symbols
Operators
Arrows
Negated Relationships
Scripts
Geometry
For example, the “Letter-Like Symbols” group contains the following symbols, some of which do not have keywords.
Symbol Name ASCII Hexadecimal Code Latin Small Letter Eth 00F0 Turned Capital F 2132 Hilbert Space 210C The Set of Natural Numbers 2115 Rational Numbers 211A Real Numbers 211D The Set of Integers 2124 Angstrom 212B Inverted Ohm Sign 2127 There Does Not Exist 2204
The ASCII hexadecimal code may be useful for a small subset of people. If you type that code into an equation, then select the code and type <alt>x (think of the “x” as standing for hexadecimal), it will convert the number to the corresponding character. You can convert only one character at a time. If you need to use an undefined symbol frequently, you should create your own definition with the Math Autocorrect function described in the section Math Autocorrect: A Useful Look-up Tool.
Conversely, if you are in the linear mode of the equation editor, you can highlight a single character and then type <alt>x to print that character’s hexadecimal code.
As a last resort, you can resort to using Symbol under the Symbols group of the standard Insert ribbon, which is completely independent of the equation editor.
# References
1. Sargent, M III, “Unicode nearly plain-tex encoding of mathematics,” Office Authoring Services, Microsoft Corporation, 2006 (http://unicode.org/notes/tn28/UTN28-PlainTextMath-v2.pdf).
2. Gardner J, “Shortcuts for the Word 2007 equation editor,” (http://dataninja.files.wordpress.com/2007/09/word07shortcuts.pdf).
[1] Thanks to John Goodhew, University of Sydney Business School for making me aware of these tools.
[2] An insane method that approximates the Angstrom symbol uses \vphantom and \smash. Type:
(\vphantom(a)<sp>\smash(A)<sp>)\above\circ)<sp><sp> and then convert to non-italic with <control>I. The result is . The \vphantom and \smash are used to place the circle at the right height. If you type simply A\above\circ<sp><sp>, you get , which places the circle too high.
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2021 AMC 12B Problems/Problem 21
Problem
Let $S$ be the sum of all positive real numbers $x$ for which$$x^{2^{\sqrt2}}=\sqrt2^{2^x}.$$Which of the following statements is true?
$\textbf{(A) }S<\sqrt2 \qquad \textbf{(B) }S=\sqrt2 \qquad \textbf{(C) }\sqrt2
Solution 1
Note that \begin{align*} x^{2^{\sqrt{2}}} &= {\sqrt{2}}^{2^x} \\ 2^{\sqrt{2}} \log_2 x &= 2^{x} \log_2 \sqrt{2}. \end{align*} (At this point we see by inspection that $x=\sqrt{2}$ is a solution.)
We simplify the RHS, then take the base-$2$ logarithm for both sides: \begin{align*} 2^{\sqrt{2}} \log_2 x &= 2^{x-1} \\ \log_2{\left(2^{\sqrt{2}} \log_2 x\right)} &= x-1 \\ \sqrt{2} + \log_2 \log_2 x &= x-1 \\ \log_2 \log_2 x &= x - 1 - \sqrt{2}. \end{align*} The RHS is a line; the LHS is a concave curve that looks like a logarithm and has $x$ intercept at $(2,0).$
There are at most two solutions, one of which is $\sqrt{2}.$ But note that at $x=2,$ we have $\log_2 \log_2 {2} = 0 > 2 - 1 - \sqrt{2},$ meaning that the log log curve is above the line, so it must intersect the line again at a point $x > 2.$ Now we check $x=4$ and see that $\log_2 \log_2 {4} = 1 < 4 - 1 - \sqrt{2},$ which means at $x=4$ the line is already above the log log curve. Thus, the second solution lies in the interval $(2,4).$ The answer is $\boxed{\textbf{(D) }2\le S<6}.$
~ ccx09
Solution 2
We rewrite the right side without using square roots, then take the base-$2$ logarithm for both sides: \begin{align*} x^{2^{\sqrt2}}&=\left(2^\frac12\right)^{2^x} \\ x^{2^{\sqrt2}}&=2^{\frac12\cdot2^x} \\ x^{2^{\sqrt2}}&=2^{2^{x-1}} \\ \log_2{\left(x^{2^{\sqrt2}}\right)}&=\log_2{\left(2^{2^{x-1}}\right)} \\ 2^{\sqrt2}\log_2{x}&=2^{x-1}. \hspace{20mm} (*) \end{align*} By observations, $x=\sqrt2$ is one solution. Graphing $f(x)=2^{\sqrt2}\log_2{x}$ and $g(x)=2^{x-1},$ we conclude that $(*)$ has two solutions, with the smaller solution $x=\sqrt2.$ We construct the following table of values: $$\begin{array}{c|c|c|c} & & & \\ [-2ex] \boldsymbol{x} & \boldsymbol{f(x)=2^{\sqrt2}\log_2{x}} & \boldsymbol{g(x)=2^{x-1}} & \textbf{Comparison} \\ [1ex] \hline & & & \\ [-1ex] 1 & 0 & 1 & \\ [1ex] \sqrt2 & 2^{\sqrt2-1} & 2^{\sqrt2-1} & f\left(\sqrt2\right)=g\left(\sqrt2\right) \\ [1ex] 2 & 2^{\sqrt2} & 2 & f(2)>g(2) \\ [1ex] 3 & 2^{\sqrt2}\log_2{3} & 2^2 & \\ [1ex] 4 & 2^{\sqrt2+1} & 2^3 & f(4) Let $x=t$ be the larger solution. Since exponential functions outgrow logarithmic functions, we have $f(x) for all $x>t.$ By the Intermediate Value Theorem, we get $t\in(2,4),$ from which $\[S=\sqrt2+t\in\left(\sqrt2+2,\sqrt2+4\right).$$ Finally, approximating with $\sqrt2\approx1.414$ results in $\boxed{\textbf{(D) }2\le S<6}.$
The graphs of $y=f(x)$ and $y=g(x)$ are shown below: $[asy] /* Made by MRENTHUSIASM */ size(1200,200); int xMin = 0; int xMax = 5; int yMin = 0; int yMax = 5; //Draws the horizontal gridlines void horizontalLines() { for (int i = yMin+1; i < yMax; ++i) { draw((xMin,i)--(xMax,i), mediumgray+linewidth(0.4)); } } //Draws the vertical gridlines void verticalLines() { for (int i = xMin+1; i < xMax; ++i) { draw((i,yMin)--(i,yMax), mediumgray+linewidth(0.4)); } } //Draws the horizontal ticks void horizontalTicks() { for (int i = yMin+1; i < yMax; ++i) { draw((-1/8,i)--(1/8,i), black+linewidth(1)); } } //Draws the vertical ticks void verticalTicks() { for (int i = xMin+1; i < xMax; ++i) { draw((i,-1/8)--(i,1/8), black+linewidth(1)); } } horizontalLines(); verticalLines(); horizontalTicks(); verticalTicks(); draw((xMin,0)--(xMax,0),black+linewidth(1.5),EndArrow(5)); draw((0,yMin)--(0,yMax),black+linewidth(1.5),EndArrow(5)); label("x",(xMax,0),(2,0)); label("y",(0,yMax),(0,2)); real f(real x) {return 2^sqrt(2)*log(x)/log(2);}; real g(real x) {return 2^(x-1);}; draw(graph(f,1,3.65),red,"y=2^{\sqrt2}\log_2{x}"); draw(graph(g,0,3.32),blue,"y=2^{x-1}"); pair A, B; A = intersectionpoint(graph(f,1,2),graph(g,1,2)); B = intersectionpoint(graph(f,2,4),graph(g,2,4)); dot(A,linewidth(4.5)); dot(B,linewidth(4.5)); label("0",(0,0),2.5*SW); label("\sqrt2",(A.x,0),2.25*S); label("t",(B.x,0),3*S); label("4",(4,0),3*S); label("4",(0,4),3*W); draw(A--(A.x,0),dashed); draw(B--(B.x,0),dashed); add(legend(),point(E),40E,UnFill); [/asy]$ ~MRENTHUSIASM
Solution 3
Note that this solution is not recommended unless you're running out of time.
Upon pure observation, it is obvious that one solution to this equality is $x=\sqrt{2}$. From this, we can deduce that this equality has two solutions, since $\sqrt{2}^{2^{x}}$ grows faster than $x^{2^{\sqrt{2}}}$ (for greater values of $x$) and $\sqrt{2}^{2^{x}}$ is greater than $x^{2^{\sqrt{2}}}$ for $x<\sqrt{2}$ and less than $x^{2^{\sqrt{2}}}$ for $\sqrt{2}, where $n$ is the second solution. Thus, the answer cannot be $\text{A}$ or $\text{B}$. We then start plugging in numbers to roughly approximate the answer. When $x=2$, $x^{2^{\sqrt{2}}}>\sqrt{2}^{2^{x}}$, thus the answer cannot be $\text{C}$. Then, when $x=4$, $x^{2^{\sqrt{2}}}=4^{2^{\sqrt{2}}}<64<\sqrt{2}^{2^{x}}=256$. Therefore, $S<4+\sqrt{2}<6$, so the answer is $\boxed{\textbf{(D) }2\le S<6}$.
~Baolan
~ pi_is_3.14
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# Character movement resetting x & y position in HTML5 game
I have an HTML5 game I'm working on and I'm at the point where I'm trying to make the character move. For the most part, the character(A rectangular square) is moving, but it's not moving from its starting position. Instead, when I hit the move key the character's x position and y position are reset to 0, 0. My question is, why is the character's position being set to 0, 0 before it starts moving?
I've included a link to the dropbox with the project so it's easier to see what's going on, but here's the movement script for reference.
initMap.js Script, lines 81 - 92
case "Player":
gameObjectsLayerCtx.fillStyle = "rgb(0, 0, 0)";
gameObjectsLayerCtx.fillRect(xPos, yPos, destWidth / 3, destHeight / 3);
window.onkeydown = function(ev){
var key = ev.key;
if(key == "w"){
yPos += 5;
gameObjectsLayerCtx.clearRect(0, 0, gameObjectsLayer.width, gameObjectsLayer.height);
gameObjectsLayerCtx.fillRect(xPos, yPos, destWidth / 3, destHeight / 3);
}
};
break;
Project
The script is the initMap.js script and the movement functionality is from lines 81 - 92
The character's starting position Where the character moves to when pressing the move key
• 2 possible causes (cant get the link to work to check). Either the player x,y is becoming undefined, or your x y isn't relative to the player characters scope and a different object is changing the values to 0. I would do a console.log(xPos,yPos) to see what is happening to the vars. – ericjbasti Oct 10 '16 at 16:08
• @ericjbasti sorry, I fixed the link, it should work now – Robert Oct 10 '16 at 22:44
## 1 Answer
Fair warning, you're going to run into quite a few issues with some of your code, but I was able to resolve this particular problem.
You have to save the players X,Y positions if you plan on changing them. By the time the window.onkeydown fires the xPos and yPos are reset to 0 and 0. Lets look at this function and look at the last 6 lines.
gameObjectsSpriteSheet.onload = function(){
var tileWidth = 0;
var tileHeight = 0;
var destWidth = 0;
var destHeight = 0;
var xPos = 150;
var yPos = 150;
for(var rows = 0; rows < GameObjects.length; rows++){
tileHeight = 247;
destHeight = tileHeight / 2;
for(var cols = 0; cols < GameObjects[rows].length; cols++){
tileWidth = 255;
destWidth = tileWidth / 2;
switch(GameObjects[rows][cols].object){
case "T":
gameObjectsLayerCtx.drawImage(gameObjectsSpriteSheet, 0, 0, tileWidth, tileHeight, xPos, yPos, destWidth, destHeight);
break;
case "Player":
gameObjectsLayerCtx.fillStyle = "rgb(0, 0, 0)";
gameObjectsLayerCtx.fillRect(xPos, yPos, destWidth / 3, destHeight / 3);
window.onkeydown = function(ev){
var key = ev.key;
if(key == "w"){
yPos += 5;
gameObjectsLayerCtx.clearRect(0, 0, gameObjectsLayer.width, gameObjectsLayer.height);
gameObjectsLayerCtx.fillRect(xPos, yPos, destWidth / 3, destHeight / 3);
}
};
break;
}
xPos += destWidth / 2;
}
xPos = 0; // After drawing the scene you reset the xPos to 0
yPos += destHeight / 2;
}
yPos = 0; // After drawing the scene you reset the yPos to 0
}
You're reseting the X and Y so you can draw the next line, but the player has no idea where it was drawn when the keypress event happens.
You can fix this by adding 2 variables to store the players X and Y positions:
var playerX = 0;
var playerY = 0;
Put this outside of the gameObjectsSpriteSheet.onload function. I put it directly after the close of that function when I did my testing.
This next bit of code is my solution with a couple comments that will hopefully explain what is going on.
gameObjectsSpriteSheet.onload = function() {
var tileWidth = 0;
var tileHeight = 0;
var destWidth = 0;
var destHeight = 0;
var xPos = 150;
var yPos = 150;
for (var rows = 0; rows < GameObjects.length; rows++) {
tileHeight = 247;
destHeight = tileHeight / 2;
for (var cols = 0; cols < GameObjects[rows].length; cols++) {
tileWidth = 255;
destWidth = tileWidth / 2;
switch (GameObjects[rows][cols].object) {
case "T":
gameObjectsLayerCtx.drawImage(gameObjectsSpriteSheet, 0, 0, tileWidth, tileHeight, xPos, yPos, destWidth, destHeight);
break;
case "Player":
playerX = xPos; // Ok we need to draw the player at this location, lets save it for later use.
playerY = yPos; // lets save this for later use as well
gameObjectsLayerCtx.fillStyle = "rgb(0, 0, 0)";
gameObjectsLayerCtx.fillRect(playerX, playerY, destWidth / 3, destHeight / 3); // Draw the player using the variables we just saved.
window.onkeydown = function(ev) {
var key = ev.key;
if (key == "w") {
playerY += 5; // Lets modify the playerX and Y instead of yPos since that variable is no longer useful.
gameObjectsLayerCtx.clearRect(0, 0, gameObjectsLayer.width, gameObjectsLayer.height);
gameObjectsLayerCtx.fillRect(playerX, playerY, destWidth / 3, destHeight / 3); // Draw the player using our new variables.
}
};
break;
}
xPos += destWidth / 2;
}
xPos = 0;
yPos += destHeight / 2;
}
yPos = 0;
}
}
// this is where I put the new variables.
var playerX = 0;
var playerY = 0;
• @Robert Just a comment for future readers. Use encapsulation, which will avoid problems like this. Basically you create an entity kind of object, playerEntity, which will have x&y's as private, thus avoiding unwanted overrides. To accomplish camera follow you just need to center where the player is, and drawing the map and all other entities will not interact with each others x&y positions. Another thing tou might want to do is stop drawing outside of viewport + a couple of tiles depending on the tile sizes. – Mujnoi Gyula Tamas Jan 20 '17 at 11:26
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# Prediction of vascular aging based on smartphone acquired PPG signals
## Abstract
Photoplethysmography (PPG) measured by smartphone has the potential for a large scale, non-invasive, and easy-to-use screening tool. Vascular aging is linked to increased arterial stiffness, which can be measured by PPG. We investigate the feasibility of using PPG to predict healthy vascular aging (HVA) based on two approaches: machine learning (ML) and deep learning (DL). We performed data preprocessing, including detrending, demodulating, and denoising on the raw PPG signals. For ML, ridge penalized regression has been applied to 38 features extracted from PPG, whereas for DL several convolutional neural networks (CNNs) have been applied to the whole PPG signals as input. The analysis has been conducted using the crowd-sourced Heart for Heart data. The prediction performance of ML using two features (AUC of 94.7%) – the a wave of the second derivative PPG and tpr, including four covariates, sex, height, weight, and smoking – was similar to that of the best performing CNN, 12-layer ResNet (AUC of 95.3%). Without having the heavy computational cost of DL, ML might be advantageous in finding potential biomarkers for HVA prediction. The whole workflow of the procedure is clearly described, and open software has been made available to facilitate replication of the results.
## Introduction
Photoplethysmography (PPG) offers a non-invasive optical measurement method and can be used for heart rate (HR) monitoring purposes. For example, by using the white light-emitting diode as light source and the phone camera as photo-detector positioned on the index fingertip, a smartphone could be used to measure the volumetric variations of blood circulation without any additional devices1,2. Smartphone ownership continues to grow rapidly, and mobile phone apps are increasingly used for screening, diagnosis, and monitoring of HR and rhythm disorders such as atrial fibrillation (AF)3,4. The benefits and harms of using mobile apps to improve cardiovascular health or screening for specific diseases have been evaluated and discussed3,5,6,7; while HR measured by smartphone apps performing PPG agrees with a validated method such as electrocardiogram (ECG) in resting sinus rhythm6, the US Preventive Services Task Force (USPSTF) found that the current evidence is insufficient to assess the balance of benefits and harms of screening for AF with ECG7.
Our motivating crowd-sourced data comes from the Heart for Heart (H4H) initiative, promoted by the Arrhythmia Alliance, the Atrial Fibrillation Association, Happitech and other partners8,9. The aim of H4H is to gather millions of cardiac measurements and to increase the pace of progress on AF diagnostic technology. The primary advantages of using this population cohort data are: abundance of PPG recordings in large samples (ca. 10,000) and relatively long sequences (90 seconds), and free access to raw PPG signals via Happitech app. On the downside, these PPG signals can be very noisy compared to those in clinical settings, due to the uninstructed and not monitored PPG captures – during the measurement an appropriate pressure should be maintained and the measuring point should be kept still. Analyzing crowd-sourced data, and not data from some clinical trials, hinders to directly evaluate PPG signals for prediction of AF. Moreover, validation of the results through confirmation of ECG is not possible. On the other hand, to monitor cardiovascular status such as arterial blood pressure or arterial stiffness, the non-invasive PPG technique can be very useful.
Aging is a major non-reversible risk factor for cardiovascular disease. Chronological aging plays a significant role in promoting processes involved in biological aging including oxidative stress, DNA methylation, telomere shortening, as well as structural and functional changes to the vasculature of the heart10. Vascular aging, in particular, is characterized by a gradual change of the vascular structure and function, and increasing arterial stiffness is considered to be the hallmark of vascular aging11,12,13. Arterial stiffness can be measured by pulse wave velocity (PWV)14, or by the use of the PPG technique15. In particular, some aging indexes (AGI) can be calculated from the second derivative of the PPG (SDPPG) waveform16,17. In our data, additional information such as age, sex, weight, height and smoking is available. Therefore, we consider the chronological age as a proxy for vascular aging, and investigate how well the PPG signals can predict and classify healthy or unhealthy vascular aging.
Figure 1 summarizes the procedure at a quick glance. Briefly, after detrending, demodulating, and denoising the PPG signals, we consider two approaches, machine learning (ML) and deep learning (DL). While ML is based on (known) extracted features from PPG, such as beat-to-beat interval, DL uses the whole signal to predict HVA, skipping the feature extraction step. The prediction performance of the final models is validated on the dataset held back from the training and testing of the model, such that AUC (area under the curve) presents an unbiased performance measure for comparing final models. Moreover, clearly documenting the whole procedure from data preprocessing, statistical analysis, to validation of the results, will enable the scientific community to improve the usage of PPG in health care, and may lead to a robust standard operating procedure. By doing so, our work contributes to promoting open science, open software (source code in Python), and open data (available on request by H4H).
## Methods
### Description of datasets used
Our database comes from the Heart for Heart initiative8, and it consists of 4769 individuals. For each subject, one file in the .csv format is provided, containing the PPG recordings and some additional information such as sex, age, weight, height, and smoking. Data matrices of PPG recordings have 7 columns: time (around 90 seconds of measurement with a sampling frequency of averagely 30 points per second), simultaneous recordings of red, green and blue light PPG, and three axes of an accelerometer. Infrared light has a more effective penetration depth in the skin compared to green light, but is more susceptible to motion artifacts1,3,18. For the current analysis, we assume that motion artifact in PPG signals obtained from the smartphone camera recording is negligible compared to those obtained by smartwatch. Considering the relative short recordings of PPG, we took two columns (time and red PPG) for further analysis. More detailed description is given in Supplementary Materials (Section 1 and Supplementary Fig. S1).
### PPG preprocessing
We first removed the trend of the raw red PPG signal by computing a centered moving average (CMA) and subtracting it from the raw signal (to perform a high-pass filter). The sliding window $$\mathbf {w}$$ (average sampling frequency) was used to have the same number of points per second for each of the different signals. Next, to account for motion and noise artifacts, demodulation of PPG signals was obtained using the detrended signal (a real-valued s) and its Hilbert transform ($$\mathscr {H}(s)$$) to generate the analytic signal (a complex-valued $$s + \mathscr {H}(s)$$). From the module of the analytic signal, the instantaneous amplitude (also called the envelope) was obtained and then smoothed with CMA. Lastly, the detrended signal was divided by the smoothed envelope, resulting in the detrended, demodulated, and denoised signal for further analysis (Fig. 2). More details can be found in Supplementary Materials (Section 2 and Supplementary Fig. S2 and S3).
### Feature extraction from the PPG signal for the ML approach
#### Peak detection
To identify the correct peak locations, we slightly modified an existing algorithm19. Our current algorithm consists of: (i) compute the CMA of the processed signal and align it to its signal; (ii) locate the maximum of a specific region of the signal continuously above its moving average; (iii) label the set of maxima as partial peaks, repeating with the window width $$\mathbf {w_i} = (0.5, 1, 1.5, 2, 2.5, 3)*sf$$ (with sf = average sampling frequency), and consider as possible maxima those points labelled 6 times as partial peaks; (iv) if two possible maxima are separated by less than 400ms, discard the smaller of the two. Having found the final peak positions, RR can be defined as the sequence of time intervals (expressed in ms) between consecutive detected peaks.
#### Features extracted from RR
From RR we can extract information about heart rate variability (HRV) and inter beat time intervals distribution: ibi (average inter beat interval); medianRR; madRR (median absolute distance); sdnn (the standard deviation); tpr (turning point ratio of RR: computed as number of local extrema divided by number of points in the signal, ranging from 0 to 1 and used as and index of randomness20); skewnessRR; kurtosisRR, etc.
#### Features extracted from RR$$\mathbf {_{diff}}$$
The features derived from $$RR_{diff}$$, the sequence of differences between successive elements of RR, are the following: sdsd (standard deviation of successive differences); rmssd (root mean square of successive differences); pnn20 (proportion of normal to normal $$> 20\hbox { ms}$$: how many elements of $$RR_{diff}$$ are larger than 20 ms); pnn50; $${{\varvec{tpr}}}_{{ diff}}$$ (tpr computed on $$RR_{diff}$$).
#### SDPPG features
The second derivative of the PPG processed signal (SDPPG) has been linked to chronological age16. During each heart beat cycle the 5 typical points (a - e) can be identified (Supplementary Fig. S1). To identify these points we computed the FDPPG (fourth derivative of PPG) for locating the zero crossings, which correspond to the inflection points in the SDPPG. Once two consecutive inflection points were found, only one local extreme can exist between them. The first and highest maximum of a beat cycle can be determined as a, and the subsequent points, (b, c, d and e) can be detected as next minima or maxima starting from the previous one (Supplementary Fig. S1). After identifying these points, the features obtained from SDPPG are given by the amplitudes, the slopes, and the time distances.
The list of 38 features extracted based on RR, RR$$\mathbf {_{diff}}$$, and SDPPG is given in Supplementary Materials (Section 3 and Supplementary Fig. S4).
#### Quality thresholding
A quality score (Q) of the signal is computed by taking into account bad demodulation (through the variance of detected peaks height) and noise (expressed as a number of local extrema that are not detected as peaks or their corresponding valleys):
\begin{aligned} Q&= \text {Variance (detected height of peaks)} \times \frac{\text {Nr.}\,\, \text {local extrema} - \text {Nr.}\,\, \text {detected local extrema} + 2}{\text {Nr.}\,\, \text {points in the signal}}. \end{aligned}
(1)
Note that +2 is used to avoid negative numbers in case of a detected peak having no valley before or after itself due to the length of its signal. The Q score indicates how noisy and badly demodulated the signal is; the higher the score the poorer the quality, such that 0 is the perfect score. The threshold value of Q was set to 0.01. By applying quality thresholding after the feature extraction phase, a flexible choice of Q is possible without repeating the feature extraction step.
### Prediction and classification of HVA
After discarding incomplete data (e.g. missing age label) and quality thresholding with $$Q < 0.01$$, the cleaned database consists of 3612 subjects: 2205 males and 1407 females, with an average age of 49 years and a standard deviation of 14.5 years. Next, the “continuous” outcome, age, was dichotomized into two classes: young (18 - 38 years, a proxy for HVA) coded 1 and old (60 - 79 years, non-HVA) coded 0. By partitioning the data into a train (2709 subjects) and a test (903 subjects) set, we train and validate the model (both the ML and DL approaches) on a training set (Fig. 1). The final model (ML) and hyperparameters (DL) were validated using the held-out test set. The age distribution of the training and test set was kept similar to that of the whole database.
#### Ridge regression (ML)
Prior to the analysis, 38 extracted features were robustly standardized by subtracting the median and dividing by the interquartile range. Since some features were highly correlated (dark red or blue colors in Fig. 3) and there may be multicollinearity issues, a penalized regression can be considered. Shrinking the coefficient values towards zero in multiple regressions allows the less contributing variables to have a coefficient close or equal to zero. To jointly select the relevant features for HVA, we applied ridge penalized regression; linear and logistic ridge regressions were employed for continuous and binary age outcomes, respectively21,22. To obtain the final model, 2/3 of the training dataset were randomly sampled and used to fit the model; for fine-tuning of the penalty parameters a 10-fold cross-validation (CV) was applied. This procedure was repeated 100 times for both continuous age (using linear ridge) and dichotomized age (using logistic ridge). The resulting rankings of the coefficients were recorded, and averaged over 100 runs. The final score for the ranking was achieved by summing up the two scores. Then we selected some combinations of the most relevant features and we evaluated their HVA/non-HVA classification performance using Support Vector Machine (SVM) classifier (previously tuned with a 3-fold CV on the training set) on the test set.
#### Convolutional neural networks (DL)
For convolutional neural networks (CNNs), the processed signals (not the extracted features) were used as input, and the dichotomized age as target. Among many CNNs, ResNet, DenseNet and WaveNet were considered. The first two have the merit of avoiding many gradient vanishing problems; they feed directly any convolutional layer using the sum (ResNet) and the concatenation (DenseNet) of all the previous layers’ output, respectively. Moreover, ResNet was found to be highly effective with some time series datasets23, while DenseNet used on PPG signals has already shown good results24. Alternatively, WaveNet was proposed to accommodate the specific nature of time sequences25, by using a different kind of time-oriented convolution, which is basically a truncated dilated convolution. CNNs were trained, epoch by epoch, using 2/3 of the training dataset. Their performance was validated using the remaining 1/3 of the training set; for each hyperparameter, many different values were tried out and the best was determined.
#### Evaluation of the performance of ML and DL approaches
The optimized models by ML and DL were validated on the held-out test set. Prediction performance of five models was compared using AUC: four covariates (weight, height, sex, smoking); the best two PPG features (a and tpr) based on ML; covariates and these two features; the best performing CNN; covariates and 38 PPG features.
## Results
The workflow is depicted in Fig. 1, and the details are described in the methods section. The extracted features from PPG are given in Supplementary Materials (Section 3). All the algorithms and analysis have been performed using Python 3.7 and its suitable Anaconda distribution26. The code for preprocessing, feature extraction, ML and DL analysis can be found on GitHub, https://github.com/Nico-Curti/cardio, and https://github.com/LorenzoDallOlio/vascular-ageing.
### Exploratory data analysis
To visualize how strong the correlations are among age, four covariates (sex, weight, height, and smoking) and 38 PPG features (extracted from the PPG signals), a heatmap based on Pearson correlation is given in Fig. 3. The strongest correlation between age and PPG features was found in a and tpr ($$r \ge 0.35$$). We further investigated how well the classifier for healthy vascular aging (HVA) would perform using all extracted PPG features; spectral embedding26 for non-linear dimensionality reduction is first employed to cluster similar features. Figure 4 shows a relatively good separation between biologically young (blue points, indicating HVA) and old (red points, indicating non-HVA), which shows a gradual transition from HVA to non-HVA.
### Application of ML and DL to predict HVA
The final scores achieved by linear and logistic ridge regression are given in Table 1. The best performing PPG features were a (from SDPPG) and tpr (from PPG). Instead of just extracting features from PPG or SDPPG signals and then applying ML to these features, we also considered several CNNs using the raw signals as input. Application of CNN is heavy, regarding the computational burden, and tuning hyperparameters (to train CNN) is difficult as there are numerous parameters to configure. In Table 2 we report the values of hyperparameters used for WaveNet, DenseNet, and ResNet. The bold-faced ones show the hyperparameters of the best model based on the validation set performances.
### Evaluation of prediction performance
To validate the results, the independent test set was used (Fig. 1). The AUC obtained using only one feature (i.e., a, slope-AC, or tpr) was around 0.8 (Supplementary Table S1). The following models are considered: (i) covariates (weight, height, sex, smoking), (ii) the best two PPG features (a and tpr) from ML, (iii) covariates and these two features, (iv) the best performing CNN, and (v) covariates and all PPG features. One should take note that the model (v) is not recommended due to the overfitting, which may not lead to such a good performance on a different dataset. For comparing the prediction performance of the five models, the area under the ROC curve (AUC) was computed, and the results are depicted in Fig. 5. By adding the PPG features (tpr and a) to the covariates, AUC increased from 0.742 to 0.947. The 12-layer ResNet model (AUC=0.953) performed similarly to including all variables (AUC=0.954).
### Sex-stratified analysis
There are differences in body weight, height, body fat distribution, heart rate, stroke volume, and arterial compliance between the two sexes. In the very elderly, age related large artery stiffness is reported to be more pronounced in women11. Therefore, based on the two features obtained by the best performing ML, we investigated whether men and women age differently regarding HVA. The model, age = a + tpr + covariates + error, was used to estimate coefficients using the training set. The obtained coefficient estimates were used to predict vascular aging using the test set. The results are shown in Fig. 6. The points in the figures are the predicted age of each male (blue) and female (red). To depict the fitted trend line the locally weighted scatterplot smoothing (LOWESS) technique is used. For a clear comparison, two fitted lines are presented (in the right panel). On average, females are healthier than males regarding vascular aging. Focusing on the individual feature, the a wave from SDPPG showed a considerable difference between men and women (Supplementary Fig. S5).
## Discussion
Our work has been motivated by the unique crowd-sourced data from the Heart for Heart initiative. Aging is the main risk factor for vascular disease. The concept of biological aging comes from the fact that individuals do not age at the same pace. In particular, vascular aging involves arterial degeneration and hardening that impairs vascular function and ultimately causes end organ damage27. Concerning biomarkers of biological vascular aging, the most common arterial stiffness measure is pulse wave velocity (PWV), in our case the PPG waves, the velocity at which the blood pressure wave moves along the arterial tree. From the statistical point of view, biological aging can be studied in two ways. Firstly, one can study individual differences as in the brain aging process to reflect advanced or delayed brain aging28. Typically a linear regression is employed to show the differences in slope (see Figure 1 in Hamczyk et al.27), or to compare two age groups using Pearson correlation coefficients29. Secondly, as in the research of healthy aging and longevity30,31, one can consider two groups of young and old. Through a classification task, one can estimate the probability of belonging to either class and determine suitable biomarkers for longevity potential. Our approach here is the second one, which can be extended to monitor other diseases such as Atrial Fibrillation. Also, publicly available PPG data that includes ‘raw’ signals are still rare. Having abundant PPG data from the general population, we aimed to scrutinize the whole process from data preprocessing of the PPG signals to the prediction of HVA using PPG.
Recently deep neural network approaches have grown in popularity24,32; there are several challenges applying DCNN (deep convolutional neural network) to health data. One needs to choose the right topology of NN: how many hidden layers, how do you trade off the number of parameters versus the amount of training data, etc. Computational power is one of the biggest limitation, when the aim is to get an instant diagnosis using a wearable device33. Moreover, the algorithms deployed are inscrutable (it is a black-box approach) and it is difficult to interpret their results compared to other ML approaches. Prior to employing ML, several steps are involved: the signals have to be detrended and demodulated, the quality of the signals has to be assessed and signals of poor quality should be removed, peak detection algorithm has to be applied, relevant features must be extracted. Instead of using some aspects extracted from the signal, the whole signal is used as input for DL; the latter might be advantageous, when certain hidden features were not included in the model for ML. Therefore, we have investigated the effectiveness of applying the computer intensive DL methods (ResNet, DenseNet, WaveNet) against the relatively simple ML method (ridge penalized regression).
The prediction performance indicated the best DL (AUC of 95.3%), 12-layers ResNet, slightly outperformed the ML (AUC, sensitivity, and specificity of 94.7%, 87.3%, and 86.3%) with the model, 2 PPG features (a and tpr) + 4 covariates (sex, weight, height, smoking). Nevertheless, ML had the merit of identifying potential biomarkers. It has been reported that features derived from the contour of PPG signals showed association with age, in particular regarding arterial elasticity and the changes in the elastic properties of the vascular system15. The feature a wave extracted from SDPPG and tpr from PPG showed negative and positive correlation with age, respectively. The SDPPG features derived from the amplitudes of the distinctive waves situated in the systolic phase of the heart cycle, quantify the acceleration of the arterial blood vessels’ walls. tpr (turning point ratio) is based on the non-parametric ‘Runs Test’ to measure the randomness in a time-series, and it is higher when series are more random: for instance, AF patients tend to have higher tpr20. Taking age as a proxy for vascular aging, both larger a and lower tpr predict healthy vascular aging (HVA). In addition vascular aging differs between the sexes. Age-related changes in vascular function generally include increasing endothelial dysfunction and arterial stiffness34. We have shown that women appeared to be healthier (younger) than men regarding vascular aging (Fig. 6). Moreover, the clear slope variation in the females of Fig. 6 might be interpreted as slowing down the vascular aging process, ultimately leading to longer life expectancy of women. The best performing ML model (a + tpr + sex + weight + height + smoking) supports these findings (Fig. 6). For predicting HVA, therefore, ML using a few PPG features together with relevant clinical information can be a viable option against computer-intensive DL approaches33. Further, this study might be considered as a proof of concept for prediction of vascular aging based on chronological age. Extending and fine-tuning of the best performing ML model may lead to a risk score for vascular aging. Adding other relevant variables (the known risk factors) into the current model can be an option. Taking PPG measurements by smartphone camera together with simple algorithm for feature extraction and prediction for vascular aging facilitates self-monitoring of individual risk score, which can be linked to smoking and eating habits.
Note that CNNs investigated here demonstrated good performances obtained by simple structures (never more than 12 hidden layers, which is feasible for a common laptop to train). When one’s interest is studying aging process27, for demonstration purpose, we slightly modified our approach as reported in Supplementary Materials (Section 4 and Supplementary Fig. S6S8). Further fine-tuning of CNN to predict a continuous outcome is needed35, but beyond the scope of this work. When accurate prediction is the main purpose and any distinct features need to be fully incorporated (as for AF detection), CNNs can be good candidates. Our results with age as a target can be employed for transfer learning approaches for a specific target such as AF.
We showed that PPG measured by smartphone has the potential for large scale, non-invasive, patient-led screening. However, current evidence is often biased due to low quality studies, black-box methods, or small sample sizes. For instance, the reliability of ultra-short HRV features (PPG measurements less than 5 minutes) remains unclear and many HRV analyses have been conducted without questioning their validity36. On the other end of the spectrum, a systematic review of assessing the balance of benefits and harms of screening for AF with smartphone acquired PPG is lacking. This work contributes to establishing generally accepted algorithm based on open data and software, which is of major importance to reproduce the procedures, and to further improve and develop methods.
## References
1. 1.
Matsumura, K., Rolfe, P. & Yamakoshi, T. iPhysioMeter: a smartphone photoplethysmograph for measuring various physiological indices. Methods Mol. Biol. 1256, 305–326. https://doi.org/10.1007/978-1-4939-2172-0_21 (2015).
2. 2.
Krivoshei, L. et al. Smart detection of atrial fibrillation. Europace 19, 753–757. https://doi.org/10.1093/europace/euw125 (2017).
3. 3.
Castaneda, D., Esparza, A., Ghamari, M., Soltanpur, C. & Nazeran, H. A review on wearable photoplethysmography sensors and their potential future applications in health care. Int. J. Biosens. Bioelectron. 4, 195–202. https://doi.org/10.15406/ijbsbe.2018.04.00125 (2018).
4. 4.
Li, K. H. C. et al. The current state of mobile phone apps for monitoring heart rate, heart rate variability, and atrial fibrillation: narrative review. JMIR Mhealth Uhealth 15, e11606. https://doi.org/10.2196/11606 (2019).
5. 5.
Chan, P. H. et al. Diagnostic performance of a smartphone-based photoplethysmographic application for atrial fibrillation screening in a primary care setting. J. Am. Heart Assoc. 5, 27444506. https://doi.org/10.1161/JAHA.116.003428 (2016).
6. 6.
De Ridder, B., Van Rompaey, B., Kampen, J. K., Haine, S. & Dilles, T. Smartphone apps using photoplethysmography for heart rate monitoring: meta-analysis. JMIR Cardio 2, e4. https://doi.org/10.2196/cardio.8802 (2018).
7. 7.
Jonas, D. E. et al. Screening for atrial fibrillation with electrocardiography: an evidence review for the u.s. preventive services task force. JAMA 320, 485–498. https://doi.org/10.1001/jama.2018.419 (2018).
8. 8.
Sudler & Hennessey. Heart For Heart. Websitehttp://www.heartrateapp.com/ (2020).
9. 9.
Happitech. Monitor your heart rhythm using only a smartphone. Smartphone Apphttp://www.happitech.com (2020).
10. 10.
Ghebre, Y. T., Yakubov, E., Wong, W. T. & Krishnamurthy, P. Vascular aging: implications for cardiovascular disease and therapy. Transl. Med. 06, 183. https://doi.org/10.4172/2161-1025.1000183 (2016).
11. 11.
Jani, B. & Rajkumar, C. Ageing and vascular ageing. Postgr. Med. J. 82, 357–362. https://doi.org/10.1136/pgmj.2005.036053 (2006).
12. 12.
North, B. J. & Sinclair, D. A. The intersection between aging and cardiovascular disease. Circul. Res. 110, 1097–1108. https://doi.org/10.1161/CIRCRESAHA.111.246876 (2012).
13. 13.
Laina, A., Stellos, K. & Stamatelopoulos, K. Vascular ageing: underlying mechanisms and clinical implications. Exp. Gerontol. 109, 16–30. https://doi.org/10.1016/j.exger.2017.06.007 (2018).
14. 14.
Nilsson, P. M. et al. Characteristics of healthy vascular ageing in pooled population-based cohort studies: the global metabolic syndrome and artery research consortium. J. Hypertens. 36, 2340–2349. https://doi.org/10.1097/HJH.0000000000001824 (2018).
15. 15.
Yousef, Q., Reaz, M. B. & Ali, M. A. The analysis of PPG morphology: investigating the effects of aging on arterial compliance. Meas. Sci. Rev. 12, 266–271. https://doi.org/10.2478/v10048-012-0036-3 (2012).
16. 16.
Pilt, K. et al. New photoplethysmographic signal analysis algorithm for arterial stiffness estimation. Sci. World J. 169035, https://doi.org/10.1155/2013/169035 (2013).
17. 17.
Ahn, J. M. New aging index using signal features of both photoplethysmograms and acceleration plethysmograms. Healthc. Informati. Res. 23, 53–59. https://doi.org/10.4258/hir.2017.23.1.53 (2017).
18. 18.
Tamura, T., Maeda, Y., Sekine, M. & Yoshida, M. Wearable photoplethysmographic sensors–past and present. Electronics 3, 282–302. https://doi.org/10.3390/electronics3020282 (2014).
19. 19.
van Gent, P., Farah, H., van Nes, N. & van Arem, B. HeartPy: HeartPy: a novel heart rate algorithm for the analysis of noisy signals. Transp. Res. F. Traffic Psychol. Behav. 66, 368–378. https://doi.org/10.1016/j.trf.2019.09.015 (2019).
20. 20.
Tang, S.-C. et al. Identification of atrial fibrillation by quantitative analyses of fingertip photoplethysmogram. Sci. Rep. 7, 45644. https://doi.org/10.1038/srep45644 (2017).
21. 21.
Hoerl, A. E. & Kennard, R. W. Ridge regression: biased estimation for nonorthogonal problems. Technometrics 12, 55–67 (1970).
22. 22.
Hastie, T., Tibshirani, R. & Friedman, J. H. J. H. The Elements of Statistical Learning :Data Mining, Inference, and Prediction (Springer, Berlin, 2001).
23. 23.
Ismail Fawaz, H., Forestier, G., Weber, J., Idoumghar, L. & Muller, P.-A. Deep learning for time series classification: a review, https://doi.org/10.1007/s10618-019-00619-1 (2019).
24. 24.
Poh, M.-Z. et al. Diagnostic assessment of a deep learning system for detecting atrial fibrillation in pulse waveforms. Heart 104, 1921–1928. https://doi.org/10.1136/heartjnl-2018-313147 (2018).
25. 25.
van den Oord, A. et al. WaveNet: a generative model for raw audio. Arxivhttp://dx.doi.org/10.6084/m9.figshare.853801 (2016).
26. 26.
Anaconda. Anaconda Software Distribution. Softwarehttp://www.anaconda.com (2016).
27. 27.
Hamczyk, M. R., Nevado, R. M., Barettino, A., Fuster, V. & Andrés, V. Biological Versus Chronological Aging: JACC Focus Seminar. J. Am. College Cardiol. 75, 919–930. https://doi.org/10.1016/j.jacc.2019.11.062 (2020).
28. 28.
Cole, J. H. & Franke, K. Predicting age using neuroimaging: innovative brain ageing biomarkers. Trends Neurosci. 40, 681–690. https://doi.org/10.1016/j.tins.2017.10.001 (2017).
29. 29.
Remondini, D. et al. Identification of a T cell gene expression clock obtained by exploiting a MZ twin design. Sci. Rep. 7, 1–8. https://doi.org/10.1038/s41598-017-05694-2 (2017).
30. 30.
Gonzalez-Covarrubias, V. et al. Lipidomics of familial longevity. Aging Cell 12, 426–434. https://doi.org/10.1111/acel.12064 (2013).
31. 31.
Beekman, M. et al. Classification for longevity potential: the use of novel biomarkers. Front. Public Health 4, 28. https://doi.org/10.3389/FPUBH.2016.00233 (2016).
32. 32.
Kwon, S. et al. Deep learning approaches to detect atrial fibrillation using photoplethysmographic signals: algorithms development study. JMIR Mhealth Uhealth 21, e12770. https://doi.org/10.2196/12770 (2019).
33. 33.
Bashar, S. K. et al. Atrial fibrillation detection from wrist photoplethysmography signals using smartwatches. Sci. Rep. 9, 15054. https://doi.org/10.1038/s41598-019-49092-2 (2019).
34. 34.
Merz, A. A. & Cheng, S. Sex differences in cardiovascular ageing. Heart 102, 825–831. https://doi.org/10.1136/heartjnl-2015-308769 (2016).
35. 35.
Biswas, D. et al. CorNET: deep learning framework for PPG-based heart rate estimation and biometric identification in ambulant environment. IEEE Trans. Biomed. Circuits Syst. 13, https://doi.org/10.1109/TBCAS.2019.2892297 (2019).
36. 36.
Pecchia, L., Castaldo, R., Montesinos, L. & Melillo, P. Are ultra-short heart rate variability features good surrogates of short-term ones? State-of-the-art review and recommendations. Healthc. Technol. Lett. 5, 94–100. https://doi.org/10.1049/htl.2017.0090 (2018).
## Acknowledgements
This work has received support from the EU/EFPIA Innovative Medicines Initiative 2 Joint Undertaking BigData@Heart grant (116074), and from the European Union’s Horizon 2020 research and innovation programme IMforFUTURE under H2020-MSCA-ITN grant agreement number 721815. Data for this study come from the Heart for Heart (H4H) initiative. Permission was obtained to use data for this study, and Happitech has shared research data. F.W.A. is supported by UCL Hospitals NIHR Biomedical Research Centre.
## Author information
Authors
### Contributions
H.W.U., G.C., and F.W.A. conceived and designed the study. L.D. and H.W.U. analyzed and interpreted the data, and drafted the manuscript. N.C. and D.R. helped with the initial part of the code for the PPG data preprocessing. All authors reviewed the manuscript.
### Corresponding author
Correspondence to Hae-Won Uh.
## Ethics declarations
### Competing interests
Y.S.H. is a shareholder at Happitech. The other authors declare that they have no competing interests.
### Publisher's note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
## Rights and permissions
Reprints and Permissions
Dall’Olio, L., Curti, N., Remondini, D. et al. Prediction of vascular aging based on smartphone acquired PPG signals. Sci Rep 10, 19756 (2020). https://doi.org/10.1038/s41598-020-76816-6
• Accepted:
• Published:
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# sylvainhalle / textidote
Spelling, grammar and style checking on LaTeX documents
Java TeX HTML Shell
Fischer96 and sylvainhalle Added CI argument to ignore the issue count as return code (#113)
* Removed SystemExit call in for main
* Revert last commit, added ci argument
Latest commit 204a7af Dec 14, 2019
Type Name Latest commit message Commit time
Failed to load latest commit information.
Completions Aug 25, 2018
Source Dec 14, 2019
docs Aug 3, 2019
.gitignore Jul 26, 2019
.travis.yml Jul 8, 2018
LICENSE Jun 11, 2018
Readme.md Aug 5, 2019
build.xml Jan 6, 2019
config.xml Aug 6, 2019
dictionary.txt Jul 2, 2018
example.tex Jun 19, 2018
mymap.txt Jun 14, 2018
sonar-project.properties Nov 6, 2019
# TeXtidote: a correction tool for LaTeX documents and other formats
Have you ever thought of using a grammar checker on LaTeX files?
If so, you probably know that the process is far from simple. Since LaTeX documents contain special commands and keywords (the so-called "markup") that are not part of the "real" text, you cannot run a grammar checker directly on these files: it cannot tell the difference between markup and text. The other option is to remove all this markup, leaving only the "clear" text; however, when a grammar tool points to a problem at a specific line in this clear text, it becomes hard to retrace that location in the original LaTeX file.
TeXtidote solves this problem; it can read your original LaTeX file and perform various sanity checks on it: for example, making sure that every figure is referenced in the text, enforcing the correct capitalization of titles, etc. In addition, TeXtidote can remove markup from the file and send it to the Language Tool library, which performs a verification of both spelling and grammar in a dozen languages. What is unique to TeXtidote is that it keeps track of the relative position of words between the original and the "clean" text. This means that it can translate the messages from Language Tool back to their proper location directly in your source file.
You can see the list of all the rules checked by TeXtidote at the end of this file.
TeXtidote also supports spelling and grammar checking of files in the Markdown format.
## Getting TeXtidote
You can either install TeXtidote by downloading it manually, or by installing it using a package.
### Under Debian systems: install package
Under Debian systems (Ubuntu and derivatives), you can install TeXtidote using dpkg. Download the latest .deb file in the Releases page; suppose it is called textidote_X.Y.Z_all.deb. You can install TeXtidote by typing:
$sudo apt-get install ./textidote_X.Y.Z_all.deb The ./ is mandatory; otherwise the command won't work. ### Manual download You can also download the TeXtidote executable manually: this works on all operating systems. Simply make sure you have Java version 8 or later installed on your system. Then, download the latest release of TeXtidote; put the JAR in the folder of your choice. ## Using TeXtidote TeXtidote is run from the command line. The TeXtidote repository contains a sample LaTeX file called example.tex. Download this file and save it to the folder where TeXtidote resides. You then have the choice of producing two types of "reports" on the contents of your file: an "HTML" report (viewable in a web browser) and a "console" report. ### HTML report To run TeXtidote and perform a basic verification of the file, run: java -jar textidote.jar --output html example.tex > report.html In Linux, if you installed TeXtidote using apt-get, you can also call it directly by typing: textidote --output html example.tex > report.html Here, the --output html option tells TeXtidote to produce a report in HTML format; the > symbol indicates that the output should be saved to a file, whose name is report.html. TeXtidote will run for some time, and print: TeXtidote v0.8 - A linter for LaTeX documents (C) 2018-2019 Sylvain Hallé - All rights reserved Found 23 warnings(s) Total analysis time: 2 second(s) Once the process is over, switch to your favorite web browser, and open the file report.html (using the File/Open menu). You should see something like this: As you can see, the page shows your original LaTeX source file, where some portions have been highlighted in various colors. These correpond to regions in the file where an issue was found. You can hover your mouse over these colored regions; a tooltip will show a message that describes the problem. If you don't write any filename (or write -- as the filename), TeXtidote will attempt to read one from the standard input. ### Plain report To run TeXtidote and display the results directly in the console, simply omit the --output html option (you can also use --output plain), and do not redirect the output to a file: java -jar textidote.jar example.tex TeXtidote will analyze the file like before, but produce a report that looks like this: * L25C1-L25C25 A section title should start with a capital letter. [sh:001] \section{a first section} ^^^^^^^^^^^^^^^^^^^^^^^^^ * L38C1-L38C29 A section title should not end with a punctuation symbol. [sh:002] \subsection{ My subsection. } ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ * L15C94-L15C99 Add a space before citation or reference. [sh:c:001] things, like a citation\cite{my:paper} .The text Each element of the list corresponds to a "warning", indicating that something in the text requires your attention. For each warning, the position in the original source file is given: LxxCyy indicates line xx, column yy. The warning is followed by a short comment describing the issue, and an excerpt from the line in question is displayed. The range of characters where the problem occurs is marked by the "^^^^" symbols below the text. Each of these warnings results from the evaluation of some "rule" on the text; an identifier of the rule in question is also shown between brackets. ### Single line report Another option to display the results directly in the console is the single line report: java -jar textidote.jar --output singleline example.tex Textidote will analyze the file like before, but this time the report looks like this: example.tex(L25C1-L25C25): A section title should start with a capital letter. "\section{a first section}" example.tex(L38C1-L38C29): A section title should not end with a punctuation symbol. "\subsection{ My subsection. }" example.tex(L15C94-L15C99): Add a space before citation or reference. "things, like a citation\cite{my:paper} .The text" Each line corresponds to a warning, and is parseable by regular expressions easily, e.g., for further processing in another tool. The file is given at the beginning of the line, followed by the position in parentheses. Then, the warning message is given, and the excerpt causing the warning is printed in double quotes (""). Note, that sometimes it may happen that a position cannot be determined. In this case, instead of LxxCyy, ? is printed. ### Spelling, grammar and style You can perform further checks on spelling and grammar, by passing the --check option at the command line. For example, to check text in English, you run: java -jar textidote.jar --check en example.tex The --check parameter must be accompanied by a two-letter code indicating the language to be used. Language Tool is a powerful library that can verify spelling, grammar, and even provide suggestions regarding style. TeXtidote simply passes a cleaned-up version of the LaTeX file to Language Tool, retrieves the messages it generates, and coverts the line and column numbers associated to each message back into line/column numbers of the original source file. For more information about the kind of verifications made by Language Tool, please refer to its website. The language codes you can use are: • de: (Germany) German, and the variants de_AT (Austrian) and de_CH (Swiss) • en: (US) English, and the variants en_CA (Canadian) and en_UK (British) • es: Spanish • fr: French • nl: Dutch • pt: Portuguese ### Using a dictionary If you have a list of words that you want TeXtidote to ignore when checking spelling, you can use the --dict parameter to specify the location of a text file: java -jar textidote.jar --check en --dict dico.txt example.tex The file dico.txt must be a plain text file contain a list of words to be ignored, with each word on a separate line. (The list is case sensitive.) If you already spell checked you file using Aspell and saved a local dictionary (as is done for example by the PaperShell environment), TeXtidote can automatically load this dictionary when invoked. More specifically, it will look for a file called .aspell.XX.pws in the folder where TeXtidote is started (this is the filename Aspell gives to local dictionaries). The characters XX are to be replaced with the two-letter language code. If such a file exists, TeXtidote will load it and mention it at the console: Found local Aspell dictionary ### Ignoring rules You may want to ignore some of TeXtidote's advice. You can do so by specifying rule IDs to ignore with the --ignore command line parameter. For example, the ID of the rule "A section title should start with a capital letter" is sh:001 (rule IDs are shown between brackets in the reports given by TeXtidote); to ignore warnings triggered by this rule, you call TeXtidote as follows: java -jar textidote.jar --ignore sh:001 myfile.tex If you want to ignore multiple rules, separate their IDs with a comma (but no space). ### Ignoring environments TeXtidote can be instructed to remove user-specified environments using the --remove command line parameter. For example: $ java -jar textidote.jar --remove itemize myfile.tex
This command will remove all text lines between \begin{itemize} and \end{itemize} before further processing the file.
### Ignoring macros
The same can be done with macros:
$java -jar textidote.jar --remove-macros foo myfile.tex This command will remove all occurrences of use-defined command \foo in the text. Alternate syntaxes like \foo{bar} and \foo[x=y]{bar} are also recognized and deleted. ### Reading a sub-file By default, TeXtidote ignores everything before the \begin{document} command. If you have a large document that consists of multiple included LaTeX "sub-files", and you want to check one such file that does not contain a \begin{document}, you must tell TeXtidote to read all the file using the --read-all command line option. Otherwise, TeXtidote will ignore the whole file and give you no advice. TeXtidote also automatically follows sub-files that are embedded from a main document using \input{filename} and \include{filename} (braces are mandatory). Any such non-commented instruction will add the corresponding filename to the running queue. If you want to exclude an \input from being processed, you must surround the line with ignore begin/end comments (see below, Helping TeXtidote). ### Removing markup You can also use TeXtidote just to remove the markup from your original LaTeX file. This is done with the option --clean: java -jar textidote.jar --clean example.tex By default, the resulting "clean" file is printed directly at the console. To save it to a file, use a redirection: java -jar textidote.jar --clean example.tex > clean.txt You will see that TeXtidote performs a very aggressive deletion of LaTeX markup: • All figure, table and tabular environments are removed • All equations are removed • All inline math expressions ($...$) are replaced by "X" • All \cite commands are replaced by "0" • All \ref commands are replaced by "[0]" • Commands that alter text (\textbf, \emph, \uline, \footnote) are removed (but the text is kept) • Virtually all other commands are simply deleted Surprisingly, the result of applying these modifications is a text that is clean and legible enough for a spelling or grammar checker to provide sensible advice. As was mentioned earlier, TeXtidote keeps a mapping between character ranges in the "cleaned" file, and the same character ranges in the original LaTeX document. You can get this mapping by using the --map option: java -jar textidote.jar --clean --map map.txt example.tex > clean.txt The --map parameter is given the name of a file. TeXtidote will put in this file the list correspondences between character ranges. This file is made of lines that look like this: L1C1-L1C24=L1C5-L128 L1C26-L1C28=L1C29-L1C31 L2C1-L2C10=L3C1-L3C10 ... The first entry indicates that characters 1 to 24 in the first line of the clean file correspond to characters 5 to 28 in the first line of the original LaTeX file --and so on. This mapping can have "holes": for example, character 25 line 1 does not correspond to anything in the original file (this happens when the "cleaner" inserts new characters, or replaces characters from the original file by something else). Conversely, it is also possible that characters in the original file do not correspond to anything in the clean file (this happens when the cleaner deletes characters from the original). ### Using a configuration file If you need to run TeXtidote with many command line arguments (for example: you load a local dictionary, ignore a few rules, apply replacements, etc.), it may become tedious to invoke the program with a long list of arguments every time. TeXtidote can be "configured" by putting those arguments in a text file called .textidote in the directory from which it is called. Here is an example of what such a file could contain: --output html --read-all --replace replacements.txt --dict mydict.txt --ignore sh:001,sh:d:001 --check en mytext.tex As you can see, arguments can be split across multiple lines. You can then call TeXtidote without any arguments like this: textidote > report.html If you call TeXtidote with command line arguments, they will be merged with whatever was found in .textidote. You can also tell TeXtidote to explicitly ignore that file and only take into account the command line arguments using the --no-config argument. ### Markdown input TeXtidote also supports files in the Markdown format. The only difference is that rules specific to LaTeX (references to figures, citations) are not evaluated. Simply call TeXtidote with a Markdown input file instead of a LaTeX file. The format is auto-detected by looking at the file extension. However, if you pass a file through the standard input, you must tell TeXtidote that the input file is Markdown by using the command line parameter --type md. Otherwise, TeXtidote assumes by default that the input file is LaTeX. ## Helping TeXtidote It order to get the best results when using TeXtidote, it is advisable that you follow a few formatting conventions when writing your LaTeX file: • Avoid putting multiple \begin{envionment} and/or \end{environment} on the same line • Keep the arguments of a command on a single line. Commands (such as \title{}) that have their opening and closing braces on different lines are not recognized by TeXtidote and will result in garbled output and nonsensical warnings. • Do not hard-wrap your paragraphs. It is easier for TeXtidote to detect paragraphs if they have no hard carriage returns inside. (If you need word wrapping, it is preferrable to enable it in your text editor.) • Put headings like \section or \paragraph alone on their line and separate them from the text below by a blank line. As a rule, it is advisable to first see what your text looks like using the --clean option, to make sure that TeXtidote is performing checks on something that makes sense. If you realize that a portion of LaTeX markup is not handled properly and messes up the rest of the file, you can tell TeXtidote to ignore a region using a special LaTeX comment: % textidote: ignore begin Some weird LaTeX markup that TeXtidote does not understand... % textidote: ignore end The lines between textidote: ignore begin and textidote: ignore end will be handled by TeXtidote as if they were comment lines. ## Linux shortcuts To make using TeXtidote easier, you can create shortcuts on your system. Here are a few recommended tips. First, we recommend you create a folder called /opt/textidote and put the big textidote.jar file there (this requires root privileges). This step is already taken care of if you installed the TeXtidote package using apt-get. ### Command line shortcut (This step is not necessary if TeXtidote has been installed with apt-get.) In/usr/local/bin, create a file called textidote with the following contents: #! /bin/bash java -jar /opt/textidote/textidote.jar "$@"
Make this file executable by typing at the command line:
sudo chmod +x /usr/local/bin/textidote
(These two operations also require root previliges.) From then on, you can invoke TeXtidote on the command line from any folder by simply typing textidote, e.g.:
textidote somefile.tex
### Desktop shortcut
If you use a desktop environment such as Gnome or Xfce, you can automate this even further by creating a TeXtidote icon on your desktop. First, create a file called /opt/textidote/textidote.sh with the following contents, and make this file executable:
#! /bin/bash
dir=$(dirname "$1")
pushd $dir java -jar /opt/textidote/textidote.jar --check en --output html "$@" > /tmp/textidote.html
popd
sensible-browser /tmp/textidote.html &
This script enters into the directory of the file passed as an argument, calls TeXtidote, sends the HTML report to a temporary file, and opens the default web browser to show that report.
Then, on your desktop (typically in your ~/Desktop folder), create another file called TeXtidote.desktop with the following contents:
[Desktop Entry]
Version=1.0
Type=Application
Name=TeXtidote
Comment=Check text with TeXtidote
Exec=/opt/textidote/textidote-desktop.sh %F
Icon=/opt/textidote/textidote-icon.svg
Path=
Terminal=false
StartupNotify=false
This will create a new desktop shortcut; make this file executable. From then on, you can drag LaTeX files from your file manager with your mouse and drop them on the TeXtidote icon. After the analysis, the report will automatically pop up in your web browser. Voilà!
### Tab completions
You can auto-complete the commands you type at the command-line using the TAB key (as you are probably used to). If you installed TeXtidote using apt-get, auto-completion for Bash comes built-in. You can also enable auto-completion for other shells as follows.
#### Zsh
Users of Zsh can also enable auto-completion; in your ~/.zshrc file, add the line
source /opt/textidote/textidote.zsh
(Create the file if it does not exist.) You must then restart your Zsh shell for the changes to take effect.
## Rules checked by TeXtidote
Here is a list of the rules that are checked on your LaTeX file by TeXtidote. Each rule has a unique identifier, written between square brackets.
### Language Tool
In addition to all the rules below, the --check xx option activates all the rules verified by Language Tool (more than 2,000 grammar and spelling errors). Note that the verification time is considerably longer when using that option.
If the --check option is used, you can add the --languagemodel xx option to find errors using n-gram data. In order to do so, xx must be a path pointing to an n-gram-index directory. Please refer to the LanguageTool page (link above) on how to use n-grams and what this directory should contain.
### Style
• A section title should start with a capital letter. [sh:001]
• A section title should not end with a punctuation symbol. [sh:002]
• A section title should not be written in all caps. The LaTeX stylesheet takes care of rendering titles in caps if needed. [sh:003]
• Use a capital letter when referring to a specific section, chapter or table: 'Section X'. [sh:secmag, sh:chamag, sh:tabmag]
• A (figure, table) caption should end with a period. [sh:capperiod]
### Citations and references
• There should be one space before a \cite or \ref command [sh:c:001], and no space after [sh:c:002].
• Do not use 'in [X]' or 'from [X]': the syntax of a sentence should not be changed by the removal of a citation. [sh:c:noin]
• Do not mix \cite and \citep or \citet in the same document. [sh:c:mix]
• When citing more than one reference, do not use multiple \cite commands; put all references in the same \cite. [sh:c:mul, sh:c:mulp]
### Figures
• Every figure should have a label, and every figure should be referenced at least once in the text. [sh:figref]
• Use a capital letter when referring to a specific figure: 'Figure X'. [sh:figmag]
### Structure
• A section should not contain a single sub-section. More generally, a division of level n should not contain a single division of level n+1. [sh:nsubdiv]
• The first heading of a document should be the one with the highest level. For example, if a document contains sections, the first section cannot be preceded by a sub-section. [sh:secorder]
• There should not be a jump down between two non-successive section levels (e.g. a \section followed by a \subsubsection without a \subsection in between). [sh:secskip]
• You should avoid stacked headings, i.e. consecutive headings wihtout text in between. [sh:stacked]
### Hard-coding
• Figures should not refer to hard-coded local paths. [sh:relpath]
• Do not refer to sections, figures and tables using a hard-coded number. Use \ref instead. [sh:hcfig, sh:hctab, sh:hcsec, sh:hccha]
• You should not break lines manually in a paragraph with \\. Either start a new paragraph or stay in the current one. [sh:nobreak]
• If you are writing a research paper, do not hard-code page breaks with \newpage. [sh:nonp]
### LaTeX subtleties
• Use a backslash or a comma after the last period in "i.e.", "e.g." and "et al."; otherwise LaTeX will think it is a full stop ending a sentence. [sh:010, sh:011]
• There should not be a space before a semicolon or a colon. If in your language, typographic rules require a space here, LaTeX takes care of inserting it without your intervention. [sh:d:005, sh:d:006]
### Potentially suspicious
• There should be at least N words between two section headings (currently N=100). [sh:seclen]
## Building TeXtidote
First make sure you have the following installed:
• The Java Development Kit (JDK) to compile. TeXtidote requires version 8 of the JDK (and probably works with later versions).
• Ant to automate the compilation and build process
Download the sources for TeXtidote from GitHub or clone the repository using Git:
git clone [email protected]:sylvainhalle/textidote.git
### Compiling
ant download-deps
Then, compile the sources by simply typing:
ant
This will produce a file called textidote.jar in the folder. This file is runnable and stand-alone, or can be used as a library, so it can be moved around to the location of your choice.
In addition, the script generates in the docs/doc folder the Javadoc documentation for using TeXtidote.
### Testing
TeXtidote can test itself by running:
ant test
Unit tests are run with jUnit; a detailed report of these tests in HTML format is availble in the folder tests/junit, which is automatically created. Code coverage is also computed with JaCoCo; a detailed report is available in the folder tests/coverage.
## About the author
TeXtidote was written by Sylvain Hallé, Full Professor in the Department of Computer Science and Mathematics at Université du Québec à Chicoutimi, Canada.
## Like TeXtidote?
TeXtidote is free software licensed under the GNU General Public License 3. It is released as postcardware: if you use and like the software, please tell the author by sending a postcard of your town at the following address:
Sylvain Hallé
Department of Computer Science and Mathematics
Univerité du Québec à Chicoutimi
555, boulevard de l'Université
Chicoutimi, QC
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# Associativity: Intuition
Associative functions can be interpreted as families of functions that reduce lists down to a single output by combining adjacent elements in any order. Alternatively, associativity can be seen as a generalization of “listyness,” which captures and generalizes the “it doesn’t matter whether you added b to c or a to c, the result is b, c regardless” aspect of lists.
There are many different ways for a function to be associative, so it is difficult to provide a single litmus test for looking at a function and telling whether it associates (aside from just checking the associative axiom directly). However, a few heuristics can be used to make a good guess.
# Associative operators as natural functions on lists
The generalized associative law says that an associative function $$f : X \times X \to X$$ gives rise to a method for combining any non-empty list of $$X$$ elements into a single output, where the order in which adjacent elements are combined doesn’t affect the result. We can flip that result around, and interpret associative operators as the pairwise versions of a certain class of “natural” functions for combining the elements of a list.
On this interpretation, we start by noting that some methods for reducing a list down to a single element can be broken down into pairwise combinations of adjacent elements, while others can’t. For example, when we’re trying to compute $$3 + 4 + 5 + 6,$$ we can pick any two adjacent elements and start by combining those using the binary version of $$+$$. But when we’re trying to compute adjacent_ones(0, 0, 1, 1, 0) to check whether the list has any two adjacent ones, we’re going to run into trouble if we only look for adjacent ones in the pairs (0, 0), (0, 1), and (1, 0).
The lists that can be reduced by pairwise combination of adjacent pairs have a nice locality property; the result can be computed only by looking at adjacent elements (without worrying about the global structure). Locality is a common idiom in physics and mathematics, so we might start by asking what sort of functions on lists have this locality property. The answer is “any list that can be broken down into pairwise combinations of elements such that the order doesn’t matter.” If we formalize that notion, we get the result that any function on lists with this locality property corresponds (in a one-to-one fashion) to an associative operation. Thus, we can view associativity as the mathematical formalization of this nice “locality” property on lists.
Empirically, this locality property turns out to be quite useful for math, physics, and in computer programming, as evidenced by the commonality of associative operators. See, for example, Group theory, or the pages on semigroups and monoids.
# Associativity as a generalization of “list”
The above interpretation gives primacy to lists, and interprets associative operators in terms of natural functions on lists. We can invert that argument by treating associativity as a generalization of what it means for something to act “list-like.”
A list $$[a, b, c, d, \ldots]$$ is a set of elements that have been combined by some “combiner” function, where the order of the elements matters, but the order in which they were combined does not matter. For example, if we combine $$a$$ with $$b$$ (forming $$[a, b]$$) and then combine that with $$c$$, then we get the same list as if we combine $$b$$ and $$c$$ into $$[b, c]$$ first, and then combine $$a$$ with that.
The very fact that we can unambiguously say “the list $$[a, b, c]$$” without worrying about the order that the elements were combined in means that lists are built out of an associative “combination” operator. On this interpretation, associativity is capturing part of the essence of listyness, and associativity in general generalizes this notion. For example, associative operators are allowed to be a little forgetful about what exact elements you combined (e.g., 3 + 4 = 2 + 5) so long as you retain the “it doesn’t matter what order you combine the things in” property. In other words, we can view associativity as “part of what it means to be list-like.”
(One particularly important property of lists — namely, that they can be empty — is not captured by associativity alone. Associative operators on sets that have an element that acts like an “empty list” are called “monoids.” For more on the idea of generalizing the notion of “list”, refer to the page on monoids.)
# Associative mechanisms
The above two interpretations give an intuition for what it means that a function is associative. This still leaves open the question of how a function can be associative. Imagine $$f : X \times X \to Y$$ as a physical mechanism of wheels and gears. Someone says “$f$ is associative.” What does that mean, in terms of the function’s physical mechanisms? What should we expect to see when we pop the function open, given the knowledge that it “is associative”?
Recall that associativity says that the two methods for combining two instantiations of the function yield the same output:
Thus, the ultimate physical test of associativity is hooking up two instantiations of $$f$$ as in the left diagram, and then checking whether dragging the mechanisms of the lower-right instantiation above the mechanisms of the upper-left instantiation (thereby reconfiguring the system according to the diagram on the right) causes the behavior of the overall system to change. What happens when the right-hand-side instantiation is given access to the middle input first versus second? Does that affect the behavior at all? If not, $$f$$ is associative.
This is not always an easy property to check by looking at the mechanisms of $$f$$ alone, and sometimes functions that appear non-associative (at first glance) turn out to be associative by apparent coincidence. In other words, there are many different ways for a function to be associative, so it is difficult to give a single simple criterion for determining associativity by looking at the internals of the function. However, we can use a few heuristics that help one distinguish associative functions from non-associative ones.
## Heuristic: Can two copies of the function operate in parallel?
$$f$$ is associative if, when using two copies of $$f$$ to reduce three inputs to one output, then changing whether the right-hand copy gets access to the middle tape first vs second does not affect the output. One heuristic for checking whether this is the case is to check whether both copies of $$f$$ can make use of the middle input at the same time, without getting in each other’s way. If so, $$f$$ is likely associative.
For example, consider an implementation of $$+$$ that gets piles of poker chips as input (where a pile of $$n$$ chips represents the number $$n$$) and computes the output by simply sweeping all the poker chips from its input belts onto its output belt. To make a function that adds three piles of chips together, you could set up two two-pile adders in the configuration of the diagram on the left, but you could also have two two-tape sweepers operating on three tapes in parallel, such that they both sweep the middle tape. This parallelization wouldn’t change the output, and thus $$+$$ is associative.
By contrast, consider a mechanism that takes wooden blocks as input, and glues them together, and nails silver-colored caps on either end of the glued block. For example, if you put in a red block on the left and a blue block on the right, you get a silver-red-blue-silver block in the output. You could set up two copies of these like the diagram on the left, but if you tired to parallelize them, you’d get into trouble — each mechanism would be trying to nail one of its caps into the place that the other mechanism was attempting to apply glue. And indeed, this mechanism is non-associative.
This heuristic is imperfect. Some mechanisms that seem difficult to parallelize are still associative. For example, consider the multiplier mechanism, which takes two poker piles as input and puts a copy of the left pile onto the output tape for every chip in the right pile. It would be difficult to parallelize two copies of this function: One would be trying to count the chips in the middle pile while the other was attempting to copy the chips in the middle pile, and the result might not be pretty. However, multiplication is associative, because a pile of $$x$$-many copies of a ($y$ copies of $$z$$)-many poker chips has the same number of chips as a pile of ($x$ copies $$y$$)-many copies of $$z$$-many poker chips.
## Heuristic: Does the output interpretation match both input interpretations?
Another (vaguer) heuristic is to ask whether the output of the function should actually be treated as the same sort of thing as the input to the function. For example, recall the adjacent_ones function from above, which checks a list for adjacent ones, and returns 1 if it finds some and 0 otherwise. The inputs to adjacent_ones are 0 and 1, and the output is 0 or 1, but the output interpretation doesn’t quite match the input interpretation: Intuitively, the output is actually intended to mean “yes there were adjacent ones” or “not here weren’t adjacent ones”, and so applying adjacent_ones to the output of adjacent_ones is possible but ill-advised. If there is a mismatch between the output interpretation and at least one of the input interpretations, then the function probably isn’t associative.
For example, imagine a person who is playing a game that works as follows. The board has three positions: red, green, and blue. The player’s objective is to complete as many clockwise red-green-blue cycles as possible, without ever backtracking in the counter-clockwise direction.
Each turn, the game offers them a choice of one of the three spaces, and they get to choose whether or not to travel to that square or stay where they are. Clearly, their preferences depend on where they currently are: If they’re on “red”, “green” is a good move and “blue” is a bad one; but if they’re on “blue” then choosing “green” is ill-advised. We can consider a binary function $$f$$ which takes their current position on the left and the proposed position on the right, and returns the position that the player prefers. For example, $$f(red,blue)=red,$$ $$f(red,green)=green,$$ $$f(blue,blue)=blue,$$ $$f(blue,green=blue).$$ In this case, the interpretation of the left input is a “player position,” the interpretation of the right input is an “offered move”, and the interpretation of the output is the resulting “player position.” The output interpretation mismatches one of the input interpretations, which implies that $$f$$ probably isn’t associative, and indeed it is not: $$f(f(red, green), blue))=blue,$$ whereas $$f(red, f(green, blue))=red.$$ The former expression can be interpreted as “where the player would be if they started at red, and were then offered green, and were then offered blue.” The latter expression doesn’t have a great interpretation, because it’s feeding the output of $$f(green, blue)$$ (a player position) in as an “offered move.”
If the interpretation of the output (in this case, “player position”) mismatches the interpretations of at least one of the inputs, then the function likely isn’t associative. However, this heuristic is also imperfect: The most obvious interpretations of the inputs and outputs to the subtraction function are “they’re all just numbers,” and subtraction still fails to associate.
## Further discussion
There are many different ways for a function to be associative, so it is difficult to give a simple litmus test. The ultimate test is always to imagine using two copies of $$f$$ to combine three outputs into one, and check whether the result changes depending on whether the left-hand copy of $$f$$ gets to run first (in which case it gets to access the second input belt at the source) or second (in which case its right-hand input is the right-hand copy’s output). For examples of functions that pass or fail this ultimate test, refer to the examples page.
• Well, we could imagine someone arguing that differences-between-numbers could be interpreted as a philosophically different entity than the numbers themselves: 3 − 5 = −2, but numbers like −2 which are in ℤ but not ℕ don’t have the same kind of easy interpretation as a number you would use to count physical objects with in the way natural numbers like 3 and 5 have.
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# How Is Server-Timing used on the web?
I was curious to see where Server-Timing was implemented on the web, so I started searching the HTTP Archive for sites using it. Interestingly enough, there were no sites in the HTTP Archive that had Server-Timing response headers before 3/1/2017. Since then it’s usage has been gradually increasing each month. As of July 2017, there are 72 sites and 352 HTTP responses containing Server-Timing headers.
Here’s a query that I wrote to find a list of sites with Server-Timing headers. You can run it on BigQuery here :
SELECT p.url, p.rank, count(*) as freq
FROM
( SELECT pageid, url, rank
FROM [httparchive:runs.latest_pages]
) p
JOIN
FROM [httparchive:runs.latest_requests]
) r
ON p.pageid = r.pageid
GROUP BY p.url, p.rank
ORDER BY freq DESC
Next I was curious to see the actual Server-Timing headers to understand how they are being used. I wrote this query to strip out the server-timing headers from the respOtherHeaders column. The query has a lot of string manipulations to extract the text. I’ve added some comments to the query to help make it easier to follow -
SELECT HOST(url), status, mimeType,
IF (
-- In respOtherHeaders, remove everything to the left of the server-timing header. Is there a ; in the remaining string?
-- Then strip out everything to the left of the server-timing header and everything to the right of the ";" character.
LEFT(
INSTR(
"server-timing")+1),
";")
),
-- If there are no ; chars, then the respOtherHeaders column ends with server-timing. Remove everything to the left of server-timing
"server-timing")+1)
FROM [httparchive:runs.latest_requests]
You can see it on BigQuery here
Looking at the Server-Timing headers, I can see 3 different types of data being sent in these headers:
Timing Metrics (such as application time, gateway time, image resize time, etc)
Server-Timing = app=133.99219512939;
Server-Timing = gateway-time=25;
Server-Timing = resizer=0.015;
server-timing = view=2.188, db=0.000, total=61.743, x-runtime = 0.069661, x-ua-compatible = ie=edge,chrome=1, x-frame-options = SAMEORIGIN
Server-Timing = app=53.393840789795;
Caching Information
server-timing = hit, cf-cache-status = HIT
server-timing = cold, cf-cache-status = MISS
server-timing = hit, cf-cache-status = HIT
server-timing = cold, cf-cache-status = MISS
Informational
server-timing = ibs_5318e9209631=1
server-timing = ibs_1964fd84c9cea3=1
server-timing = ibs_1880bf4b0f6311=1
Looking at all of the data in the 7/1/2017 tables, I can see that the current examples in the wild are split between these 3 types of data, although the caching ones are mostly on images, the informational ones are set on JS and responses w/ no content (ie, 204s -likely beacon responses), and the actual timing metrics are spread across a wide range of content.
It’ll be interesting to see how this evolves as Server-Timing adoption increases over time.
Originally published at https://discuss.httparchive.org/t/how-is-server-timing-used-on-the-web/1034
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Sales Toll Free No: 1-855-666-7446
# Percent Composition Calculator
Top
This online Percentage composition calculator determines the mass of each element present in the compound, expressed in percentage. To find the percent composition, we have to find the mass of each element then the total molar mass of the compound.
Percentage Composition Formula :
Percentage Composition = $\frac{Mass \ of \ each \ element}{Molecular \ mass \ of \ compound }$
## Steps
Step 1 : Determine the molecular mass of the compound by using the atomic mass given in the periodic table.
Step 2 : Then, determine the mass of each element. Then by using the percentage composition formula given above, substitute the values and get the percentage of particular element.
## Problems
Below are the problems based on percent composition .
### Solved Examples
Question 1: Determine Percentage composition of carbon in $CO_{2}$ .
Solution:
Step 1 : Given elements are C and O.
Atomic mass of Carbon C = 12.01 amu and Oxygen O = 16.00 amu.
Molecular mass of compound = 12.01 + 2 $\times$ 16.00
= 44.01 amu.
Step 2 : Mass of Carbon in $CO_{2}$ = 12.01 amu.
Percentage composition of Carbon in $CO_{2}$ is,
= $\frac{Mass \ of \ each \ element}{Molecular \ mass \ of \ compound }$ $\times$ 100 %
= $\frac{12.01 \ amu}{44.01 \ amu}$ $\times$ 100%
= 40 %.
Question 2: Calculate the Percentage composition of sodium and chlorine in $NaCl$ ?
Solution:
Step 1 : Given elements are Na and Cl.
Atomic mass of Sodium Na = 22.99 amu and Chlorine Cl = 35.45 amu.
Molecular mass of compound = 22.99 amu + 35.45 amu
= 58.44 amu.
Step 2 : Mass of Sodium in $NaCl$ = 22.99 amu.
Percentage composition of sodium in NaCl is,
= $\frac{Mass \ of \ each \ element}{Molecular \ mass \ of \ compound }$ $\times$ 100 %
= $\frac{22.99 \ amu}{58.44 \ amu}$ $\times$ 100%
= 39.34 %.
Mass of Chlorine in $NaCl$ = 35.45 amu.
Percentage composition of chlorine in NaCl is,
= $\frac{Mass \ of \ each \ element}{Molecular \ mass \ of \ compound }$ $\times$ 100 %
= $\frac{35.45 \ amu}{58.44 \ amu}$ $\times$ 100%
= 60.66 %.
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Disclaimer: This is an example of a student written essay.
Any scientific information contained within this essay should not be treated as fact, this content is to be used for educational purposes only and may contain factual inaccuracies or be out of date.
# Bioremediation of Halogenated Aliphatic Compounds
✅ Paper Type: Free Essay ✅ Subject: Sciences ✅ Wordcount: 6332 words ✅ Published: 8th Feb 2020
### Introduction
A Halogenated compound is a chemical compound in which one or more hydrogen atoms have been replaced by a halogen atom such as - fluorine, chlorine, and bromine (ILO, 2011). Due to their excellent ability to dissolve oils, fast evaporation rate, and chemical stability, they have been widely used in industrial, commercial, and agricultural fields. They are used as pesticides, plasticizers, paint and printing-ink components, textile auxiliaries, refrigerants, and flame retardants (Chaudhry and Chapalamadugu, 1991). They are also widely used for cleaning purposes as dry cleaning fluid, degreasing solvent, and ink and paint strippers.
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Despite their efficacy, halogenated compounds require additional attention and cost to pretreat before they can be disposed. They are incompatible with other waste because they are toxic, mutagenic, and carcinogenic (Chaudhry and Chapalamadugu, 1991), thus, posing a threat to environment and causing human health problems. Humans are usually exposed to halogenated compounds by oral inhalation or dermal contact, as they are prevalent in drinking water and groundwater. Once exposed to humans, they cause not only cancer but also effect on nervous system, and injury of vital organ, particularly liver (Stellman, 1998). Due to the detrimental effects of halogenated compounds, their use has been forbidden worldwide. Even trace amounts of halogenated compounds present in effluents are of concern.
Therefore, renowned researchers have been doing studies to degrade halogenated compounds, and they have found efficient and effective methods to remediate sites contaminated by halogenated compounds by utilizing certain bacteria. In this context, this paper intends to provide background knowledge about halogenated compounds and a selected compound to aid better understanding of this paper. In addition, this paper introduces bioremediation pathways used in selected articles and discusses their approach and conclusion of the studies, and demonstrate writer’s own critical thinking on the papers.
### Body
Halogenated compounds are composed of two major groups: halogenated aromatic compounds and halogenated aliphatic compounds. Most halogenated aliphatic compounds are chemically stable, and lack a benzene ring typical in halogenated aromatic compounds (Flowers, 2017). This paper mainly focuses on halogenated aliphatic compound, and its biodegradation pathway. Therefore, the sections in this paper intend to provide general information about aliphatic compounds, and introduce proposed degradation pathways by utilizing certain bacteria on selected compounds from the articles. Furthermore, the following sections scrutinize the approach and conclusion on published papers in order to demonstrate depth of author’s understanding about dehalonization - process of replacing halogen atoms to hydrogen atoms - pathway. In the end of this paper, it reveals writer’s independent assessments on selected papers.
### I.Properties of Aliphatic Compounds
All of aliphatic compounds are divided into two categories; saturated aliphatic compounds and unsaturated aliphatic compounds. Saturated aliphatic compounds known as alkanes only contain single bonds. In contrast, unsaturated aliphatic compounds contains multiple double bonds, known as alkenes, and triple bonds, known as alkynes (Abozenadah et al., 2017). Both saturated and unsaturated aliphatic compounds are entirely linked carbon and hydrogen atoms, and contained straight-chain. Once hydrogen atoms are replaced by halogen atoms, it is called halogenated aliphatic compound. Moreover, unsaturated aliphatic compounds are linked less hydrogen atoms than saturated aliphatic compounds, therefore, this property makes alkenes and alkynes relatively reactive (Flowers, 2017).
Figure 1. Skeletal structure of Saturated Alkane and Unsaturated Alkene
Aliphatic compounds will react differently with halogen atoms based on their physical feature, therefore, comprehending their physical feature is very important. The physical features of alkane and alkene are relatively similar. The boiling point of alkane and alkene increase with increasing molecule weight. They are insoluble in water, however, they are soluble in organic solvents.
Name Molecular Formula Molecular Weight (g/mol) Melting Point (${\mathbf{℃}}$) Boiling Point (${\mathbf{℃}}$) methane CH4 16 -182 -162 ethane C2H6 30 -183 -89 propane C3H8 44 -188 -42 butane C4H10 58 -138 -0.5 pentane C5H12 72 -130 36 hexane C6H14 86 -95 69 heptane C7H16 100 -91 98
Table 1. Physical Properties of Alkane Group (Mehendale, 2017)
Name Molecular Formula Molecular Weight (g/mol) Melting Point (${\mathbf{℃}}$) Boiling Point (${\mathbf{℃}}$) ethene C2H4 28 -169 -104 propene C3H6 42 -185 -47 1-butene C4H8 56 -185 -6 1-pentene C5H10 70 -138 30 1-hexene C6H12 84 -140 63 1-heptene C7H14 98 -119 94
Table 2. Physical Properties of Alkene Group (Abozenadah et al., 2017)
### 1.Properties of Chloroethenes
Chloroethenes, unsaturated aliphatic compounds contained double bonds, are mainly used in the industrial fields as paint remover, solvent, and chemical intermediate for fibers and pesticides productions. They were produced about approximately 20 to 25 million tonnes/year in late 80s, and 90% of the total production was vinyl chloride (Agteren et al., 1998).
Chloroethene Vinyl Chloride (VC) Trichloroethene (TCE)
Figure 3. Chemical structure of Chloroethene and selected Chloroethene compounds
(Callahan et al., 1979, Agteren et al., 1998)
They are transferred into atmospheric environment through volatilization, and once they are emitted into atmosphere, they are photolyzed with certain wavelength. World Health Organization (WHO) informs half-life, required time to be half of the substances, of vinyl chloride is about 20 hours, and they are the most unstable compound among chloroethene.
Name Molecular weight (g/mol) Melting Point (${\mathbf{℃}}$) Boiling Point (${\mathbf{℃}}$) Density at 20${\mathbf{℃}}$ tetrachloroethene (PCE) 165.8 -22.7 121.3 1.62 trichloroethene (TCE) 131.4 -73 86.7 1.46 cis-1,2-dichloroethene (cis-1,2-DCE) 96.95 -81 60 1.28 trans-1,2-dichloroethene (trans-1,2-DCE) 96.95 -50 47.5 1.26 1,1-dicholoroethene (1,1-DCE) 96.95 -122.5 31.9 1.218 vinyl chloride (VC) 62.5 -153.8 -13.37 0.9121
Table 3. Physical Properties of Alkene Group (Agteren et al., 1998)
Tri- and perchloroethylene are the most prevailingly used in varied field. Chloroethene compound are persistence, therefore, they can be detected in groundwater under manufacture sites decades as a result even after introduction of those compounds have stopped (Agteren et al., 1998).
### 2.Natural bioremediation pathway of Chloroethenes compound
Several steps of PCE dechlorination were taken happens naturally under anaerobic condition - oxygen absence and other terminal electron acceptor are presented condition, and this section discuss about natural bioremediation pathway of PCE compounds suggested by Agterne et al. (1998).
The most significant step of chloroethene conversion is chloride atoms naturally replace to hydrogen. Tetrachloroethene (PCE) is reductively dehalorinated to trichloroethene (TCE) and 1,2-dichloroethene (DCE, mainly cis-1,2-DCE). In addition, 1,1-dicholoroethene (1,1-DCE) is reduced to vinyl chloride (VC). VC, which is end product proposed by Agrerne et al. (1998), is the most toxicity compound, and it is not naturally degraded to ethane or ethane. Therefore, it is easily accumulated at contaminated site.
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The reference covered anaerobic biodegradation is relatively outdated resource, nevertheless, it includes comprehensive as well as well-structured mechanisms about entire biodegradation pathway with providing chemical structure of all compounds on the pathway to aid better understanding. In addition, it clearly reveals limitation of natural bioremediation under anaerobic condition, which is end product under anaerobic condition with most toxic cannot achieve further degrade naturally. Though this contains constructive pathways of nature biodegradation, it does not proposed further research should take implementation in the future for dealing with VC residue under anaerobic condition. Furthermore, they were missed further VC biodegradation by utilizing certain bacteria proposed by Maymó-Gatell et al. (1997). Including further research to remediate residue VC compounds makes this reference more reliable resource to be foundation stone of PCE bioremediation.
### 3.Bioremediation pathways of Chloroethene with certain microorganisms
This section includes three prominent research papers about Chloroethene bioremediation pathways in varied time (I was trying to describe first paper was published in 1997, second paper was published in 2008, and last paper was published in this year). It helps to understand how researchers develop their work to determine the overall pathway of Chloroethene degradation. In addition, all three papers used with different types of bacterium or bacteria consortia, therefore, it is easy to evaluate its effectiveness and reliability of their paper as well as their experimental design and conclusion of their paper.
### 3-1) Isolation of a Bacterium that Reductively Dechlorinates Tetrachloroethene to Ethene
Maymó-Gatell et al. (1997) suggest that PCE can achieve complete dechlorination by natural microbial community and mixed microbial enrichment culture under anaerobic condition. They assumed isolated bacteria utilized hydrogen as electron donor and PCE as an electron acceptors for their growth. Therefore, when hydrogen is consumed, chloride supposed to be produced, and irregular coccus and a short rod bacteria are presented under the same condition, they can conclude those bacteria can dechlorination PCE products. They drove their result by stoichiometric comparison between hydrogen consumption and chlorine elimination. In order to avoid competition to consume hydrogen with methanogens and acetogen, they only put methanol-PCE and hydrogen with a mixture called ABSS [2 mM acetate, 0.05 mg of vitamin B12 per liter and 25% anaerobic digester sludge supernatant]. In addition, they only can observe the morphology of the stain under medium containing vancomycin (100mg/liter) or ampicillin (3g/liter), and those are the condition for PCE dechlorination. Under contained 20 mg/liter of tetracyline (a eubacterial protein-synthesis inhibitor) medium, PCE dechlorination wouldn’t take happened.
Figure 4. Proposed biodegradation pathway from PCE to ETH by strain 195
(Maymó-Gatell et al., 1997)
Due to strain 195 is classified into eubacteria, typically don’t have the cell wall, unlike Bacteria and Archaea do, certain nutrients - cholesterol and horse serum - require by mycoplasmas. In order to get quantitative information of bacterium, they measured the strain by direct microscopic cell counts and cell protein during metabolite pathway from PCE to VC and ethene. In addition, they found strain only grow about 5 days with 19.2 doubling times. However, after they cease to grow, they still can PCE dechlorination, and 90% of the PCE were used VC and ethane production. Furthermore, degradation pathway from VC to ethene or ethane only can initiate after PCE depletion, and the rate of conversion is faster than conversion rate from methanol-PCE to VC. Strain 195 only grow condition under hydrogen and PCE present. In order to determine morphology of the strain, they examined it through the electron microscopic method, and they found that the strain is irregular coccoid cells with the unusual cell wall, which resembled with S-layer protein typically Archaea have. In order to distinguish the presence of peptidoglycan cell wall, they tested it by fluorescently labeled wheat germ agglutinin. This binds to N-acetyl-glucosamine and N-acetylneuraminic acid, which gram-positive eubacterium Clostridium pasteurianum WF and gram-negative eubacterium E.coli DH5ɑ have.
Figure 5. Phylogenetic tree by 16S ribosomal DNA sequence and classification of strain 195 (Maymó-Gatell et al., 1997)
As a result, there is no binding on whole cell of stain, therefore, it is impossible to determine whether the existence of the cell wall with fluorescently labeled wheat germs agglutinin. To classify its phylogenetic position, it was tested by 16S ribosomal DNA sequence. Even strain didn’t cluster within any other phylogenetic lines, the branch of 195 strain is included cyanobacteria and planctomycetes. Moreover, strain shares high similarity with Clostridium butyricum with closer DNA distance. However, strain has few affiliations for other gram-positive branch. Therefore, they just classified strain 195 into eubacterial branches at that time.
Experimental design used in this research was not included uncertainty to compete for hydrogen with methanogen and acetogens. Even they knew the mechanism of microbial consortia, it cannot be assumed the correlation of coexistence between strain 195 and methanogens and acetogens through their research. In addition, there are other microbial group can utilize hydrogen as electron donor for their metabolite - sulfur oxidizer, nitrogen oxidizer, and iron oxidizer.
In order to get accurate information about rate of conversion, other microbial group should be included in their experiment design. With lacking of sufficient information on clone library, researchers couldn’t elucidate classification of strain 195 at that time, and they were put strain 195 into eubacterial branches based on DNA distance with vogue. Now we all know strain 195 is Dehalococcoides mccartyi strain 195, which contains multiple dehalogenase genes, therefore, it has the capacity to achieve complete dechlorination from PFC to VC. Furthermore, this study was used mixture called ABSS. However, this paper doesn’t reveal specific reason and role of being used ABSS in the experiment. It makes vague to me whether it was used as solvent for halogenated compound, or created a suitable environment for microbes for enrichment experiment.
### 3-2) Adaptation of aerobic, Ethene-Assimilating Mycobacterium Strains to Vinyl Chloride as a Growth substrate
Although researchers found several VC-assimilating bacteria, such as Dehalococcoides spp. under anaerobic condition, application have not been implemented in real world. In addition, microbes can easily enrich under aerobic - oxygen presents and plays a role as electron acceptor in reaction - condition, therefore, this study was conducted to test possibility of utilization VC as a growth substrate by currently isolated ethene-assimilating Mycobacterium strains. VC and ethene transform into VC epoxide (chlorooxirane) and epoxyethane (ethylene oxide) with alkene monooxygenase (AkMO) enzyme, therefore, all the microorganisms, which contained AkMO enzyme, can play a role as VC-assimilating bacteria. In addition, VC epoxide can reach further degradation, and epoxyethane can utilize as substrate into metabolite pathway with application of epoxyalkane: coenzyme M transferase (EaCoMT). Furthermore, ethene from anaerobic condition can be utilized as the only carbon and energy source by ethene-assimilating bacteria - Pseudomonas aeruginosa strain DL1 - under the condition with exceed 30 mg/L in soil.
Currently isolated stains (Mycobacterium JS622, JS623, JS624 and JS625) contain genes shared similarities with EaCoMT gene. Moreover, after varied incubation periods, they observed VC depletion, and they assumed VC were utilized as their substrate to grow. To verify uncertainty on VC utilization as substrate, researchers tested strains growth with 20mM of acetate. In order to verify purity of the culture, experiment was used plate with 1/10-strength trypticase soy agar and 1% glucose (TSAG). Both samples of VC and ethene were analyzed by flame ionization detection, and empty space was evaluated by gas chromatography. A spectrophotometer was measured growth rate of the samples by optical density at 600 nano wavelength (OB600). In order to check purity of the strain, a bead-beating DNA extraction method were used with modification of the samples and it analyzed by PCR. (‘The Taq PCR Master Mix (Quiagen) kit was used for PCR with 16S rRNA gene primers while the Taq PCR Core Mix (Qiagen) was used for REP-PCR. Sequencing data were compared to 16S rRNA gene sequences deposited in Genbank. BioEdit and ClustalX were used for alignment and analysis of DNA sequences.- I think those idea should be included, but have no idea how to rephrase it :( Could you help me on modification of this sentence?). VC adaptation experiment when through with exthene-grown cultures, acetate-grown cultures, and TSAG-grown cultures, which harvested at mixexponential phase. For ethene-grown cultures, multiple strains were feed on MSM as well as initial VC concentration of 0.8-1.0mM and incubated at room temperature. For acetate-grown cultures, strains were grown on acetate, which one was VC provided and other one is not. For TSAG-grown cultures, they were incubated at 30℃. In addition, after estimating 7 to 14 days colonies were shown on the plate, and it successfully obtained one TSAG-cultures with sufficient biomass.
The researchers were observed that VC consumption with ethene-grown JS622, JS623, JS624 and JS625 cultures. They consumed VC as their substrate about 14 days. There is no further observation on VC degradation as well as stabilization of OB600 after VC depletion. Correlation between VC consumption and OB600 pattern indicated cometabolic VC biodegradation. TSAG-culture strains also were observed VC consumption with relatively slower growth rate than ethene-culture strains.
Unfortunately, both cultures didn’t possesed capability of utilization of VC as growth substrate. However, after varied VC adaptation period depending on types of culture, ethene-grown cultures initiated VC consumption with similar patterns for all cultures. TSAG-grown cultures also required varied VC adaptation period to initiate VC consumption. In addition, the correlation between VC consumption and OB600 pattern indicated that TSAG-grown cultures also consumed VC concentration. Moreover, higher VC concentration required longer incubation period than initial VC concentration.
Figure 6.Observed adaptation time in Mycobacterium strains (Yang & Mattes, 2008)
Due to the long incubation periods, this study validated purity of VC-adapted strains. They compared morphology of strains with VC-unadapted ethene-grown cultures. Most of the strains shown identical morphology with VC-unadapted ethene-grown cultures, however, VC-adapted JS623 shown different morphology and color even with 16S rRNA sequencing and REP-PCR method. The results with 16S rRNA sequencing shown VC-adapted and unadapted cultures have 100% identical gene, therefore, VC-adapted cultures used in experiment had purity.
The results of this paper shown that Mycobacterium stains have capability to degrade VC compounds, and could apply for remediating VC contaminated site as promising ethene-assilimiting bacterium. In addition, the requirement of varied VC adaptation period suggested that composition of fields microbial community would effect on period of VC adaptation. Moreover, this experiment was conducted with low concentration of oxygen; it may refers that ethene-assimilating strains can do VC degradation under groundwater condition with low oxygen concentration as well as high oxygen concentration. Additionally, they found large-plasmid in VC-assimilating bacteria, however, they didn’t cover the role of large-plasmid and didn’t put into their consideration to drive its result.
This paper clearly stated the objective on their experiment to reveal limitations of previous work from others - other VC-assimilating bacteria, such as Dehalococcoides spp, have limitation on VC degradation under aerobic condition. In addition, they constructively designed their experiment to compare VC-adapted and undapted cultures, and accurately understood mechanisms of VC degradation in terms of involved enzyme activity. However, they couldn’t clearly reveal morphology of VC-adapted JS623 strain with 16S rRNA sequencing and REP-PCR. I think amplified fragment length polymorphism (AFLP) can be utilized as alternative way to identify morphology of VC-adapted JS623. AFLP employs similar banding patterns as REP-PCR, but this is based digestion genomic DNA with larger number of bands. It may help to validate disparate morphology on VC-adapted JS623. Furthermore, the result didn’t include present of large-plasmid. Plasmid contains gene expression encode protein. If they accounted for present of large-plasmid in their consideration, they could determine reasons on morphology differences.
### 3-3) Reductive dechlorination of high concentrations of chloroethenes by a Dehalococcoides Mccartyi strain 11G
A study conducted at the National University of Singapore and published on 18 October, 2018 was successfully able to show the reductive dechlorination of high concentrations of chlorethenes by a Dehalococcoides Mccartyi strain 11G. Reductive dechlorination has been established as a reliable in situ treatment for chloroethenes in anaerobic and anoxic conditions. Dehalococcoides is the most effective genus that can detoxify chloroethenes to the harmless compound ethene. This study isolated the Dehalococcoides Mccartyi strain 11G and found that it has tolerance to elevated chloroethene concentrations.
The authors of this study established a microcosm with mangrove sediment in defined DCB-1 medium and amended with acetate as a carbon source and and hydrogen as an electron donor exhibited complete dechlorination of TCE at high concentrations (up to 3.0 mM) to ethene, with the production of cis-DCE and trace amounts of VC as intermediates under strictly anaerobic conditions after 90 days’ incubation. To achieve complete dechlorination, 3 mM TCE rather than higher concentrations was spiked in the following consecutive sub-culturing, and a highly enriched culture which maintained TCE-to-ethene dechlorinating capability was obtained. This enrichment culture exhibited faster dechlorination (dechlorinating 3.0mM TCE within 40 days) than the original microcosm. Only three major populations were present in the enrichment, among which, Dehalococcoides, an obligate OHRB, became the dominant genus, accounting for 63.52% of the community, with Clostridium (17.95%) and Sulfurospirillum (12.94%) making up the rest.
Serial dilution-to-extinction was performed to separate potential TCE dechlorinating bacteria from the highly enriched culture. After five successive serial dilutions, complete conversion of TCE to ethene was repeatedly detected in 10−1–10−7 dilutions. Twenty randomly picked clones from a clone library analysis of a sample from the 10−7 dilution were identified as D. mccartyi and shared an identical sequence. After tests like Fluorescence microscopy revealed uniform disc-shaped morphology, which is similar to that of other Dehalococcoides strains (Löffler et al. 2013) and real-time quantitative PCR (Polymerase Chain Reaction) was applied to quantify the cell growth using bacteria- and Dehalococcoides-specific primer pairs, these results suggest that a pure D. mccartyi strain, designated D. mccartyi strain 11G, was obtained.
While the authors of the study did a great job of identifying and demonstrating the high tolerance to chloroethene of the 11G strain using practical experiments and existing literature to support their findings, the presentation of the paper was slightly complex and the writing and presentation of the paper was slightly flawed. The authors referred to figures that were not published in the paper and used acronyms that were not elaborated upon making the reading and understanding experience puzzling.
### Conclusion
Chemical compounds are the double-edged swords. It helps us to live our life with enrichment and aids the reactions human cannot manipulate. However, as halogenated compounds, it also can detrimentally destroy environment, ecosystem, and corrupted water quality. Thus, I think this paper alters our attention on use of chemical compounds as well as appreciation on mechanisms of nature. Because microorganisms, even people are unknown, play a crucial role on remediate our nature, contaminated by human. Consequently, I think it is important to discovery mechanisms of their roles and pathways to aid their activity for achieving efficient result on bioremediation.
In this context, estimating and evaluating interaction in microbial community is imperative to perceive. Because it may help to be coexisted in same habitat as syntrophy, however, it can be treated to be living in same habitat. As mentioned in body paragraph, some microorganisms - methanogens and acetogens - compete for consuming hydrogen as electron acceptor with strains to degrade halogenated compounds. Therefore estimating total amount of required electron acceptor is essential step should take before evaluating effectiveness of remediation.
In addition, utilizing bacteria consortia can produce higher efficient result on degradation than employing single strain. Hence, based on current observation and understanding, it would be worth to construct bacteria consortia to remediate contaminated site by halogenated aliphatic compound.
References
1. International Labour Organization content manager. (2011, August 03). Hydrocarbon, Aliphatic and Halogenated. Retrieved from http://www.iloencyclopaedia.org/part-xviii-10978/guide-to-chemicals/104/hydrocarbons-aliphatic-and-halogenated?fbclid=IwAR0De7eMlNcEPuqf7kMc9K4laSqWGMnZRRbsxDLnLdajwUBdpuLk-N2QcpI
2. G. Rasul Chaudhry, & S. Chapalamadugu. (1991). Biodegradation of Halogenated Organic Compounds. Microbiological review, pp 59-79
3. Stellman, J. M. (1998). HYDROCARBONS, HYLOGENATED AROMATIC. Encyclopaedia of Occupational Health and Safety, Vol. 4, pp 104·286-288
4. Paul Flowers., Klaus Theopold., Richard Langley., William R. PhD Robinson., & OpenStax. (2017). Chemistry. OpenStax, OpenStax Chemistry, chapter. 20
5. Abozenadah, H., Bishop, A., Bittner, S., Lopez, O., Wiley, C., and Flatt, P.M. (2017) Consumer Chemistry: How Organic Chemistry Impacts Our Lives. Attribution-Non-Commercial-Share Alike, chapter 7&8
6. H.M. Mehendale. (2010). Comprehensive Toxicology, 2nd ed. Vol: 7.19. Elsevier Science, pp 459-474
7. Martin H. van Agteren., Stytze Keuning., & Dick B. Janssen. (1998). Handbook on Biodegradation and Biological Treatment of Hazardous Organic Compound. Kluwer Academic Publishers, pp 77-167
8. Xavier Maymó-Gatell., Yueh-tyng Chien., James M. Gossett., & Stephen H. Zinder. (1997). Isolation of Bacterium that Reductively Dechlorinates Tetrachloroethene to Ethene. Science, Vol. 276
9. Yang, Oh Jin, & Timothy E. Mattes. (2008). Adaptation of aerobic ethane-assimilating Mycobacterium Strains to Vinyl Chloride as a growth substrate. Environ. Sci. Technol, 42, 4784-4789
10. Siyan Zhao, & Jianzhong He. (2018). Reductive dechlorination of high concentrations of chloroethenes by a Dehalococcoides mccartyi strain 11G. FEMS Microbiology Ecology, 95
11. Madigan, M. T., Martinko, J. M., Bender, K. S., Buckley, D. H., & Stahl, D. A. (2018). Brock biology of microorganisms, 5th ed., Boston: Pearson
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# Simultaneously whitening correlated vectors
I have a $P \times K$ matrix $\mathbf{Z} = \begin{bmatrix}\mathbf{z_1} & \mathbf{z_1} & \cdots & \mathbf{z_K}\end{bmatrix}$. I want to whiten the columns of $\mathbf{Z}$ so that the covariance of the columns is an identity matrix. I have two cases here.
1. When the columns are uncorrelated and identically distributed, then the $P \times P$ covariance matrix $M$ of the columns of $\mathbf{Z}$ is Toeplitz. In my problem, I can also obtain expression of the $m$th diagonal element of $\mathbf{M}$. Then, I can whiten the measurements by multiplying $\mathbf{Z}$ on the left by $\mathbf{M}^{-1/2}$.
2. In the second case, the columns are not uncorrelated. I can still obtain the expressions of cross-correlations $R_{i_1,i_2}[j_1,j_2]$ between any two $(i_1, j_1)$th and $(i_2, j_2)$th elements of $\mathbf{Z}$. When the columns were uncorrelated, i.e. the first case, then $R_{i_1,i_2}[j_1,j_2]$ is same for any $j_1=j_2$ and zero otherwise. However, this is no longer the case now for $R_{i_1,i_2}[j_1,j_2]$. So, how can I simultaneously whiten the correlated columns $\mathbf{z_1}$, $\cdots$ $\mathbf{z_K}$?
From what I read, it looks like I should first use Principal Component Analysis (PCA) to decorrelate the columns and then apply the whitening process for uncorrelated columns as described above in the first case. However, two aspects are not clear to me:
1. Most PCA examples I found in online resources show decorrelation of only two vectors. However, here, I want to simultaneously decorrelate $K$ correlated column vectors.
2. If I apply PCA, then it should result in a modified expression of $R_{i_1,i_2}[j_1,j_2]$ that I can later use to construct the matrix $M$. It is not clear to me how should I be able to obtain such an expression using some PCA-related properties.
Any help would be greatly appreciated.
-ryan
[Update] In the above, I have assumed the means of the elements of $\mathbf{Z}$ are zero. For non-zero means, in both correlated and uncorrelated cases, I understand that I should first element-wise subtract the respective means from matrix $\mathbf{Z}$ before moving forward with decorrelation and whitening.
• There are several unclear or confusing points in your question. One ia about Toeplitz. Why not just diagonal cov. matrix? Another is that using PCA to decorrelate one vector (?). – ttnphns Sep 7 '17 at 10:30
• PCA is regularly used to whiten variables. The whitened result is called standardized PC scores. – ttnphns Sep 7 '17 at 10:32
• @ttnphns I am referring to the covariance matrix of a particular column of $\mathbf{Z}$. So $\mathbf{M} = E\{\mathbf{z_i}^T\mathbf{z_i}\}$. This is Toeplitz and is same for all columns or $\forall i$. However, if I compute the cross-covariance matrix of any two columns, then it will be diagonal because the columns themselves are uncorrelated. – ryan80 Sep 7 '17 at 16:03
• @ttnphns Regarding your comment on PCA, I made an edit in the question. I meant to ask for a PCA example which shows decorrelating more than two random vectors. – ryan80 Sep 7 '17 at 16:07
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Find Paper, Faster
Example：10.1021/acsami.1c06204 or Chem. Rev., 2007, 107, 2411-2502
On non‐orientable surfaces embedded in 4‐manifolds
Journal of Topology (IF1.582), Pub Date : 2021-05-10, DOI: 10.1112/topo.12188
We find conditions under which a non‐orientable closed surface smoothly embedded into an orientable $4$‐manifold $X$ can be represented by a connected sum of an embedded closed surface in $X$ and an unknotted projective plane in a $4$‐sphere. This allows us to extend the Gabai $4$‐dimensional light bulb theorem and the Auckly–Kim–Melvin–Ruberman–Schwartz ‘one is enough’ theorem to the case of non‐orientable surfaces.
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# Math Help - Crossover rate - How to solve r?
1. ## Crossover rate - How to solve r?
Hi all!
My math problem is about a Finance class, but it is really the math that gets me into trouble. Please see the equation:
(15/(1+r))+(10/(1+r)^2)+(8/(1+r)^3)-25=(2.5/(1+r))+(2.5/(1+r)^2)+(2.5/(1+r)^3)
I guess the first step would be to get it all to one side of the equation. In particular, I am wondering what to do with the denominators. Can I add them up?
Please note I'm only allowed to use a basic (i.e. non-graphic) calculator.
Thank you!
Sebastiaan
2. ## Re: Crossover rate - How to solve r?
Originally Posted by sebastiaan
Hi all!
My math problem is about a Finance class, but it is really the math that gets me into trouble. Please see the equation:
$\dfrac{15}{(1+r)}+\dfrac{10}{(1+r)^2}+\dfrac{8}{(1 +r)^3}-25=\dfrac{2.5}{(1+r)}+ \dfrac{2.5}{(1+r)^2}+\dfrac{2.5}{(1+r)^3}$
I guess the first step would be to get it all to one side of the equation. In particular, I am wondering what to do with the denominators. Can I add them up?
Please note I'm only allowed to use a basic (i.e. non-graphic) calculator.
Thank you!
Sebastiaan
You can't add the denominators because they are not the same. I would multiply both sides by the LCD of $(1+r), (1+r)^2 \text{ and } (1+r)^3$ to clear the denominator.
You can then collect like terms
3. ## Re: Crossover rate - How to solve r?
Do you mean that for the right side of the equation I could end up with:
((2.5^6)/((1+r)^6))+((2.5^3)/((1+r)^6))+((2.5^2)/((1+r)^6))
So that all denominators share the common '^6'?
Thanks a lot!
4. ## Re: Crossover rate - How to solve r?
Originally Posted by sebastiaan
Do you mean that for the right side of the equation I could end up with:
((2.5^6)/((1+r)^6))+((2.5^3)/((1+r)^6))+((2.5^2)/((1+r)^6))
So that all denominators share the common '^6'?
Thanks a lot!
Something I should have mentioned in the last post: $r \neq -1$.
Nope, you can't raise both sides to the power. To use an example: $\dfrac{2}{3} \neq \dfrac{2^2}{3^2}$. If you multiply both sides of your equation by the LCD you will clear the denominator making the equation easier to solve.
Do you know what the LCD of $(1+r), (1+r)^2 \text{ and } (1+r)^3$ is? Hint: 1+r and (1+r)^2 are both factors of (1+r)^3
5. ## Re: Crossover rate - How to solve r?
Thanks! Helped a lot
6. ## Re: Crossover rate - How to solve r?
You really only need to multiply both sides by $(1+ r)^3$ since this will cancel $1+ r$ and $(1+ r)^2$ also.
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# Simultaneous Localization and Mapping (SLAM)
In SLAM, the robot acquires a map of its environment while simultaneously localizing itself relative to this map. It is a significantly more difficult problem compared to Robot Localization and Occupancy Grid Mapping.
There are 2 main forms of SLAM:
Online SLAM
estimating the posterior over the momentary pose along with the map: $$p(x_t, m | z_{1:t}, u_{1:t})$$
full SLAM
posterior over the entire path $$x_{1:t}$$ along with the map: $$p(x_{1:t}, m | z_{1:t}, u_{1:t})$$
The online SLAM algorithm is the result of integrating out past poses in the full SLAM problem
## EKF SLAM
EKF SLAM uses a number of approximations and limiting assumptions:
Feature-based maps
Maps are composed of a small number of point landmarks. The method works well when the landmarks are relatively unambiguous.
Gaussian Noise
The noise in motion and perception is assumed to be Gaussian.
Positive Measurements
It can only process positive sightings of landmarks, ignoring negative information.
### SLAM with Known Correspondence
The key idea is to integrate landmark coordinates into the state vector. This corresponds to the continuous portion of the SLAM problem. EKF SLAM combines the state vector as such:
\begin{aligned} y_{t} &=\left(\begin{array}{c}{x_{t}} \ {m}\end{array}\right) \end{aligned}
and calculates the online posterior:
$$p\left(y_{t} | z_{1: t}, u_{1: t}\right)$$
### EKF SLAM with Unknown Correspondences
It uses an incremental maximum likelihood estimator to determine correspondences.
### Feature Selection and Map Management
EKF SLAM requires several additional techniques to be robust in practice. First, one needs to deal with outliers in the measurement space. One technique is to maintain a provisional landmark list. Instead of adding the landmark immediately, it is added to this list, and when the uncertainty has shrunk after repeated observations of the landmark, it is added in.
## EIF SLAM
Unlike EKF SLAM, the extended information form SLAM algorithm (EIF SLAM) solves the full SLAM problem. EIF represents the posterior gaussian in its canonical representation form, with the precision matrix and information state vector (See Information Filter).
EIF SLAM is also not incremental: it calculates posteriors over a robot path. It is best suited for problems where a map needs to be built from data of fixed size, and can afford to hold the data in memory until the map is built.
Suppose we are given a set of measurements $$z_{1:t}$$ with associated correspondence variables $$c_{1:t}$$, and a set of controls $$u_{1:t}$$. Then the EIF SLAM algorithm operates as follows:
1. Construct the information matrix and information vector fromt he joint space of robot poses $$x_{1:t}$$ and map $$m = \{m_j\}$$.
2. Each measurement leads to a local update of $$\Omega$$ and $$\xi$$. This is because information is an additive quantity.
The key insight is that information is sparse. Specifically,
• Measurements provide information of a feature relative to the robot’s pose at the time of measurement, forming constraints between pairs of variables.
• Motion provides information between two subsequent poses, also froming constraints
EIF SLAM records all this information, through links that are defined between poses and features, and pairs of subsequent poses. However, this information representation does not provide estimates of the map or robot path.
Maps are recovered via an iterative procedure involving 3 steps:
1. Construction of a linear information form through Taylor expansion
2. Reduction of this linear information form
3. Solving the resulting optimization problem
### Sparse EIF SLAM
The sparse EIF SLAM only maintains a posterior over the present robot pose and the map. Hence, they can be both run online, and are efficient. Unlike EKFs, they also maintain information representation of all knowledge.
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1. Apr 19, 2009
### chemstudent09
I'm not even sure if I am asking this the right way, but..
Does the Hartree approximation basically assume that electrons are independent and do not interact with each other?
Also, do Exchange integrals essentially measure the interaction between two different electrons?
Thanks.
2. Apr 20, 2009
### alxm
Well, the Hartree (or Hartree-Fock) method makes several approximations, but not that the electrons are non-interacting. If that were the case the resulting wave function of an atom would simply be a superposition of solutions to the hydrogen-like atom.
HF improves on that by including exchange (the effect of the Pauli principle), and the electrostatic interaction between the electrons (the Coulomb integral). So it includes two forms of electron-electron interaction. (although exchange is not, strictly speaking, an interaction, but a boundary condition placed on the solutions to the S.E.) The exchange integrals have no classical analog. They're simply a direct consequence of preserving the known boundary condition (antisymmetry).
However, HF does assume that the kinetic energy of the electrons are independent. Or in other words, that the motion of the electrons are independent. But they're not, since electrons 'avoid' each other due to their charges. So you have a coupling (termed correlation) between the kinetic and potential energy, which is neglected. Each electron moves in a 'mean field' of the charge of every other electron, but the non-linear dynamical effects of their interactions is not included.
The other approximation, implicit in HF, is that it's a single-determinant description of the system.
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The Catenary Problem
Introduction
A chain with uniformly distributed mass hangs from the endpoints $$(0,1)$$ and $$(1,1)$$ on a 2-D plane. Gravitational force acts in the negative $$y$$ direction. Our goal is to find the shape of the chain in equilibrium, which is equivalent to determining the $$(x,y)$$ coordinates of every point along its curve when its potential energy is minimized.
This is the famous catenary problem.
A Discrete Version
To formulate as an optimization problem, we parameterize the chain by its arc length and divide it into $$m$$ discrete links. The length of each link must be no more than $$h > 0$$. Since mass is uniform, the total potential energy is simply the sum of the $$y$$-coordinates. Therefore, our (discretized) problem is
$\begin{array}{ll} \underset{x,y}{\mbox{minimize}} & \sum_{i=1}^m y_i \\ \mbox{subject to} & x_1 = 0, \quad y_1 = 1, \quad x_m = 1, \quad y_m = 1 \\ & (x_{i+1} - x_i)^2 + (y_{i+1} - y_i)^2 \leq h^2, \quad i = 1,\ldots,m-1 \end{array}$
with variables $$x \in {\mathbf R}^m$$ and $$y \in {\mathbf R}^m$$. This discretized version which has been studied by Griva and Vanderbei (2005) was suggested to us by Hans Werner Borchers.
The basic catenary problem has a well-known analytical solution (see Gelfand and Fomin (1963)) which we can easily verify with CVXR.
## Problem data
m <- 101
L <- 2
h <- L / (m - 1)
## Form objective
x <- Variable(m)
y <- Variable(m)
objective <- Minimize(sum(y))
## Form constraints
constraints <- list(x[1] == 0, y[1] == 1,
x[m] == 1, y[m] == 1,
diff(x)^2 + diff(y)^2 <= h^2)
## Solve the catenary problem
prob <- Problem(objective, constraints)
result <- solve(prob)
We can now plot it and compare it with the ideal solution. Below we use alpha blending and differing line thickness to show the ideal in red and the computed solution in blue.
xs <- result$getValue(x) ys <- result$getValue(y)
catenary <- ggplot(data.frame(x = xs, y = ys)) +
geom_line(mapping = aes(x = x, y = y), color = "blue", size = 1) +
geom_point(data = data.frame(x = c(xs[1], ys[1]), y = c(xs[m], ys[m])),
mapping = aes(x = x, y = y), color = "red")
ideal <- function(x) { 0.22964 *cosh((x -0.5) / 0.22964) - 0.02603 }
catenary + stat_function(fun = ideal , colour = "brown", alpha = 0.5, size = 3)
A more interesting situation arises when the ground is not flat. Let $$g \in {\mathbf R}^m$$ be the elevation vector (relative to the $$x$$-axis), and suppose the right endpoint of our chain has been lowered by $$\Delta y_m = 0.5$$. The analytical solution in this case would be difficult to calculate. However, we need only add two lines to our constraint definition,
constr[[4]] <- (y[m] == 0.5)
constr <- c(constr, y >= g)
to obtain the new result.
Below, we define $$g$$ as a staircase function and solve the problem.
## Lower right endpoint and add staircase structure
ground <- sapply(seq(0, 1, length.out = m), function(x) {
if(x < 0.2)
return(0.6)
else if(x >= 0.2 && x < 0.4)
return(0.4)
else if(x >= 0.4 && x < 0.6)
return(0.2)
else
return(0)
})
constraints <- c(constraints, y >= ground)
constraints[[4]] <- (y[m] == 0.5)
prob <- Problem(objective, constraints)
result <- solve(prob)
to obtain the new result.
The figure below shows the solution of this modified catenary problem for $$m = 101$$ and $$h = 0.04$$. The chain is shown hanging in blue, bounded below by the red staircase structure, which represents the ground.
xs <- result$getValue(x) ys <- result$getValue(y)
ggplot(data.frame(x = xs, y = ys)) +
geom_line(mapping = aes(x = x, y = y), color = "blue") +
geom_point(data = data.frame(x = c(xs[1], ys[1]), y = c(xs[m], ys[m])),
mapping = aes(x = x, y = y), color = "red") +
geom_line(data.frame(x = xs, y = ground),
mapping = aes(x = x, y = y), color = "brown")
Session Info
sessionInfo()
## R version 3.6.0 (2019-04-26)
## Platform: x86_64-apple-darwin18.5.0 (64-bit)
## Running under: macOS Mojave 10.14.5
##
## Matrix products: default
## BLAS/LAPACK: /usr/local/Cellar/openblas/0.3.6_1/lib/libopenblasp-r0.3.6.dylib
##
## locale:
## [1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
##
## attached base packages:
## [1] stats graphics grDevices datasets utils methods base
##
## other attached packages:
## [1] ggplot2_3.1.1 CVXR_0.99-6
##
## loaded via a namespace (and not attached):
## [1] gmp_0.5-13.5 Rcpp_1.0.1 highr_0.8
## [4] compiler_3.6.0 pillar_1.4.1 plyr_1.8.4
## [7] R.methodsS3_1.7.1 R.utils_2.8.0 tools_3.6.0
## [10] digest_0.6.19 bit_1.1-14 evaluate_0.14
## [13] tibble_2.1.2 gtable_0.3.0 lattice_0.20-38
## [16] pkgconfig_2.0.2 rlang_0.3.4 Matrix_1.2-17
## [19] yaml_2.2.0 blogdown_0.12.1 xfun_0.7
## [22] withr_2.1.2 dplyr_0.8.1 Rmpfr_0.7-2
## [25] ECOSolveR_0.5.2 stringr_1.4.0 knitr_1.23
## [28] tidyselect_0.2.5 bit64_0.9-7 grid_3.6.0
## [31] glue_1.3.1 R6_2.4.0 rmarkdown_1.13
## [34] bookdown_0.11 purrr_0.3.2 magrittr_1.5
## [37] scales_1.0.0 htmltools_0.3.6 scs_1.2-3
## [40] assertthat_0.2.1 colorspace_1.4-1 labeling_0.3
## [43] stringi_1.4.3 lazyeval_0.2.2 munsell_0.5.0
## [46] crayon_1.3.4 R.oo_1.22.0
R Markdown
References
Gelfand, I. M., and S. V. Fomin. 1963. Calculus of Variations. Prentice-Hall.
Griva, I. A., and R. J. Vanderbei. 2005. “Case Studies in Optimization: Catenary Problem.” Optimization and Engineering 6 (4): 463–82.
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# Can fixed point division be implemented using a divider that outputs quotient and remainder?
From what I have seen, division is a highly expensive operation in terms of time or area (tradeoff). It is usually implemented as an operation of continuous subtraction of a number from another number to get the quotient bits.
While I understand how addition, subtraction and multiplication are implemented, there is some confusion surrounding division. I have 3 interrelated questions.
Q1: If division is merely recursion of subtractions until we are left with remainder, how would one get a fixed point output i.e output with both integer and fraction parts; since an integer quotient represents how many times we did subtraction in a loop.
Q2: How would one deal with recurring decimal numbers as quotient? I assume through round-off i.e we do not care if result is recurring decimal or not, we just calculate the result to certain digits of fractional part.
Q3: Provided that I have a divider IP that outputs remainder and quotient, how would I get the fractional part of the output since the remainder does not actually equal the fractional part?
• Is this a hardware question? (Sounds like a software one, but the title makes me unsure.) The rest reads to me as though you are struggling to write code to generate a display-formatted decimal value, with a decimal point and fractional digits. Not a real division-only question. What's the purpose here? Why are you asking these particular questions? What's the goal? – jonk Nov 29 '16 at 7:03
• I want to understand how to carry out division with specific precision i.e fixed point division. I have an IP block in Quartus ii called lpm_div that gives quotient and remainder. However, it is not clear how to get fractional part of quotient to specific decimal places. – quantum231 Nov 29 '16 at 9:26
• Where at all possible, I would try to implement your division as a multiplication by the reciprocal of the denominator. Alternatively, arrange things so that any divisions are by a power of two. – scary_jeff Dec 2 '16 at 12:25
I don't know anything about that IP block. But since you don't yet have a better answer (mine won't be all that good), I'll offer some general advice. I'll draw on some vague assumptions I'll make:
• You can define the width of the numerator and denominator, separately.
• You can define the width of the quotient and remainder, separately.
I usually normalize both the numerator and denominator using a barrel shifter prior to any division. This counts the number of shifts required and I save those aside. As a result of this step: $2\cdot denominator > numerator \ge \tfrac{1}{2}\cdot denominator$.
Assume that the significant bits of your numerator are the same size as the denominator. But also consider a numerator width to the IP block that is twice the size of the denominator. The above normalized numerator is placed in the upper half of the double-width bus. You make a comparison. If the denominator is larger then shift the widened numerator down by one bit. This will guarantee that your quotient will fit in the same width as the denominator.
The upshot of the above is that the numerator bus width is twice that of the other three. An incoming numerator is placed in the upper half, but may be shifted down one bit. Both numerator and denominator are normalized, prior to division.
But let's say you don't use a barrel shifter and don't care to normalize. Then consider setting the numerator width of the IP block at twice the width of the other three. Your external numerator then goes into the upper half of that width (lane shift) and zeros always go into the lower half (optimize the logic for that case.)
Then the fractional part simply using the same divider but now by taking the resulting remainder and dividing it by the original divisor you used before.
You really haven't provided enough information AND I'm also ignorant of your IP block. So that's the best I can offer. Hopefully, someone else will do better.
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Article Contents
Article Contents
# Stationary solutions for some shadow system of the Keller-Segel model with logistic growth
• From a viewpoint of the pattern formation, the Keller-Segel system with the growth term is studied. This model exhibited various static and dynamic patterns caused by the combination of three effects, chemotaxis, diffusion and growth. In a special case when chemotaxis effect is very strong, some numerical experiment in [1],[22] showed static and chaotic patterns. In this paper we consider the logistic source for the growth and a shadow system in the limiting case that a diffusion coefficient and chemotactic intensity grow to infinity. We obtain the global structure of stationary solutions of the shadow system in the one-dimensional case. Our proof is based on the bifurcation, singular perturbation and a level set analysis. Moreover, we show some numerical results on the global bifurcation branch of solutions by using AUTO package.
Mathematics Subject Classification: Primary: 35B32, 37G40, 35K57; Secondary: 35B36.
Citation:
• [1] M. Aida, T. Tsujikawa, M. Efendiev, A. Yagi and M. Mimura, Lower estimate of the attractor dimension for a chemotaxis growth system, J. London Math. Soc., 74 (2006), 453-474.doi: 10.1112/S0024610706023015. [2] W. Alt and D. A. Lauffenburger, Transient behavior of a chemotaxis system modelling certain types of tissue inflammation, J. Math. Biol., 24 (1987), 691-722.doi: 10.1007/BF00275511. [3] P. Biler, Local and global solvability of some parabolic system modelling chemotaxis, Adv. Mathe. Sci. and Appl., 8 (1998), 715-743. [4] N. Chafee and E. F. Infante, A bifurcation problem for a nonlinear partial differential equation of parabolic type, Appl. Anal., 4 (1974), 17-37.doi: 10.1080/00036817408839081. [5] E. J. Doedel, R. C. Paffenroth, A. R. Champneys, T. F. Fairgrieve, Y. A. Kuznetsov, B. E. Oldeman, B. Sandstede and X. Wang, AUTO 2000, Continuation and bifurcation software for ordinary differential equations. [6] S.-I. Ei, H. Izuhara and M. Mimura, Spatio-temporal oscillations in the Keller-Segel system with logistic growth, Physica D, 277 (2014), 1-21.doi: 10.1016/j.physd.2014.03.002. [7] C. Gai, Q. Wang and J. Yan, Qualitative analysis of stationary Keller-Segel chemotaxis models with logistic growth, preprint, arXiv:1312.0258. [8] D. D. Hai and A. Yagi, Numerical computations and pattern formation for chemotaxis-growth model, Sci. Math. Jpn, 70 (2009), 205-211. [9] Y. Kabeya and W.-M. Ni, Stationary Keller-Segel model with the linear sensitivity, RIMS Kokyuroku, 1025 (1998), 44-65. [10] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.doi: 10.1016/0022-5193(70)90092-5. [11] N. Kurata, K. Kuto, K. Osaki, T. Tsujikawa and T. Sakurai, Bifurcation phenomena of pattern solution to Mimura-Tsujikawa model in one dimension, Math. Sci. Appl., 29 (2008), 265-278. [12] K. Kuto, K. Osaki, T. Sakurai and T. Tsujikawa, Spatial pattern in a chemotaxis-diffusion-growth model, Physica D, 241 (2012), 1629-1639.doi: 10.1016/j.physd.2012.06.009. [13] K. Kuto and T. Tsujikawa, Stationary patterns for an adsorbate-induced phase transition model: II. Shadow system, Nonlinearity, 26 (2013), 1313-1343.doi: 10.1088/0951-7715/26/5/1313. [14] K. Kuto and T. Tsujikawa, Bifurcation structure of steady-states for bistable equations with nonlocal constraint, Discrete Continuous Dynam. Systems, Supplement volume, (2013), 455-464. [15] K. Kuto and T. Tsujikawa, Limiting structure of steady-states to the Lotka-Volterra competition model with large diffusion and advection, J. Differential Equations, 258 (2015), 1801-1858.doi: 10.1016/j.jde.2014.11.016. [16] K. Kuto and T. Tsujikawa, Bifurcation structure of steady-states for generalized Allen-Cahn equations with nonlocal constraint, preprint. [17] D. A. Lauffenburger and C. R. Kennedy, Localized bacterial infection in a chemotaxis-diffusion-growth model, J. Math. Biol., 16 (1983), 141-163. [18] C.-S. Lin, W.-N. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations, 72 (1988), 1-27.doi: 10.1016/0022-0396(88)90147-7. [19] P. K. Maini, M. R. Myerscough, K. H. Winters and J. D. Murray, Bifurcating spatially heterogeneous solutions in a chemotaxis model for biological pattern formation, Bull. Math. Biol., 53 (1991), 701-719. [20] M. Mimura and T. Tsujikawa, Aggregating pattern dynamics in a chemotaxis model including growth, Physica A, 230 (1996), 499-543.doi: 10.1016/0378-4371(96)00051-9. [21] K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Analysis, Theory, Methods and Applications, 51 (2002), 119-144.doi: 10.1016/S0362-546X(01)00815-X. [22] K. J. Painter and T. Hillen, Spatio-temporal chaos in a chemotaxis model, Physica D, 240 (2011), 363-375.doi: 10.1016/j.physd.2010.09.011. [23] R. Schaaf, Global behaviour of solution branches for some Neumann problems depending on one or several parameters, J. Reine Angew. Math., 364 (1984), 1-31.doi: 10.1515/crll.1984.346.1. [24] R. Schaaf, Global Solution Branches of Two-Point Boundary Value Problems, Lecture Notes in Mathematics, 1458, Springer-Verlag, Berlin, 1990. [25] T. Senba and T. Suzuki, Some structures of the solution set for a stationary system of chemotaxis, Adv. Math. Sci. Appl., 10 (2000), 191-224. [26] J. Shi, Semilinear Neumann boundary value problems on a rectangle, Trans. Amer. Math. Soc., 354 (2002), 3117-3154.doi: 10.1090/S0002-9947-02-03007-6. [27] J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.doi: 10.1080/03605300701319003. [28] T. Tsujikawa, Global structure of the stationary solutions for the limiting system of a chemotaxis-growth model, to appear in RIMS Kokyuroku, 2014.
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# FTR underfunding leaves PJM power traders out of pocket
## Shortfall in biggest power market exceeds $1 billion since 2010 The world's biggest power market has a problem. For over a decade, PJM – the wholesale electricity market that serves 13 eastern US states and the District of Columbia – has allowed market participants to trade financial transmission rights (FTRs), instruments that allow companies to hedge against the costs of congestion on the PJM grid. But in recent years, PJM's mechanism for funding payments to FTR holders has broken down, and that has many market participants upset, including both hedgers and financial traders. From 2010, when the problem first emerged, to January 2014, firms holding FTRs received roughly$1.1 billion less in FTR revenues than they should have, according to an analysis of data from PJM and Monitoring Analytics, PJM's Pennsylvania-based independent market monitor. Yet the problem shows no signs of going away. Though PJM formed an ‘FTR Task Force' to study the problem, the group failed to agree on a proposed solution to address the market design flaw behind the FTR funding shortfall, due to a deadlock among its members.
Ohio-based utility FirstEnergy grew so exasperated it filed a series of complaints about the problem to the US Federal Energy Regulatory Commission (Ferc). By the end of 2013, the company had suffered a total revenue shortfall of more than \$65 million on its FTR portfolio, according to Don Moul, vice-president of commodity operations at FirstEnergy Solutions, a subsidiary of the company that supplies power to around two million customers.
"FTRs are an important tool to help mitigate congestion risk in the PJM energy market," says Moul. "This is particularly true for suppliers that use generation to hedge their retail load obligations. When FTRs are underfunded, this reduces the effectiveness of this risk mitigation tool."
PJM expresses sympathy with FTR holders and says it is trying to find a path forward. The underfunding problem "continues to be a very concerning issue for PJM," says Stu Bresler, Pennsylvania-based vice-president for market operations at PJM. "We continue to do everything we can on a day-to-day basis to make sure that FTRs are funded to the greatest level possible."
###### The biggest question is, if FTR holders aren’t responsible for real-time congestion, who should be?
The ABCs of FTRs
Like other regional transmission organisations (RTOs) across the US, PJM runs a nodal electricity market. Each of the numerous points in its grid has a different locational marginal price (LMP) for power. The LMP contains both an energy component – a system-wide value that depends on overall supply and demand in the network – and a congestion component, specific to each node, which increases when it becomes more costly to deliver power to that location.
Any FTR contract specifies a path from one node to another, and how much the FTR should pay out depends on the difference in day-ahead LMPs along that path. Typically, FTRs are associated with paths that run from a source to a sink – that is, a point where power is generated to a point where it is consumed – and they make money when the LMP at the sink exceeds the LMP at the source. However, there are also so-called ‘counterflow' FTRs that run in the opposite direction.
PJM holds periodic auctions to sell FTRs of various terms, ranging from one month to three years. Besides generators and power suppliers, which use FTRs to hedge their congestion costs, participants in the auctions include hedge funds and proprietary trading firms. Such players are drawn to FTRs because they are a purely financial instrument, which do not require the ownership of physical assets, and because they lend themselves to quantitative trading strategies.
Whenever PJM auctions off FTRs, it runs an elaborate model to ensure that the revenue it expects to collect from congestion payments will be adequate to cover payouts to FTR holders. Crucially, though, the model focuses on congestion in PJM's day-ahead market, and not in the real-time market, so when unexpected conditions arise in real time, it can scramble PJM's plans and cause underfunding.
Under PJM's current market rules, the pool of money used to pay out FTR holders comes from three sources: surplus revenues from the FTR auctions, day-ahead congestion payments and real-time congestion payments. In past years, that has generally worked well. From July 2006 to February 2010, FTRs were 100% funded, according to PJM data. But starting in 2010, the payout ratio collapsed. In the year from June 2012 to May 2013, it sank as low as 67.8%. Since then, it has recovered somewhat, reaching about 80% in January 2014, according to data from PJM and Monitoring Analytics.
PJM blames the drop-off on real-time congestion, which in recent years has often been a negative number. Positive congestion means PJM is collecting money from market participants for congestion, while negative congestion means PJM is paying money out. The reasons why that number has turned negative revolve around the dramatic changes seen in the US power industry over the past half-decade.
Gone with the wind
One of the key industry shifts that has contributed to the FTR underfunding problem has been the huge build-out of wind generation in the Midwest, within the territory of PJM's neighbouring RTO, the Midcontinent Independent System Operator (Miso). Wind power is intermittent and can create unexpected congestion in the real-time market that is difficult, if not impossible, to model.
Wind power from Miso often manifests itself in PJM in the form of ‘loop flows', with power flowing across the border from Miso into the western parts of PJM, before heading back out into Miso again. That clogs up PJM's transmission lines. "It's really Miso generation to Miso load transfers that are essentially utilising the PJM transmission facilities to get there," says Bresler. "It takes transmission room that PJM market participants can't use, so we can't collect congestion for it."
The shale gas revolution, which has made natural gas a more attractive fuel for power generation than coal, has also contributed to the shift. According to an April 2012 report by PJM on the causes of FTR underfunding, the relative decline in competitiveness of coal-fired power plants in the west of PJM, compared with natural gas-fired plants in the east of PJM, has led to "reduced internal PJM west to east flows" and altered long-established patterns of congestion.
There are other, more mundane reasons, which have fuelled real-time congestion in PJM. According to Bresler, these include a major build-out of new transmission capacity, which has caused construction-related outages; an unusually large number of maintenance outages; and finally, the derating of existing transmission lines, in which grid operators downgrade a line's official capacity.
The ultimate solution to FTR underfunding will be the revamping of PJM's transmission grid to account for all of the changed flows, Bresler believes. "The best fix is going to be once the construction projects are finished and we actually have more transmission system capability," he says. "That's really going to be the end-all, be-all fix."
Market participants acknowledge there are physical reasons behind the FTR underfunding issue, but they argue that PJM must also fix its market design. "PJM's answer fails to address the problem," says FirstEnergy's Moul. "It's not what's happening on the system, or what PJM is doing to address conditions on the system. It's the fact that the tariff requires PJM to pay the money that should go to FTR holders to someone else."
FirstEnergy and a number of other FTR holders support modifying the mechanism PJM uses to fund its FTRs – namely, by removing real-time congestion from the mix, so FTR holders are not on the hook when real-time congestion turns negative. But that has proved contentious, and when market participants have discussed making such a fix, their talks have ended in deadlock.
The difficulty is determining a fair way to allocate the costs of real-time congestion, says Jason Barker, director of wholesale market development at Exelon, the Chicago-based electricity firm.
"The biggest question is, if FTR holders aren't responsible for real-time congestion, who should be?" Barker asks. "In the absence of a known root cause for real-time congestion, it seems to make sense to socialise it across the load base, which would bring FTRs back closer to being fully funded, make them a more effective hedge, and ultimately take [the] risk out of the pricing of the load."
However, when the FTR Task Force presented proposals along those lines to PJM members in December 2011 and December 2012, they went down in defeat. According to Barker, most market participants are simply unwilling to bear any additional costs associated with real-time congestion, so the status quo has remained in place. "There are winners and losers whenever you change the market rules," he says.
FTR holders' main hope right now rests on FirstEnergy's latest complaint to Ferc. Filed on February 15, 2013, the complaint argues that PJM's rules on the funding of FTRs are "unjust and unreasonable" because they unfairly force FTR holders to bear the burden of real-time congestion costs. In an order issued on June 5, 2013, Ferc dismissed the complaint, siding with PJM's argument that 100% revenue adequacy for FTRs was a goal, not a guarantee. Since then, however, a number of major FTR traders and utilities have asked Ferc to reconsider its decision. At the time Energy Risk went to press, it was unclear whether Ferc would grant the FirstEnergy complaint a rehearing.
PJM rejects the claim that its rules are unjust and unreasonable – but it concedes that there are probably more rational ways to allocate the costs of real-time congestion. "We believe that the FirstEnergy proposal would be a beneficial market design change," says Bresler. "Assigning negative balancing congestion to FTR holders probably doesn't make a whole lot of sense anymore."
Living with underfunding
For the time being, FTR holders must wrestle with the possibility that the instruments may not pay out as much as they theoretically should.
Some financial traders have responded by exiting the market altogether. "Our company has pulled out from the PJM FTR market because of the underfunding," says a Pennsylvania-based trader with one proprietary trading firm. "You can't hedge what percentage will actually be paid out. If you can only get 50% of your FTR payment, then there's really no way you can price that into a bid and effectively buy or sell an FTR. It's not worth the risk to do that."
Physical market participants, which use FTRs to hedge against specific congestion risks in PJM, have less choice in the matter. "It's certainly fair to say that FTR underfunding has affected the vitality and the usefulness of FTRs as a congestion hedge, and that obviously we've had to make business decisions that reflect the degraded value of FTR payouts," says Exelon's Barker.
Market sources suggest that one beneficiary of the FTR underfunding problem is Nodal Exchange, the Virginia-based futures exchange that specialises in power. Nodal Exchange offers contracts for energy plus congestion at selected nodes in PJM, a product sometimes described as an ‘FTR lookalike' contract. Such contracts can be used to hedge congestion risks, but without the risk of underfunding, according to Paul Cusenza, chief executive of Nodal Exchange. "There are certain entities that look at [FTR lookalike contracts] as a mechanism to get more certain pricing than would otherwise be available," he says.
Meanwhile, PJM says it is still grappling with FTR underfunding and continues to examine potential market design fixes. "It's not something that we have stopped analysing," says Bresler. "We certainly continue to look at this and see if there are ways we can make it better."
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### Agnostic Pointwise-Competitive Selective Classification
Pointwise-competitive classifier from class F is required to classify identically to the best classifier in hindsight from F. For noisy, agnostic settings we present a strategy for learning pointwise-competitive classifiers from a finite training sample provided that the classifier can abstain from prediction at a certain region of its choice. For some interesting hypothesis classes and families of distributions, the measure of this rejected region is shown to be diminishing at a fast rate, with high probability. Exact implementation of the proposed learning strategy is dependent on an ERM oracle that can be hard to compute in the agnostic case. We thus consider a heuristic approximation procedure that is based on SVMs, and show empirically that this algorithm consistently outperforms a traditional rejection mechanism based on distance from decision boundary.
### Online Learning: Random Averages, Combinatorial Parameters, and Learnability
We develop a theory of online learning by defining several complexity measures. Among them are analogues of Rademacher complexity, covering numbers and fat-shattering dimension from statistical learning theory. Relationship among these complexity measures, their connection to online learning, and tools for bounding them are provided. We apply these results to various learning problems. We provide a complete characterization of online learnability in the supervised setting.
### From Stochastic Mixability to Fast Rates
Empirical risk minimization (ERM) is a fundamental learning rule for statistical learning problems where the data is generated according to some unknown distribution $\mathsf{P}$ and returns a hypothesis $f$ chosen from a fixed class $\mathcal{F}$ with small loss $\ell$. In the parametric setting, depending upon $(\ell, \mathcal{F},\mathsf{P})$ ERM can have slow $(1/\sqrt{n})$ or fast $(1/n)$ rates of convergence of the excess risk as a function of the sample size $n$. There exist several results that give sufficient conditions for fast rates in terms of joint properties of $\ell$, $\mathcal{F}$, and $\mathsf{P}$, such as the margin condition and the Bernstein condition. In the non-statistical prediction with expert advice setting, there is an analogous slow and fast rate phenomenon, and it is entirely characterized in terms of the mixability of the loss $\ell$ (there being no role there for $\mathcal{F}$ or $\mathsf{P}$). The notion of stochastic mixability builds a bridge between these two models of learning, reducing to classical mixability in a special case. The present paper presents a direct proof of fast rates for ERM in terms of stochastic mixability of $(\ell,\mathcal{F}, \mathsf{P})$, and in so doing provides new insight into the fast-rates phenomenon. The proof exploits an old result of Kemperman on the solution to the general moment problem. We also show a partial converse that suggests a characterization of fast rates for ERM in terms of stochastic mixability is possible.
### Statistical Learning Theory: Models, Concepts, and Results
Statistical learning theory provides the theoretical basis for many of today's machine learning algorithms. In this article we attempt to give a gentle, non-technical overview over the key ideas and insights of statistical learning theory. We target at a broad audience, not necessarily machine learning researchers. This paper can serve as a starting point for people who want to get an overview on the field before diving into technical details.
### Fast Rates for Online Prediction with Abstention
In the setting of sequential prediction of individual $\{0, 1\}$-sequences with expert advice, we show that by allowing the learner to abstain from the prediction by paying a cost marginally smaller than $\frac 12$ (say, $0.49$), it is possible to achieve expected regret bounds that are independent of the time horizon $T$. We exactly characterize the dependence on the abstention cost $c$ and the number of experts $N$ by providing matching upper and lower bounds of order $\frac{\log N}{1-2c}$, which is to be contrasted with the best possible rate of $\sqrt{T\log N}$ that is available without the option to abstain. We also discuss various extensions of our model, including a setting where the sequence of abstention costs can change arbitrarily over time, where we show regret bounds interpolating between the slow and the fast rates mentioned above, under some natural assumptions on the sequence of abstention costs.
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# Projectivity and faithfully flatness (module theory) [closed]
Is it true that every projective module is faithfully flat, if not what is a counter example.
Thanks!
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## closed as too localized by Pete L. Clark, Hailong Dao, S. Carnahan♦Aug 8 '11 at 3:06
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$0$ is a counterexample, but I think that you want to exclude it. Every finitely generated projective module is locally free of finite rank and thus faithfully flat (or trivial). In the general case, we may approximate our module by finitely generated projective modules, at least if the ground ring is Dedekind. – Martin Brandenburg Aug 6 '11 at 15:54
A flat (right) module is faithfully flat iff $R/mR \neq 0$ for all (left) maximal ideals of $R$. Now take $R = \mathbb Z/(6)$, $M=\mathbb Z/(2)$. – Hailong Dao Aug 6 '11 at 15:56
Should be $M/mM \neq 0$ above! – Hailong Dao Aug 6 '11 at 16:35
Thanks, I found myself similar example: $R=\mathbb{Z}_{35}$ Then $R=\mathbb{Z}_7\times\mathbb{Z}_5$, thus both $\mathbb{Z}_5$ and $\mathbb{Z}_7$ are projective $R$-modules and $\mathbb{Z}_7\otimes_R\mathbb{Z}_5=0$. The same can be doe with $\mathbb{Z}_6$ (and any $\mathbb{Z}_{pq}$ for p, q distinct prime number.s – Marcin Szamotulski Aug 7 '11 at 0:47
If you like algebraic geometry, you can consider vector bundles of non-constant rank over a disconnected space. – S. Carnahan Aug 8 '11 at 3:10
Let $k$ be a field and consider the ring $k\times k$. There are two (indecomposable) projectives. Are they faithfully flat?
Perhaps this was a rhetorical question, but no: for the same reason as in Hailong Dao's comment. The point is that when $Spec R$ is disconnected (in the commutative case), you can play this kind of game. So the answer in this case is that a projective is faithfully flat iff its support is all of $Spec R$. – Donu Arapura Aug 6 '11 at 18:49
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# rolling dices and coins problem
• September 1st 2007, 03:27 PM
rbenito
rolling dices and coins problem
I have other probability problem, could anyone help me with this one?
Die A has five olive faces and one lavender face; die B has three faces of each of these colors. A fair coin is flipped once. If it falls heads, the game continues by throwing die A alone; if it falls tails, die B alone is used to continue the game. However awful their face color may be, it is known that both dice are fair.
a. Determine the probability that the nth throw of the die results in olive.
b. Determine the probability that both the nth and (n+1)st throw of the die results in olive.
c. If olive readings result from all the first n throws, determine the conditional probability of an olive outcome on the (n+1)st toss. Interpret your result for large values of n.
• September 1st 2007, 05:13 PM
galactus
Here's a start.
The probability of rolling olive on any roll is $(\frac{1}{2})(\frac{5}{6})+(\frac{1}{2})(\frac{3}{ 6})=\frac{2}{3}$
• September 4th 2007, 09:37 AM
rbenito
generalize for 'n' die tosses
How can I generalize for large values of 'n'?
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Note
Last update 04/05/2021
# List of currently implemented elements¶
SuperflexPy provides four levels of components (elements, units, nodes and network) for constructing conceptual hydrological models. The components presented in the page Organization of SuperflexPy represent the core of SuperflexPy. These components can be extended to create customized models.
Most of the customization efforts will be required for elements (i.e., reservoirs, lag, and connection elements). This page describes all elements that have been created and shared by the community of SuperflexPy. These elements can be used to construct a wide range of model structures.
This section lists elements according to their type, namely
• Reservoir
• Lag elements
• Connections
Within each section, the elements are listed in alphabetical order.
## Reservoirs¶
### Interception filter¶
This reservoir is used to simulate interception in models, including GR4J. Further details are provided in the page GR4J.
from superflexpy.implementation.elements.gr4j import InterceptionFilter
#### Inputs¶
• Potential evapotranspiration $$E^{\textrm{in}}_{\textrm{POT}}\ [LT^{-1}]$$
• Precipitation $$P^{\textrm{in}}\ [LT^{-1}]$$
#### Outputs from get_output¶
• Net potential evapotranspiration $$E^{\textrm{out}}_{\textrm{POT}}\ [LT^{-1}]$$
• Net precipitation $$P^{\textrm{out}}\ [LT^{-1}]$$
#### Governing equations¶
$\begin{split}& \textrm{if } P^{\textrm{in}} > E^{\textrm{in}}_{\textrm{POT}}: \\ & \quad P^{\textrm{out}} = P^{\textrm{in}} - E^{\textrm{in}}_{\textrm{POT}} \\ & \quad E^{\textrm{out}}_{\textrm{POT}} = 0 \\ \\ & \textrm{if } P^{\textrm{in}} < E^{\textrm{in}}_{\textrm{POT}}: \\ & \quad P^{\textrm{out}} = 0 \\ & \quad E^{\textrm{out}}_{\textrm{POT}} = E^{\textrm{in}}_{\textrm{POT}} - P^{\textrm{in}}\end{split}$
### Linear reservoir¶
This reservoir assumes a linear storage-discharge relationship. It represents arguably the simplest hydrological model. For example, it is used in the model HYMOD to simulate channel routing and lower-zone storage processes. Further details are provided in the page HYMOD.
from superflexpy.implementation.elements.hymod import LinearReservoir
#### Inputs¶
• Precipitation $$P\ [LT^{-1}]$$
#### Outputs from get_output¶
• Total outflow $$Q\ [LT^{-1}]$$
#### Governing equations¶
$\begin{split}& \frac{\textrm{d}S}{\textrm{d}{t}}=P - Q \\ & Q=kS\end{split}$
### Power reservoir¶
This reservoir assumes that the storage-discharge relationship is described by a power function. This type of reservoir is common in hydrological models. For example, it is used in the HBV family of models to represent the fast response of a catchment.
from superflexpy.implementation.elements.hbv import PowerReservoir
#### Inputs¶
• Precipitation $$P\ [LT^{-1}]$$
#### Outputs from get_output¶
• Total outflow $$Q\ [LT^{-1}]$$
#### Governing equations¶
$\begin{split}& \frac{\textrm{d}S}{\textrm{d}{t}}=P - Q \\ & Q=kS^{\alpha}\end{split}$
### Production store (GR4J)¶
This reservoir is used to simulate runoff generation in the model GR4J. Further details are provided in the page GR4J.
from superflexpy.implementation.elements.gr4j import ProductionStore
#### Inputs¶
• Potential evapotranspiration $$E_{\textrm{pot}}\ [LT^{-1}]$$
• Precipitation $$P\ [LT^{-1}]$$
#### Outputs from get_output¶
• Total outflow $$P_{\textrm{r}}\ [LT^{-1}]$$
#### Secondary outputs¶
• Actual evapotranspiration $$E_{\textrm{act}}\ [LT^{-1}]$$ get_aet()
#### Governing equations¶
$\begin{split}& \frac{\textrm{d}S}{\textrm{d}{t}}=P_{\textrm{s}}-E_{\textrm{act}}-Q_{\textrm{perc}} \\ & P_{\textrm{s}}=P\left(1-\left(\frac{S}{x_1}\right)^\alpha\right) \\ & E_{\textrm{act}}=E_{\textrm{pot}}\left(2\frac{S}{x_1}-\left(\frac{S}{x_1}\right)^\alpha\right) \\ & Q_{\textrm{perc}} = \frac{x^{1-\beta}}{(\beta-1)}\nu^{\beta-1}S^{\beta} \\ & P_{\textrm{r}}=P - P_{\textrm{s}} + Q_{\textrm{perc}}\end{split}$
### Routing store (GR4J)¶
This reservoir is used to simulate routing in the model GR4J. Further details are provided in the page GR4J.
from superflexpy.implementation.elements.gr4j import RoutingStore
#### Inputs¶
• Precipitation $$P\ [LT^{-1}]$$
#### Outputs from get_output¶
• Outflow $$Q\ [LT^{-1}]$$
• Loss term $$F\ [LT^{-1}]$$
#### Governing equations¶
$\begin{split}& \frac{\textrm{d}S}{\textrm{d}{t}}=P-Q-F \\ & Q=\frac{x_3^{1-\gamma}}{(\gamma-1)}S^{\gamma} \\ & F = \frac{x_2}{x_3^{\omega}}S^{\omega}\end{split}$
### Snow reservoir¶
This reservoir is used to simulate snow processes based on temperature. Further details are provided in the section Dal Molin et al., 2020, HESS.
from superflexpy.implementation.elements.thur_model_hess import SnowReservoir
#### Inputs¶
• Precipitation $$P\ [LT^{-1}]$$
• Temperature $$T\ [°C]$$
#### Outputs from get_output¶
• Sum of snow melt and rainfall input $$=P-P_{\textrm{snow}}+M\ [LT^{-1}]$$
#### Governing equations¶
$\begin{split}& \frac{\textrm{d}S}{\textrm{d}{t}}=P_{\textrm{snow}}-M \\ & P_{\textrm{snow}}=P\quad\textrm{if } T\leq T_0;\quad\textrm{else } 0 \\ & M = M_{\textrm{pot}}\left(1-\exp\left(-\frac{S}{m}\right)\right) \\ & M_{\textrm{pot}}=kT\quad\textrm{if } T\geq T_0;\quad\textrm{else } 0 \\\end{split}$
### Unsaturated reservoir (inspired to HBV)¶
This reservoir specifies the actual evapotranspiration as a smoothed threshold function of storage, in combination with the storage-discharge relationship being set to a power function. It is inspired by the HBV family of models, where a similar approach (but without smoothing) is used to represent unsaturated soil dynamics.
from superflexpy.implementation.elements.hbv import UnsaturatedReservoir
#### Inputs¶
• Precipitation $$P\ [LT^{-1}]$$
• Potential evapotranspiration $$E_{\textrm{pot}}\ [LT^{-1}]$$
#### Outputs from get_output¶
• Total outflow $$Q\ [LT^{-1}]$$
#### Secondary outputs¶
• Actual evapotranspiration $$E_{\textrm{act}}$$ get_AET()
#### Governing equations¶
$\begin{split}& \frac{\textrm{d}S}{\textrm{d}{t}}=P - E_{\textrm{act}} - Q \\ & \overline{S} = \frac{S}{S_{\textrm{max}}} \\ & E_{\textrm{act}}=C_{\textrm{e}}E_{\textrm{pot}}\left(\frac{\overline{S}(1+m)}{\overline{S}+m}\right) \\ & Q=P\left(\overline{S}\right)^{\beta}\end{split}$
### Upper zone (HYMOD)¶
This reservoir is part of the HYMOD model and is used to simulate the upper soil zone. Further details are provided in the page HYMOD.
from superflexpy.implementation.elements.hymod import UpperZone
#### Inputs¶
• Precipitation $$P\ [LT^{-1}]$$
• Potential evapotranspiration $$E_{\textrm{pot}}\ [LT^{-1}]$$
#### Outputs from get_output¶
• Total outflow $$Q\ [LT^{-1}]$$
#### Secondary outputs¶
• Actual evapotranspiration $$E_{\textrm{act}}\ [LT^{-1}]$$ get_AET()
#### Governing equations¶
$\begin{split}& \frac{\textrm{d}S}{\textrm{d}{t}}=P - E_{\textrm{act}} - Q \\ & \overline{S} = \frac{S}{S_{\textrm{max}}} \\ & E_{\textrm{act}}=E_{\textrm{pot}}\left(\frac{\overline{S}(1+m)}{\overline{S}+m}\right) \\ & Q=P\left(1-\left(1-\overline{S}\right)^{\beta}\right)\end{split}$
## Lag elements¶
All lag elements implemented in SuperflexPy can accommodate an arbitrary number of input fluxes, and apply a convolution based on a weight array that defines the shape of the lag function.
Lag elements differ solely in the definition of the weight array. The nature (i.e., number and order) of inputs and outputs depend on the element upstream of the lag element.
The weight array can be defined by giving the area below the lag function as a function of the time coordinate. The maximum lag $$t_{\textrm{lag}}$$ must also be specified. The weights are then given by differences between the values of the area at consecutive lags. This approach is shown in the figure above, where the weight $$W_i$$ is calculated as the difference between areas $$A_i$$ and $$A_{i-1}$$.
### Half triangular lag¶
This lag element implements the element present in the case study Dal Molin et al., 2020, HESS and used in other Superflex studies.
from superflexpy.implementation.elements.thur_model_hess import HalfTriangularLag
#### Definition of weight array¶
The area below the lag function is given by
$\begin{split}&A_{\textrm{lag}}(t) = 0 & \quad \textrm{for } t \leq 0\\ &A_{\textrm{lag}}(t) = \left(\frac{t}{t_{\textrm{lag}}}\right)^2 & \quad \textrm{for } 0< t \leq t_{\textrm{lag}}\\ &A_{\textrm{lag}}(t) = 1 & \quad \textrm{for } t > t_{\textrm{lag}}\end{split}$
The weight array is then calculated as
$w(t_{\textrm{i}}) = A_{\textrm{lag}}(t_{\textrm{i}}) - A_{\textrm{lag}}(t_{\textrm{i-1}})$
### Unit hydrograph 1 (GR4J)¶
This lag element implements the unit hydrograph 1 of GR4J.
from superflexpy.implementation.elements.gr4j import UnitHydrograph1
#### Definition of weight array¶
The area below the lag function is given by
$\begin{split}&A_{\textrm{lag}}(t) = 0 & \quad \textrm{for } t \leq 0\\ &A_{\textrm{lag}}(t) = \left(\frac{t}{t_{\textrm{lag}}}\right)^\frac{5}{2} & \quad \textrm{for } 0< t \leq t_{\textrm{lag}}\\ &A_{\textrm{lag}}(t) = 1 & \quad \textrm{for } t > t_{\textrm{lag}}\end{split}$
The weight array is then calculated as
$w(t_{\textrm{i}}) = A_{\textrm{lag}}(t_{\textrm{i}}) - A_{\textrm{lag}}(t_{\textrm{i-1}})$
### Unit hydrograph 2 (GR4J)¶
This lag element implements the unit hydrograph 2 of GR4J.
from superflexpy.implementation.elements.gr4j import UnitHydrograph2
#### Definition of weight array¶
The area below the lag function is given by
$\begin{split}&A_{\textrm{lag}}(t) = 0 & \quad \textrm{for } t \leq 0\\ &A_{\textrm{lag}}(t) = \frac{1}{2}\left(\frac{2t}{t_{\textrm{lag}}}\right)^\frac{5}{2} & \quad \textrm{for } 0< t \leq \frac{t_{\textrm{lag}}}{2}\\ &A_{\textrm{lag}}(t) = 1 - \frac{1}{2}\left(2-\frac{2t}{t_{\textrm{lag}}}\right)^\frac{5}{2} & \quad \textrm{for } \frac{t_{\textrm{lag}}}{2}< t \leq t_{\textrm{lag}}\\ &A_{\textrm{lag}}(t) = 1 & \quad \textrm{for } t > t_{\textrm{lag}}\end{split}$
The weight array is then calculated as
$w(t_{\textrm{i}}) = A_{\textrm{lag}}(t_{\textrm{i}}) - A_{\textrm{lag}}(t_{\textrm{i-1}})$
## Connections¶
SuperflexPy implements four connection elements:
• splitter
• junction
• transparent element
In addition, customized connectors have been implemented to achieve specific model designs. These customized elements are listed in this section.
### Flux aggregator (GR4J)¶
This element is used to combine routing, exchange and outflow fluxes in the GR4J model. Further details are provided in the page GR4J.
from superflexpy.implementation.elements.gr4j import FluxAggregator
#### Inputs¶
• Outflow routing store $$Q_{\textrm{RR}}\ [LT^{-1}]$$
• Exchange flux $$Q_{\textrm{RF}}\ [LT^{-1}]$$
• Outflow UH2 $$Q_{\textrm{UH2}}\ [LT^{-1}]$$
#### Main outputs¶
• Outflow $$Q\ [LT^{-1}]$$
#### Governing equations¶
$\begin{split}& Q = Q_{\textrm{RR}} + \max(0;Q_{\textrm{UH2}} - Q_{\textrm{RF}}) \\\end{split}$
|
## Nothing is impossible (?)
Something’s been bothering me for a while now. While my statistical intuition is usually quite sharp, in this case I’m wrong, although I never really heard a convincing argument why exactly. So I’ll try to explain my problem, and hopefully someone with a sound argument could reply. Basically, my confusion starts with the sentence ‘Some event that has a chance of $0$ of happening, can still happen.’ Let $P(x)$ be the chance that $x$ happens. To me `$P(x) = 0$‘ is equivalent to the statement: ‘We are certain that $x$ will not be the case’. So saying that $P(x) = 0$ doesn’t exclude $x$ from happening feels just like a contradictio in terminis, to me. For example, the chance that you throw a (normal) die and a $7$ turns up*. But this is a silly example, because everybody agrees that the chance of this happening is $0$ and also that it is impossible. But now, consider the following: If we pick a random number, the chance we pick a certain number $N$ is clearly $0$. Since this holds for every $N$, you might argue that this implies I’m wrong, because apparently something happened which had a chance of $0$. To me this just implies that it’s not possible to choose a number randomly. A different example: we flip a coin until heads turns up. Is it possible that this game never ends? To me, it’s not. Of course, it could take arbitrarily long, but it can’t take an infinite number of flips. Can it?
While trying to emphatize with the people who believe that ‘impossible events are possible’, I came up with the following: let’s say we pick a random real number from the unit interval. Since there are uncountably many reals and only a countable number of rationals, the chance to pick a fraction is $0$. But you could still say: ‘well, it’s only random. And fractions actually do exist in abundance. So there is no obvious way that it shouldn’t be possible to choose a fraction by accident’. While this idea makes sense to some extent, I believe it’s wrongheaded. Mostly because, like I said above, this just seems to be based on the belief that it is actually possible to pick a number in some truly random way.
In any case, this is not meant as a convincing argument against the idea of the impossible becoming possible. I’m just trying to show where I get confused, and I am in need of some explanation. And after all, I’ve never seen anything in my life happen that had a zero chance of happening. So the burden of proof lies with the people who claim that $2$ is a random integer ;).
*maybe it actually could happen by some weird quantum effect, but mathematically speaking, it’s not an issue
PS. Terry Tao is going to set up another mini-Polymath Project next month. Check it out.
### 4 Responses to “Nothing is impossible (?)”
1. nhung Says:
I like this blog very much^_^.
2. My idea Says:
😉
3. My idea Says:
;;) :)) :((
4. My idea Says:
:-b :-d
While I clearly appreciate your smilies, I have to ask you to make your comments slightly more senseful in the future, b:-d). W
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# Frequency, Time Period And Angular Frequency
In wave mechanics, any given wave enfolds parameters like- frequency, time period, wavelength, amplitude etc. The time period is the time taken by a complete cycle of the wave to pass a point, Frequency is the number of complete cycle of waves passing a point in unit time. Angular frequency is angular displacement of any element of the wave per unit time.
Consider the graph shown below. It represents the displacement y of any element for a harmonic wave along a string moving in the positive x-direction with respect to time. Here, the element of the string moves up and down in simple harmonic motion.
The relation describing the displacement of the element with respect to time is given as:
y (0,t) = a sin (–ωt), here we have considered the inception of from x=0
y (0,t) = -a
As we know sinusoidal or harmonic motion is periodic in nature, i.e. the nature of the graph of an element of the wave repeats itself at a fixed duration. In order to mark the duration of periodicity following terms are introduced for sinusoidal waves.
Following is the table explaining other related concepts of waves:
## What is Time Period?
In sinusoidal wave motion, as shown above, the particles move about the mean equilibrium or mean position with the passage of time. The particles rise till they reach the highest point that is the crest and then continue to fall till they reach the lowest point that is the trough. The cycle repeats itself in a uniform pattern. The time period of oscillation of a wave is defined as the time taken by any element of the string to complete one such oscillation. For a sine wave represented by the equation:
y (0, t) = -a sin(ωt)
The time period formula is given as:
$T=\frac{2\pi }{\omega }$
## What is Frequency?
We can define frequency of a sinusoidal wave as the number of complete oscillations made by any element of the wave per unit time. For a sinusoidal wave represented by the equation:
y (0,t) = -a
The formula of the frequency with the SI unit is given as:
Formula $f=\frac{1}{T}=\frac{\omega }{2\pi }$ SI unit Hertz
One Hertz is equal to one complete oscillation taking place per second.
## What is Angular Frequency?
For a sinusoidal wave, the angular frequency refers to the angular displacement of any element of the wave per unit time or the rate of change of the phase of the waveform. It is represented by ω. Angular frequency formula and SI unit are given as:
Formula $\omega=\frac{2\pi }{T}=2\pi f$ SI unit rads-1
Where,
• ω = angular frequency of the wave.
• T = time period of the wave.
• f = ordinary frequency of the wave.
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# 500 ml of "0.3 M AgNO"_(3) are added to 600 ml of "0.5 M KI" solution. The ions which will move towards the cathode and anode respectively are
Agl//Ag^(+)and NO_(3)^(-)Agl//l^(-) and K^(+)K^(+) and K^(+)K^(+) and Agl//l^(-)
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# ggfan v0.1.0
0
0th
Percentile
## Summarise a Distribution Through Coloured Intervals
Implements the functionality of the 'fanplot' package as 'geoms' for 'ggplot2'. Designed for summarising MCMC samples from a posterior distribution, where a visualisation is desired for several values of a continuous covariate. Increasing posterior intervals of the sampled quantity are mapped to a continuous colour scale.
# ggfan
Jason Hilton 14 November, 2017
## Summarise a distribution through coloured intervals
This package is a ggplot-based implementation of some of the functionality of the fanplot package by Guy Abel. Fanplot provides methods to visualise probability distributions by representing intervals of the distribution function with colours. Plotting samples from posterior distributions obtained through MCMC is a particular aim. A ggplot-based implementation is useful as it allows leverage of the power of ggplot2 features allowing, for example, facetting and easy theming.
## Installation
The package has recently been submitted to CRAN. Installation directly from CRAN will be possible if and when the package is accepted: install.packages("ggfan") Otherwise, the package can be installed directly from github using the devtools package: devtools::install_github("jasonhilton/ggfan").
## Quick Start
The provided fake_df data gives and example of the type of data you might want to plot with ggfan: it consists of 1000 samples of an outcome variable of interest at each value of a covariate x. We can plot this simply using standard ggplot2 syntax and geom_fan. Convenient ggplot features such as themes, colour scales and facetting can also be used.
ggplot(fake_df, aes(x=x, y=y)) + geom_fan() + theme_minimal() + scale_fill_distiller(palette="Spectral")
## Further Information
A full description of the functionality of ggfan is provided in the vignette, provided here.
A brief example of using ggfan to plot stan output is given in a second vignette here
## Functions in ggfan
Name Description geom_fan Fan plot visualising intervals of a distribution gp_model_fit A stan_fit object used in the ggfan_stan vignette, containing posterior samples from a latent gaussian process model. This is provided as data to avoid having to conduct computationally expensive sampling when producing the vignettes. GeomIntervalPath See ggplot2-ggproto stat_interval Line plot visualising intervals of a distribution GeomIntervalPoly See ggplot2-ggproto stat_sample Plots a randomly chosen sample of the specified groups using geom_line StatInterval See ggplot2-ggproto StatIntervalFctr See ggplot2-ggproto StatSample See ggplot2-ggproto calc_quantiles Calculate quantiles of a tidy dataframe geom_interval Line plot visualising intervals of a distribution ggfan Fanplots for ggplot2 fake_df Fake dataset intended to resemble a set of MCMC samples of a variable over one covariate (perhaps time), No Results!
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# Disable Implicit Casts¶
Consider the following simple function:
void foo(uint32_t x);
The problem with foo is that I can call it with anything which can be implicitly cast to a uint32_t with no complaints from the compiler. It is perfectly valid to call foo with a floating-point number: foo(5.9f). The compiler will happily add an implicit cast to transform the argument to a 5. Most compilers have warnings that can be turned on to catch this sort of thing, but if you are a library writer or you are working on a project where turning on these warnings would be a hassle, then -Wconversion is not a reliable option.
I want to opt-in to disabling implicit casts for a single function. To do this for foo, add the following declaration:
template <typename T>
void foo(T) = delete;
When the compiler sees a statement like foo(8.3), it searches for functions by name and it finds two valid candidates for the name foo: The templated one and the regular one. Since we have more than one available function matching the name, we have to disambiguate by the argument types. The definition template <typename T> foo can be validly instantiated by the expression foo(8.3) with T=double. The instantiated function signature foo(double) is a better match than foo(uint32_t), so the compiler decides that you mean to call the templated function. Now we ask the question: Is it legal to call this function? That’s when <tt>= delete</tt> comes in – you can not call a deleted function. Bam! No more implicit conversions allowed.</p>
Note
The question of “Is it legal to call this?” really is the last question the compiler asks. This can lead to far more surprising results in cases like this:
void bar(int x)
{ }
class base
{
private:
void bar() = delete;
};
class derived : public base
{
public:
derived()
{
bar(8);
}
};
At first glance, the expression bar(8) in derived::derived would call the free function which accepts an integer, but it turns out only base::bar is found in the name-matching stage. The free function bar we probably intended to call is never even evaluated for argument type match. For more fun times with name lookups, check out the awesome article Guru of the Week #30: Name Lookup.
## Multiple Arguments¶
This same technique works for multiple argument functions, too:
void baz(size_t x, double y);
void baz(const std::string& s, const std::vector<uint32_t>& v);
template <typename T0, typename T1>
void baz(T0, T1) = delete;
This works for the same reasons the single-argument foo works, except with two arguments. An interesting question is: Why do you need both T0 and T1? Why not make the definition more like:
template <typename T>
void baz(T, T) = delete;
That definition says that for a template to match that pattern, both arguments must be the same; this can lead to many unexpected consequences. Consider the expression baz(5.6, 1). The type for T is ambiguous between double and int, which is a substitution failure (not an error), so template <typename T> baz will not be matched. This leaves baz(size_t, double) and baz(const std::string&, const std::vector<uint32_t>&). There is an implicit conversion of 5.6 to a size_t and of 1 to a double, so baz(size_t, double) is selected. The lesson? Use a separate template parameter for each function argument you want to disable implicit casting for.
So what if you have arguments where you do not care if there is an implicit cast? You can do that, too!
void fizz(uint32_t x, const std::string& s);
template <typename T>
void fizz(T, const std::string&) = delete;
## Swashbuckling with Variadic Templates¶
Let’s say I have a whole host of functions:
void* buzz(const something& s);
double buzz(int, double, float, long long);
int buzz(const char*, int);
long long buzz(unsigned int, long long);
Yuck! All these functions have different return types and different numbers of arguments (arity). I do not want any implicit conversions happening on any of them. Now, I could use the template trick for each arity of buzz, but there are two major problems with doing that. If someone comes in and adds a function of arity 3 (say buzz(int, double, void*)), we will not have a function deletion covering it. More importantly, writing = delete so frequently is a lot of work (and a good C++ pirate is lazy). Here, variadic templates come to the rescue!
template <typename... T>
void buzz(T...) = delete;
Now, any attempted call to a function named buzz which does not exactly match a non-template option will show up as a deleted function. You probably noticed that we do not care about the return type at all. This means that expressions needing a specific return type from the result will fail, but that does not matter, since the expression is invalid anyway due to the = delete business. That means we can wrap this whole deal up in a quite nice-looking macro:
#define DISABLE_IMPLICIT_CASTS(func_name) \
template <typename... T> \
void func_name(T...) = delete
Then again, the = delete looks fine to me.
|
## Baltic OI '14 P5 - Portals
View as PDF
Points: 15 (partial)
Time limit: 1.0s
Memory limit: 256M
Problem type
##### Baltic Olympiad in Informatics: 2014 Day 2, Problem 2
There is a cake placed in a labyrinth and you desperately want to eat it. You have a map of the labyrinth, which is a grid of rows and columns. Each grid cell contains one of the following characters:
• # which denotes a wall block,
• . which denotes an open square,
• S which denotes an open square of your current location,
• C which denotes an open square with the cake.
You may only walk on the open squares and move from one open square to another if they share a side. Additionally, the rectangular area depicted on the map is completely surrounded by wall blocks.
In order to reach the cake faster you have acquired a portal gun from Aperture Science™, which operates as follows. At any time it can fire a portal in one of the four directions up, left, down and right. When a portal is fired in some direction, it will fly in that direction until it reaches the first wall. When this happens, a portal will be spawned on the wall block, on the side that faces you.
At most two portals can exist at any given time. If two portals are already placed in the labyrinth, then one of them (selected by you) will be removed immediately upon using the portal gun again. Firing a portal at an existing portal will replace it (there may be at most one portal per side of wall block). Note that there may be two portals placed on different sides of the same wall block.
Once two portals are placed in the labyrinth you can use them to teleport yourself. When standing next to one of the portals, you can walk into it and end up at the open square next to the other portal. Doing this takes as much time as moving between two adjacent squares.
You may assume that firing portals does not take time and moving between two adjacent squares or teleporting through portals takes one unit of time.
Given the map of the labyrinth together with your starting location and the location of the cake, calculate the minimum possible time needed for you to reach the cake.
#### Constraints
Every open square has at least one wall block adjacent to it.
#### Input Specification
The first line of the input contains two space-separated integers: the number of rows in the map , and the number of columns . The next lines describe the map. Each of these lines contains characters: #, ., S or C (whose meaning is described above).
It is guaranteed that characters S and C each appear exactly once in the map.
#### Output Specification
The output should contain a single integer — the minimum time that is needed to reach the cake from the starting location.
You may assume that it is possible to reach the cake from your starting location.
#### Sample Input
4 4
.#.C
.#.#
....
S...
#### Sample Output
4
#### Explanation for Sample
One quickest sequence of moves is as follows:
1. Move right.
2. Move right, shoot one portal up and one portal down.
3. Move through the bottom portal.
4. Move one square right and reach the cake.
|
12345>>
1.
Which among the following equations represents in the reduction reaction taking place in lead accumulator at the positive electrode, while it is being used as a source of electrical energy?
A) $Pb\rightarrow Pb^{2+}$
B) $Pb^{4+}\rightarrow Pb$
C) $Pb^{2+}\rightarrow Pb$
D) $Pb^{4+}\rightarrow Pb^{2+}$
2.
A molecule of stachyose contains how many carbon atoms?
A) 6
B) 12
C) 18
D) 24
3.
The amino acid, which is basic in nature is
A) histidine
B) tyrosine
C) proline
D) valine
4.
Which of the following co-ordinate complexes is an exception to EAN rule?
(Given atomic number Pt=78, Fe=26, Zn=30, Cu=29)
A) $[Pt(NH_{3})6]^{4+}$
B) $[Fe(CN)_6]^{4-}$
C) $[Zn(NH_3)_4]^{2+}$
D) $[Cu(NH_3)_4]^{2+}$
5.
What is the actual volume occupied by water molecules present in 20 cm3 of water ?
A) 20 $cm^{3}$
B) 10 $cm^{3}$
C) 40 $cm^{3}$
D) 24.89 $cm^{3}$
12345>>
|
# Lecture 21: deterministic finite automata
• We started with a second example proof by structural induction; this has been added to the end of the notes for lecture 20
• We defined deterministic finite automata, and the extended transition function
• Review exercises:
• Write the definition of $$\hat{δ}$$.
• Let $$M$$ be a machine with one final state $$q$$; and a transition function that takes $$q$$ back to $$q$$ on every character of $$Σ$$. Find $$L(M)$$
• If every $$x \in L$$ is accepted by $$M$$, is $$L$$ recognized by $$M$$?
• If $$L$$ is recognized by $$M$$, is every string in $$L$$ accepted by $$M$$?
• Draw an automaton that recognizes the set of even length strings, the set of all strings, the empty set, etc.
## Overview
An automaton is an extremely simple model of a computer or a program. The automata we will study examine an input string character by character and either say "yes" (accept the string) or "no" (reject the string).
Automata are defined by state transition diagrams. Here is one example:
This automaton processes strings containing the characters 0 and 1. It contains 4 states, $$q_{ee}$$, $$q_{eo}$$, $$q_{oe}$$ and $$q_{oo}$$.
While processing a string $$x$$, the machine starts at the beginning of $$x$$, and in the start state $$q_{ee}$$ (as indicated by the arrow pointing to $$q_{ee}$$). As it processes each character, it follows the corresponding transitions (arrows). When it has finished processing the string, if it is in a final state ($$q_{eo}$$ in this case, as indicated by the double circle), it accepts $$x$$; otherwise it rejects $$x$$.
For example, while processing $$1000110$$, the machine will start in state $$q_{ee}$$, then transition in order to states $$q_{eo}$$ (after processing the 1), $$q_{oo}$$ (after the first 0), then $$q_{eo}$$, $$q_{oo}$$, $$q_{eo}$$, $$q_{ee}$$, and finally end up in $$q_{eo}$$. Since $$q_{eo}$$ is an accepting state, the string $$1000110$$ is accepted.
Although this model of computation is very limited, it is sophisticated enough to demonstrate several kinds of analysis that apply to more sophisticated models:
• we'll talk about translating "programs" (automata) from one representation to another, and proving that those translations are correct. This is analagous to building and verifying compilers
• we'll show how to optimise automata, again proving that our optimizations don't change the behavior of the program
• we'll show that there are specifications (sets of strings) that can't be recognized by any automaton. Similar results apply to fully general models of computation, and have important practical implications.
• we'll talk about non-determinism, an important concept when reasoning about programs.
## Definitions
Before we do any of that, we need to formalize the informal definition of an automaton and its operation.
Definitions: A deterministic finite automaton $$M$$ is a 5-tuple $$M = (Q,Σ,δ,q_0,F)$$, where
• $$Q$$ is a finite set, called the set of states of $$M$$;
• $$Σ$$ is a finite set called the alphabet of $$M$$ (elements of $$Σ$$ are called characters)
• $$δ$$ is a function $$δ : Q \times Σ → Q$$, called the transition function. In the picture, there is a transition from $$q$$ to $$q'$$ on input $$a$$ if $$δ(q,a) = q'$$.
• $$q_0 \in Q$$ is the start state
• $$F \subseteq Q$$ is the set of final states
In the example diagram above,
• $$Q = \{q_{ee}, q_{eo}, q_{oe}, q_{oo}\}$$
• $$Σ = \{0,1\}$$
• $$δ$$ is given by the following table:
$$q$$ $$a$$ $$δ(q,a)$$
$$q_{ee}$$ 0 $$q_{oe}$$
$$q_{ee}$$ 1 $$q_{eo}$$
$$q_{eo}$$ 0 $$q_{oo}$$
$$q_{eo}$$ 1 $$q_{ee}$$
$$q_{oe}$$ 0 $$q_{ee}$$
$$q_{oe}$$ 1 $$q_{oo}$$
$$q_{oo}$$ 0 $$q_{eo}$$
$$q_{oo}$$ 1 $$q_{oe}$$
• $$q_0 = q_{ee}$$
• $$F = \{q_{eo}\}$$
Given an automaton $$M$$, we define the extended transition function $$\hat{δ} : Q \times \Sigma^{\bf *} → Q$$. Informally, $$\hat{δ}(q,x)$$ tells us where $$M$$ ends up after processing the entire string $$x$$; contrast the domain with that of $$δ$$, which processes only a single character. This distinction is important: since $$δ$$ only processes characters, its domain is finite, so the description of the machine is finite; but $$\hat{δ}$$ (which is not part of the description of the machine) can process an infinite number of strings.
Definition: Formally, we define the extended transition function $$\hat{δ} : Q \times Σ^* → Q$$ inductively by $$\hat{δ}(q,ε) = q$$, and $$\hat{δ}(q,xa) = δ(\hat{δ}(q,x), a)$$.
The first part of this definition says that processing the empty string doesn't move the machine; the second part says that to process $$xa$$, you first process $$x$$, and then take one more step using $$a$$ from wherever $$x$$ gets you.
|
# Math Help - Overlapping Triangles. Trick Question?
1. ## Overlapping Triangles. Trick Question?
This is on my homework sheet. Still can't figure it out :S... Help?
Find L-D to 5 Decimal Places:
- Donny
2. Looking at the bottom triangle, you can see that:
$\cos 50^{\circ} = \frac{\text{LD}}{\text{LR}} = \frac{\text{LD}}{80 \text{ cm}}$
Assuming you can use a calculator, then: $\text{LD} = 80 \text{ cm} \times \cos 50^{\circ}$
|
## Cryptology ePrint Archive: Report 2020/876
Direct Sum Masking as a Countermeasure to Side-Channel and Fault Injection Attacks
Claude Carlet and Sylvain Guilley and Sihem Mesnager
Abstract: Internet of Things is developing at a very fast rate. In order to ensure security and privacy, end-devices (e.g. smartphones, smart sensors, or any connected smartcards) shall be protected both against cyber attacks (coming down from the network) and against physical attacks (arising from attacker low-level interaction with the device). In this context, proactive protections shall be put in place to mitigate information theft from either side-channel monitoring or active computation/data corruption. Although both countermeasures have been developing fast and have become mature, there has surprisingly been little research to combine both.
In this article, we tackle this difficult topic and highlight a viable solution. It is shown to be more efficient than mere fault detection by repetition (which is anyway prone to repeated correlated faults). The presented solution leverages the fact that both side-channel protection and fault attack detection are coding techniques. We explain how to both prevent (higher-order) side-channel analyses and detect (higher-order) fault injection attacks. The specificity of this method is that it works end-to-end'', meaning that the detection can be delayed until the computation is finished. This simplifies considerably the error management logic as there is a single verification throughout the computation.
Category / Keywords: implementation / Security, privacy, Internet of Things, side-channel analysis, fault injection attacks, countermeasure, high-order, coding theory, direct sum masking (DSM)
Original Publication (with minor differences): Security and Privacy in the Internet of Things 2019
DOI:
10.3233/AISE200008
|
# [NTG-context] Using hyphen in math mode
Brian R. Landy brian at landy.cx
Tue Aug 16 18:56:04 CEST 2016
Hi, I was wondering if there is a way to access the hyphen symbol
(0x002D) while in math mode? I want to avoid the standard remapping to
Unicode minus 0x2212. It would also be workable if I could select the
math font without switching to math mode.
The problem with an approach like \math{\text{-}} is that it switches
the font back to the text font and does not use the math font. I.e, I'm
trying to get the true hyphen symbol set in whatever font is currently
defined for math mode.
Best regards,
Brian
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# How Siemens Is Tackling the Smart Grid
Siemens (s SI), the German energy and engineering giant, has been slower to move into the smart grid than fellow competitors like GE and Swiss-based ABB (s ABB). But over the past year Siemens has seemed to be making up for lost time, announcing a flurry of new projects and partnerships, and saying that it wants to double its current growth rate in the smart grid sector to capture €6 billion ($8.48 billion) in global business over the next five years, compared to its current €1 billion ($1.41 billion) in estimated smart grid related revenues in the fiscal year ending Sept. 2009.
So how exactly does Siemens plan to generate all that money? Well, as with fellow giant energy conglomerate General Electric (s GE), it’s sometimes easier to point to what Siemens is not doing than what it is doing in industries where it has a broad and sprawling interest. Here are our takeaways for what Siemens is, and isn’t, doing for the smart grid.
1). Concentrating on end-to-end network infrastructure: Siemens is one of the few corporations out there that can lay claim to almost every share of the world’s current grid infrastructure, building everything from gas and wind turbines to high-voltage transmission cables to sensors and controls that monitor and manage the delivery of power to homes and businesses.
The company is taking a similar approach for the smart grid. Right now the company is working on transmission grid sensors and controls, as well as high-voltage direct current transmission lines that can carry power from massive offshore wind farms like the London Array or for other specialty applications, such as an underwater cable project it is building to carry power across the San Francisco Bay. Beyond the physical infrastructure of transmission, Siemens has also developed an open platform to manage the wholesale sale and delivery of power between utilities and grid operators (see Greentech Media).
Farther down into the distribution grid — the network of lower-voltage lines that carry power from substations into neighborhoods and to individual businesses and homes — Siemens is doing a number of projects with utilities including Texas’ Oncor, New York’s Consolidated Edison and New England’s Northeast Utilities. These projects combine grid sensors and control systems with communications and back-end software to manage it all.
2). Providing software to unify the pieces of the smart grid puzzle: One of Siemens’ most recent smart grid projects — a stimulus-funded distribution automation system being built for Kansas City Power & Light — helps highlight Siemens’ software-centric approach. There are a confusing array of different equipment, networking and other technology vendors supplying products for such projects, but Siemens would like to make sure it can provide utilities with a few key pieces of software to manage them all, Dave Pacyna, senior vice president of Siemens Energy’s North American transmission and distribution division, told us in an interview this week.
“The more time goes on, the more we’re convinced that all these applications being talked about — like demand response programs, like fault location services and applications, like microgrid control and operation — all these kinds of things are ultimately going to be most efficiently deployed when they can come through one or two common software platforms,” Pacyna said. Siemens already makes software to manage the complex network of transmission and distribution assets.
3). Find partners that can help fill in the gaps: Because Siemens is so large and has so many broad smart grid offerings, it needs to partner with some of the smaller firms to deliver the most advanced technology across the network. Pacyna told us he sees managing smart meter data as a key software platform for the smart grid, but Siemens doesn’t make that kind of software. Instead, Siemens is relying on eMeter, a well-funded startup with several large-scale contracts to help utilities manage their millions of new smart meters. Siemens also led a \$12.5 million investment in eMeter in 2008, and the two are now involved in several meter data management projects with European utilities.
Siemens recently announced a smart grid partnership and investment in BPL Global, a maker of smart distribution grid systems. Siemens will be using BPL’s technology to control distributed generation resources like rooftop solar panels, as well as allow utilities to directly power down loads at customers’ premises to better balance the grid’s flow of electricity.
Siemens is also working with startup Viridity Energy to combine Siemens’ decentralized energy management system with Viridity’s system for managing “virtual power plants,” a collection of loads and distributed generation resources at office parks, university campuses or other discrete entities. While Siemens has a thriving building energy management business, it hasn’t yet linked those technologies with its smart grid network management systems in a comprehensive way, and BPL and Viridity’s capabilities could be seen as ways to bridge that gap.
4). The smart meter is not the linchpin of the smart grid: Unlike GE, which has built up a substantial smart meter business, Siemens sold off its metering business to a private equity firm about seven years ago, and the business unit’s technology ended up as part of Landis+Gyr. Perhaps that’s why Siemens has de-emphasized the role of the smart meter in its smart grid literature, relegating its function to the term “intelligent billing” in comparison to the “Grid Intelligence” rubric it assigns to grid infrastructure and controls.
Rather than compete in the wholesale smart meter business, Siemens has partnered with Landis+Gyr to develop common standards to ensure their systems are interoperable, as well as with smart meter networking providers such as Silver Spring Networks. But as far as the meter’s role in the broader smart grid, “We would agree that while the smart meter is an important enabler to what most people want to achieve, it’s nothing more than an enabler,” Pacyna said.
However, Siemens does have a small smart metering business, called Automated Metering and Information System, or AMIS. Based on technology from VA Tech, an Austrian company Siemens bought in 2005, it’s being tested out by Austrian utility Energie AG in a 1,000-household pilot project which could grow to up to 400,000 customers by 2014. But Pacyna told us that Siemens has no plans to deploy or market the AMIS smart meter solution beyond its current narrow focus in Central Europe.
Image courtesy of Siemens.
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# What are the characteristics of a hexagon?
The hexagon is defined as a polygon that has six sides and six interior angles. The interior angles of a hexagon add up to 720°. Taking different characteristics, it is possible to distinguish some types of hexagons. For example, depending on the length of the sides, we can have regular and irregular hexagons, and depending on the contour, we can have convex and concave hexagons.
Here, we will look at a definition of hexagons and a description of the most relevant types of hexagons. In addition, we will learn about some of the fundamental characteristics of these geometric figures. Finally, we will get to know its most important formulas and use them to solve some regular hexagon problems.
##### GEOMETRY
Relevant for
Learning about the characteristics of hexagons.
See characteristics
##### GEOMETRY
Relevant for
Learning about the characteristics of hexagons.
See characteristics
## Definition of a hexagon
A hexagon is defined as a polygon with 6 sides and 6 interior angles. Recall that a polygon is a closed two-dimensional (2D) figure that is made up of straight segments. All the sides of the hexagon meet each other end to end to form a shape.
Taking into account the different characteristics that hexagons can have, we can distinguish the following types of hexagons:
• Regular and irregular
• Convex and concave
### Irregular and regular hexagons
A regular hexagon is characterized by having sides with the same length and angles with the same measure. On the other hand, irregular hexagons have sides with different lengths and angles of different measures.
### Convex and concave hexagons
A convex hexagon is a geometric figure in which all its vertices are pointing outward. On the other hand, a concave hexagon is a geometric figure in which at least one vertex is pointing inward.
## Fundamental characteristics of a hexagon
Hexagons have the following fundamental characteristics:
• The sum of all its interior angles is 720°.
• Regular hexagons have all six sides with the same length.
• Regular hexagons have all six angles with the same measure.
• Each internal angle in a regular hexagon measures 120°.
• The total number of diagonals in a regular hexagon is 9.
• The sum of all exterior angles is 360 ° and each exterior angle measures 60°.
## Important hexagon formulas
The following are the most important formulas for regular hexagons.
### Formula for the perimeter of a regular hexagon
A regular hexagon has all its sides of the same length, so its perimeter is given by:
where s is the length of one of the sides of the hexagon.
### Formula for the area of a regular hexagon
The area of a regular hexagon is:
where s is the length of one of the sides and a is the length of the apothem.
### Formula of the apothem of a regular hexagon
The apothem of a hexagon can be found using the following formulas:
where s is the length of one side of the hexagon.
## Examples of hexagon problems
### EXAMPLE 1
• A hexagon has sides of length 12 m. What is its perimeter?
Solution: We substitute $latex s=12$:
$latex p=6s$
$latex p=6(12)$
$latex p=72$
The perimeter is 72 m.
### EXAMPLE 2
• A hexagon has sides of length 10 m. What is its area?
Solution: We have $latex s=10$. Therefore, we use this value in the area formula:
$latex A= \frac{3\sqrt{3}}{2}{{s}^2}$
$latex A= \frac{3\sqrt{3}}{2}{{(10)}^2}$
$latex A=259.8$
The area of the hexagon is 259.8 m².
### EXAMPLE 3
• A hexagon has sides of length 8m. What is its apothem?
Solution: We use the apothem formula with $latex s=8$:
$latex a= \frac{\sqrt{3}s}{2}$
$latex a= \frac{\sqrt{3}(8)}{2}$
$latex a=6.93$
The length of the apothem is 6.93 m.
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## Effective Altruism ForumEA Forum
Global health and development
# 108
Disclaimer: We (Sam Nolan, Hannah Rokebrand and Tanae Rao) do not represent GiveWell, or any of the organisations we mention. All opinions and errors are our own. This notebook is a part of a submission GiveWell's Change Our Mind Contest.
Work Done: This is about 300 hours of work done between Sam Nolan, Hannah Rokebrand and Tanae Rao, excluding the development of Squiggle by the Quantified Uncertainty Research Institute.
Epistemic Status: Somewhat confident. Despite out best efforts, it's likely there are some inconsistencies remaining in our work. We have aimed to make our claims and reasoning as easy to follow as possible. Uncertainty is quantified, but we are uncertain about many of our uncertainties! We are aware of several errors and areas of improvement, and try to highlight these explicitly. If you disagree with any of these claims, feel free to comment either on the EA Forum or in Observable. We intend to keep this notebook updated with any adjustments that need to be made.
Summary: We have quantified the uncertainty in GiveWell CEAs by translating them into Squiggle. This allows us to put a number on how uncertain we are about each of the cost-effectiveness parameters, and therefore the overall cost-effectiveness value of an intervention.
Quantifying uncertainty allows us to calculate the value of future research into the cost-effectiveness of interventions, and quantitatively see how funding this research might compare to donating to the current expected top charities.
As an example of this, we use our uncertainty quantification to calculate the Expected Value of Perfect Information on all GiveWell Top Charities, and find that the cost would outweigh the benefits if GiveWell were to spend more than 11.25% of their budget on research for this information. We then break down this value of information into the value of research into individual charities.
We also demonstrate how to estimate the value of information for individual parameters within a model, using the Catherine Hamlin Fistula Foundation as a case study. We believe this kind of analysis could help give a clearer idea of what, specifically, it would likely be cost-effective to research.
## Introduction
GiveWell writes cost-effectiveness analyses to calculate and present how much good we can expect to do by donating to different charities. One of the most criticised components of their process for this is that they do not quantify uncertainty, meaning we don't know how sure they are about the estimated effectiveness of the charities. Quantifying uncertainty offers several benefits. Here, we will focus on one of the most significant of these - value-of-information analysis. We will also touch on how quantifying uncertainty can also improve calibration over time, communication and correctness.
We go through an example methodology for value-of-information analysis. We are uncertain about this exact methodology but believe value of information could be a helpful measure in the research pipeline, whether formally or informally calculated.
## Uncertainty in the Cost-Effectiveness of GiveWell Charities
Our methodology for quantifying uncertainty in each of GiveWell's Top Charity CEAs are in the Supplemental Notebooks section below, alongside sensitivity analysis, per-country breakdowns, and additional considerations. In quantifying the uncertainty in these, we have attempted to be as faithful as possible to GiveWell's current CEAs. The purpose of this decision was primarily to investigate the impact of uncertainty quantification alone. We discuss other alterations that could be made in the Possible improvements to GiveWell models section.
After quantifying uncertainty in GiveWell top charities, we have found that the cost-effectivenesses of GiveDirectly, Against Malaria Foundation, Malaria Consortium, New Incentives, and Helen Keller International (all from GiveWell's top charity list) in 'doubling of consumption year equivalents per dollar' (DCYE/$) are distributed as follows. What can we do with this information? It communicates the conventional decisions - GiveWell's top charities are at least 10x as effective as GiveDirectly, with Helen Keller International claiming the highest expected value. If we aim to donate to the charity with the highest expected value, all else being equal, why go to all the fuss of quantifying uncertainty? Information about uncertainty is important because GiveWell doesn't only make decisions about what to donate to, but more frequently about what to research. Quantifying uncertainty in research provides inbuilt information about whether and where further research in that area is needed. Knowing how uncertain we are about existing research allows us to calculate what research directions are (and perhaps more importantly, are not) worth pursuing in the future. This is called the 'Value of Information' and can be calculated by Value of Information Analysis. ## Value of Information We can calculate the cost-effectiveness of charities by how many ‘units of value' they create per dollar, but we're uncertain about exactly how cost-effective each charity is. One way to navigate this is to present effectiveness values as probability distributions, showing how likely each possible value is the be the actual cost-effectiveness value for a given charity. If we want to donate to the charity that can do the most good with our money, we could a) donate to the charity we expect to be the most cost-effective based on our current best guesses (which, in this case, would be Helen Keller International); or b) do more research so we can (hypothetically) know the actual effectiveness of the charities. For simplicity, let's imagine we could eliminate all uncertainty about all charities with this research. We call the value we expect to get from knowing the effectiveness of a charity for sure, “Expected Value of Perfect Information,” or just "Value of Information." It’s equal to the difference between the amount of good we expect we could do with that knowledge and what we expect we could do without it. The concept of the Value of Information is intuitive and many questions relating to it can be answered without calculations. In the context of decision-making, information is more valuable when: • It is more likely to change our mind about something, in a way that allows us to create more value with the decision • The decision being made has higher consequences - a greater potential for producing value or harm, given the resources devoted to it "Cost Ratio" is the maximum portion of a decision-maker's budget they could spend on obtaining this perfect information without the cost outweighing the expected benefit. Of course, this hypothetical research project does not perfectly reflect reality. No research project ever completely eliminates uncertainty. For this reason, the cost ratio could be considered a good upper bound for the amount worth spending on research into a given area. Douglas W. Hubbard of Hubbard Decision Research recommends spending but a tenth of this amount. We call case (a) - where we donate to the expected best charity without researching - the "uniformed state." In this state, we only know the effectiveness of each charity as a probability distribution. The maximum cost-effectiveness we could expect to obtain in the uninformed state (U), in units of value per dollar, is simply the highest expected value of all the interventions considered: If are random variables representing the cost-effectiveness for each of different interventions (such as those presented in the probability distributions above), and represents the intervention with the highest expected value - the one we expect to be the best before researching, Where finds the expected value (mean) of , and finds the highest of these, considering every possible choice for . (Links for more on max and E[X] notation) We call case (b) - where we research all of the charities to find their true effectiveness - the "informed state." In this state, we know all the actual cost-effectiveness value for each of the charities, and the maximum value per dollar we could expect to generate would be the effectiveness of the charity with the highest actual cost-effectiveness value. This expectation of value obtainable in the informed state can be calculated, but in practice, it is generally easier and just as useful to approximate it. To do this, we simulate a possible informed state by taking a sample of each random variable. This process can be iterated many times to produce a distribution of possible informed states and the values they yield. In each iteration, we would donate to the charity with the highest cost-effectiveness, and reap the value of that charity.. This process can be iterated many times to produce a distribution of possible informed states and the values they yield. In each iteration, we would donate to the charity with the highest cost-effectiveness, and reap the value of that charity. The maximum cost-effectiveness we could expect to generate in the informed state (I) is the expected value of this distribution of values obtained assuming we chose the most effective charity for each iteration (). Expected Value of Perfect Information (EVPI) about the true cost-effectiveness of an intervention, , in units of value per dollar, is how much effectiveness we could expect to gain in the informed state compared to the uninformed: The amount of value we can expect to generate in the uninformed state, with the amount of money we have available is given by where is the amount donated. The amount of value we can expect to generate in the informed state, with the amount of money we have to available , and the cost of research , is given by: Where is the cost of research and is the amount donated. If research is worthwhile (cost does not outweigh benefits), we can expect to generate no less value in the informed state, than in the uninformed state: From this, we can find a maximum cost ratio, cost of research : total amount available, for which net value is likely to be added by research. For instance, if we have$100,000 to donate, and a cost ratio of 20%, then the maximum amount it would be worth spending on research to eliminate all uncertainty would be $20,000. This should be considered an upper bound for research expenditure because in reality, we can only ever reduce uncertainty, which is necessarily less valuable. Consequently, we recommend a more conservative approach of spending closer to a tenth of the cost ratio on research (value of imperfect information). In this case, that means spending no more than$2,000 on research.
We use calculus to show what these values mean and how we could calculate them in terms of probability density functions, and look at the special case of investigating only one charity at a time in the appendix.
We can also sample real distributions to calculate this in Squiggle!
voi(c) = mean(SampleSet.mapN(c, {|xs| max(xs)}))
vou(c) = max(map(c, {|x| mean(x)}))
vpi(x) = voi(x) - vou(x)
charities = [givedirectly, helen_keller_international, new_incentives, malaria_consortium, amf_effectiveness]
vpi(charities) // Returns 7.74·10-3
This is the expected units of value per dollar gained by eliminating all uncertainty in all charities. However, whether we should research to reduce uncertainty is dependent on two factors:
• Research costs - information that would cost more of our finite resources to acquire should be justified by a higher value of information.
• Size of investment - the greater the amount of money or other resources a decision is directing, the more important it is for that decision to be informed.
Now, we can calculate the cost ratio, or proportion of GiveWell's budget it would be worth devoting to research that would reduce uncertainty in their Top Charities, relative to the cost of this research:
vpi_cost(x) = {
my_voi = voi(x)
cost_ratio = (my_voi - vou(x)) / my_voi
cost_ratio
}
vpi_cost(charities) // Returns 0.113
This means that GiveWell should spend no more than 11.3% of their budget on researching this their current Top Charities. This number is assuming that research would acquire perfect information. As this would not be the case and research would only decrease uncertainty, 10% of this value, or 1.13% of GiveWell's budget, may be a more appropriate amount.
We can also calculate the value of information on each GiveWell Top Charity individually. We do this by assuming the charities that we don't want to research have already reduced all their uncertainty and are set to their expected value, such that there is no value of researching any charity other than that/those selected[1].
This indicates that it's valuable to reduce the uncertainty of current GiveWell top charities (at least, assuming the uncertainties specified in the supplemental notebooks). However, GiveWell is also focused on new interventions. Below is an example of how we evaluate the value of information for a new intervention.
## Evaluating New Interventions, a Case Study with the Catherine Hamlin Fistula Foundation
GiveWell has previously investigated the Fistula Foundation, evaluating the effectiveness of surgically treating obstetric fistulas in low-income countries, seemingly considering both the quality of life detriments and loss of income engendered by fistulas. However, the Catherine Hamlin Fistula foundation, which may be easily overlooked because of its similarity to the Fistula Foundation in name and approach to treatment, is a different charity which focuses on treatment but also prevention and rehabilitation.
Quantifying uncertainty allows us to evaluate this charity, using rough numbers and informed guesses to give us an idea of cost-effectiveness. This can then inform a quick value of information analysis indicating how valuable further research into this intervention is likely to be.
Hannah Rokebrand has used this method to do a rough three-day analysis of CHFF's cost-effectiveness.
CHFF expected value: 0.057 doubling of consumption years equivalent
The above probability distributions show the probable cost-effectiveness values of three GiveWell top charities and the Catherine Hamlin Fistula Foundation in 'doubling of consumption year equivalents per dollar'. We can use these distributions to calculate the value of reducing all uncertainty about CHFF.
cost_ratio_fistula = vpi_cost([SampleSet.fromDist(mx(vou(charities))), fistula_foundation]) // returns 0.17
This means, assuming our quantification of uncertainty, GiveWell should spend no more than 17% of their budget acquiring information about CHFF overall.
However, research projects often focus on specific variables within a CEA. We can find the most useful parameters to focus on in this case by calculating Expected Value of Perfect Partial Information.
## Expected Value of Perfect Partial Information
Once we've found that CHFF is worth researching, we can calculate the value of using our finite resources to reduce uncertainty in only a selection of the parameters that contribute to its cost-effectiveness, to maximise the benefit of researching this intervention.
We take the cost-effectiveness of CHFF to be a function , which operates on a n parameters to produce a final cost-effectiveness probability distribution.
Again, the value per dollar we can expect to generate in the "uninformed state" is the expected cost-effectiveness of the charity with the highest expected value, in this case, we will take that to be HKI. We will again refer to this charity as .
In the informed state, we would know the exact value of one of the parameters, say . We would then donate to the charity with the highest expected value given this information - either HKI or CHFF.
We then want to find the expected value of the informed state over all possible values of .
If is the expected value excluding the variable , and returns a distribution of expected values over different possible values of Y_iYi)
We can then estimate the Expected Value of Partial Perfect Information (EVPPI) about CHFF from eliminating uncertainty about one parameter by calculating the difference between these values .
We can run these calculations in Squiggle, but this is a somewhat expensive operation and requires running the entire model ff thousands of times (for every sample). To simplify, we've approximated the value of information on each parameter by assuming , although this is not necessarily the case.
mapDict(x, f) = Dict.fromList(map(Dict.keys(x), {|key| [key, f(x[key])]}))
mapSet(x, key, value) = Dict.merge(x, Dict.fromList([[key, value]]))
to_mean(x) = mapDict(x, mean)
vpi_analysis(f, params, other) = {
mean_params = to_mean(params)
Dict.fromList(map(Dict.keys(params), {|key| [key, {
possible_results = f(mapSet(mean_params, key, params[key]))
vou = max([mean(possible_results), other])
possible_decisions = SampleSet.map(possible_results, {|s| max([s, other])})
voi = mean(possible_decisions)
vpi = voi - vou
vpi / voi
}]}))
}
The calculations run in Squiggle find the cost ratio of researching each parameter contributing to our calculations of CHFF's cost-effectiveness. These indicate that the average number of midwives trained per year (midwives_trained_per_year) and portion_of_career_midwife_not_with_chff are the most impactful parameter for us to research to bring us closer to the true cost-effectiveness of CHFF.
## Other Benefits of Uncertainty Quantification
Although we are fans of the value of information, there are several other benefits to uncertainty quantification, some of which we'll mention here.
### Communication
Every measurement or observation we make is subject to uncertainty, and cannot know our calculated values to a greater level of certainty than the observed values from which they were calculated. Researchers commonly use error bars to help communicate the degree of this uncertainty so that we can compare and weigh different sources of evidence and consider the probability of results arising from the factors to which they are attributed, rather than random chance.
Quantifying uncertainty in the form of distributions also retains a dimensionality obscured by error bars. Distributions have a shape and can be combined in calculations to give us the shapes of distributions for calculated parameters, far more easily and precisely than current conventional methods of accounting for uncertainty.
In a post explaining a name change of GiveWell's fund for the most rigorously evaluated charities meeting its cost-effectiveness bar, a new criteria of inclusion in the fund is highlighted: "that we have a high degree of confidence in our expectations about the impact of their programs."
It may also be noteworthy that GiveWell has "introduced a new giving option, the All Grants Fund. The All Grants Fund supports the full range of GiveWell’s grantmaking and can be allocated to any grant that meets our cost-effectiveness bar — including opportunities outside of our top charities and riskier grants with high expected value."
We believe quantifying uncertainty may ease decisions about inclusion and exclusion from the Top Charities list, as well as communication about these decisions. This communication could potentially highlight uncertain but high expected value opportunities for individuals or organisations with lower risk aversion to consider.
### Calibration
In some types of research, we need to estimate parameters that are difficult or impossible to measure, such as how we should weigh different sources of evidence, or the value we assign to lives saved compared to economic gains. Charity cost-effectiveness analyses are a prime example of this. Many of GiveWell's parameters are "best guesses" based on the information available to them at the time. If these values were forecast as "we are 95% confident that this value lies between x and y," rather than "we expect this value to be z, but this is a rough estimate," these forecasts could be updated by further research and the scale of these updates could be used to calibrate future estimates and form a record of reliability for GiveWell's estimates, as we do with other types of forecasters.
### Correctness
GiveWell aims to maximise the expected value of the charity to which they most heavily donate (with reasonable confidence), but they calculate the final expected values of charities by doing calculations with the expected values of the parameters contributing to it. This assumes that, if is a function that gives the cost-effectiveness of an intervention from n random variable parameters,
Although it's a very practical assumption (so practical, we use it as an approximate in our Expected Value of Partial Perfect Information), it is only true when is a linear map, which is usually not the case. All the CEAs discussed here have expected values of all the parameters set to GiveWell values, however, because of the nature of the distributions, the final expected value changes slightly.
## Discussions
In this section, we'll go through a list of discussion points that came up in working with GiveWell models, but are not directly relevant to uncertainty quantification.
### Possible Improvements to our Models
Model Uncertainty and Optimizer's Curse
None of our models, particularly CHFF, consider the possibility of the model itself being incorrect, and do not adjust to a prior. This may be worthwhile to avoid optimizer's curse.
One improvement we hope to make in our Value of Information Analysis is a better model of how GiveWell makes allocation decisions. We have been unable to find publicly-available, detailed information about how GiveWell decides where to allocate its funds, although an overview of their system is available. We expect GiveWell may provide grants for the intervention they expect to have the greatest impact in each country they target, in which case the country-level estimates presented in our supplemental notebooks may be more applicable than the global aggregates used here. Pairing country-level data with the funding gap for each charity in each region would likely yield a higher value of information estimates than those in this notebook.
More Considerate Choices of Distributions
Currently, our uncertainty quantifications have been shaped by an aesthetic bias. This means that if a cost-effectiveness calculation presented itself with a significant probability in the negative, or extremely long tails, we assumed we were in error and 'fixed' the source of this unaesthetic presentation. It may be however, that the distributions are not nearly so aesthetic. When quantifying uncertainty, we found that the maths around 'other actors' and counterfactual donations could create somewhat poorly-behaved distributions.
Our analyses also currently use beta distributions for proportions and lognormal distributions for most other parameters. These choices was may not have been the most appropriate. We'd in interested in considering other distributions in the future.
Moral Uncertainty
GiveWell CEAs typically quantify the good done by an intervention in terms of the intervention's primary objective, e.g. lives saved. However, to make the cost-effectiveness of different interventions comparable, these are then converted into a general "unit of value."
GiveDirectly is a charity with a highly researched and economic-based (objective) impact, which gives it a comparatively low-uncertainty cost-effectiveness value. Because of this, GiveWell generally measures the effectiveness of other programs they evaluate against GiveDirectly, and incorporate its units ('doubling of consumption years') in the universal units of value, into which all other programs' cost-effectivenesses are converted - 'doubling of consumption year equivalents per dollar' (DCYE per dollar).
This conversion process requires answering tough questions like "how much do we value saving the life of a four year-old child compared to that of his 32 year-old mother, compared doubling the spending power of a family in poverty?"
For example, the value GiveWell assigns averting one death of a child under 5 from malaria is 116.9 DCYEs. This implies GiveWell's CEA's value averting this death equivalently to doubling consumption for 116.9 person-years. As explained in this talk, these figures are determined by a combination of GiveWell staff opinion, a GiveWell donor survey, and an IDInsight survey of beneficiary preferences (low-income individuals in Ghana and Kenya).
These moral weights are often the most contentious part of a GiveWell CEA, and can hold considerable uncertainty. One way of doing this would be worldview diversification, where we look into a range of possible values for moral parameters and optimise for the highest expected choice-worthiness.
Future lines of research related to moral uncertainty where quantified uncertainty could be useful include:
Incorporating uncertainty about beneficiary preferences. When participants in the IDInsight survey were asked about the value of life, their answers differed significantly depending on how the question was phrased. From an 'individual perspective,' averting an under-5 death was considered worth the equivalent of $40,763, while the 'community perspective' phrasing elicited a value of$91,049.
High uncertainty, exacerbated by methodological concerns, may deter GiveWell from weighting survey data more heavily. We are not arguing that survey data should be weighted more heavily, but incorporating quantified uncertainty into the CEA may make analysts more comfortable with weighing the data more heavily where warranted.
• Determining the value of information on the moral weights parameters. Obtaining this moral information if it is shown to be an effective use of GiveWell researchers' time, might include commissioning further beneficiary preference surveys, seeking preferences from a larger sample of stakeholders, spending more time on high-level considerations (e.g., using WELLBYs).
Correlated Uncertainties
To get estimates comparable to each other, GiveWell CEAs use many variables that are the same across different charities. These include moral variables such as the discount rate and the moral weights, as well as a number or adjustment factors.
Because each of our CEAs was built in its separate notebook. Variables that should have more correlated uncertainty do not correlate. This means that our value of information analysis, considers the situation where one charity has a higher moral weight for the value of a child under 5 than a different charity, which would be impossible and unfair. These inconsistencies also have implications unrelated to VoI.
To account for this, you may want to adjust our value of information estimates downwards. However, we're not sure about the extent of this adjustment.
Squiggle is capable of developing models with correlated uncertainty, however, we ran into performance issues with developing individual notebooks, and having all the notebooks relate to each other would likely only increase the problem.
### Possible improvements to GiveWell models
Integration over age brackets
Nuno Sempere is to be credited with this idea, and may be able to explain it better. As a language, Squiggle makes integration a lot easier. Nuno has an example here.
Some of GiveWell's CEAs focus on age-dependent interventions, such as infant vaccination, and breakdown the impact of these by age group. We have considered converting this format into one where impact is continuously modelled across all ages. However, in our experience, this has proved difficult to apply to GiveWell CEAs for two main reasons.
• GiveWell CEAs, particularly for age-dependent interventions, frequently cite IMHE data which presents statistics like disease prevalence and mortality for discrete (typically 4-year) age brackets, rather than as a continuous trends. Even if this date were presented more continuously, it's it would be in a form that could be easily incorporated into a formulaic model of impact by age.
• GiveWell CEAs use the units of doubling of consumption year equivalents, which to our knowledge, are not used by any other major organisation and do not have a continuous formula behind them. Because the 'equivalent' component of a doubling of consumption year equivalent is calculated by a combination of the continuous DALY formula, and discrete data about large age brackets and the values of surveyed stakeholders, it is difficult to model these units continuously.
This is not to say that the process would be impossible, but more work is needed for us to understand how to consistently do it in a way that would add information that isn't present in the discrete presentation.
Dimensional Analysis
Using Dimensional Analysis, it's possible to identify implicit assumptions within a model that could be incorrect or missing. Sam Nolan has done this with GiveDirectly, identifying two implicit assumptions, and highlighting the potential for a continuous Present Value formula.
A Continuous Present Value formula models present value where there are no discrete payouts, but continuous payouts over time. This may model the situation better, as the present value is supposed to represent benefits that accrue over time and do not occur annually. However, the change has minimal impact on overall estimates of cost-effectiveness.
## Supplemental Notebooks
Each intervention has its observable notebook that goes through the CEA calculations. As most top charities operate in multiple countries, the distributions shown in this notebook are global means. If we knew exactly what portion of a charity's budget would go to each country, we could refine our cost-effectiveness estimates or do a different analysis where we consider the most effective intervention-country pairing.
The supplemental notebooks, which account for most of our work on this project, and on which this notebook is built, are indexed below. They are in no way complete, and we are still trying to make them as accurate as they can be.
### Uncertainty Quantification of GiveDirectly
Author: Sam Nolan
Sam Nolan has written a post on the EA forum about this model. This model brings up some interesting questions with implicit parameters, and the use of logarithmic vs isoelastic utility which increases GiveDirectly's utility by 11%. This is one of complete models.
It would be interesting to include Happier Lives Institute's evaluations of cash transfers in future.
### Uncertainty Quantification of New Incentives
Author: Hannah Rokebrand
New incentives are likely the most in development-notebook. It currently has very large tails, and we are unsure whether this is due to New Incentives having a large theory of change or an oversight on our part.
### Uncertainty Quantification of the Against Malaria Foundation
Author: Tanae Rao
This has an associated EA Forum post.
### Uncertainty Quantification of Helen Keller International
Author: Sam Nolan
This model is close to complete. It may have minor errors. The written work is still in progress.
### Uncertainty Quantification of the Malaria Consortium
Author: Hannah Rokebrand
Uncertainty quantification for Malaria Consortium is complete, but the written component is still in progress.
### CEA of the Catherine Hamlin Fistula Foundation
Author: Hannah Rokebrand
The flow on effects of CHFF's program are difficult to calculate and with a high degree of certainty. However, early results calculated under high uncertainty seem to show this intervention to worth further research.
This is a very rough calculation, missing a number of variables which may affect the value generated. Several uncertainty bounds are also rough estimates, or the results of very brief research. This post still needs a lot of refinement.
It would also be interesting to look at the cost-effectiveness of the three main areas of CHFF, prevention, treatment and rehabilitation and reintegration. Unfortunately, data about the total cost of these individual areas does not seem to be available, only CHFF's total expenses.
### Value of Information on GiveDirectly
Author: Sam Nolan
This has an associated EA Forum post. This is a simple example of the value of information that I made to demonstrate the concept. This notebook doesn't serve a role in any of the calculations or results.
## Appendix: Value of Information with Calculus:
This is an example of a special case of VoI where we are comparing the cost-effectiveness of charities (no consideration of risk or profit, only value per dollar), investigating only one at a time (rather than eliminating all uncertainty in all options at once), although this method could be extended by comparing each of n charities to the new expected best charity for every iteration. Calculating the value of information for all charities in a set is important because it we always be greater than the value of information from investigating only one charity, and as such may be used as un upper bound for the possible value to be gained from researching the charities in the set. This is particularly useful for larger sets, and can be calculated almost instantly using Squiggle. However, in practice we are more likely to be making decisions about investigating one intervention at a time, and the concepts explored here may be more intuitive from this application.
Case (a) (Without Research)
The units of value per dollar we could expect to obtain in the uninformed state (U):
If is a set of random variables representing the cost-effectiveness for each of n different interventions (such as those presented in the probability distributions above), is the probability density function of the intervention represented by , and represents the intervention with the highest expected value - the one we expect to be the best before researching,
Case (b) (With Research)
The units of value per dollar we could expect to obtain in the informed state (I):
Expected Value of Perfect Information (EVPI) about the true cost-effectiveness of in units of value per dollar, is how much more effectiveness we could expect to get in the informed state than in the uninformed:
The Cost Ratio, i.e. upper bound of the proportion of the total money we have available that we should be willing to invest in research, can be found by assuming that research is only worthwhile if we expect to obtain no less value in the informed state than in the uninformed state:
1. ^
In the observable version of this post, there is a widget you can use to calculate how much value there is in researching different collections of interventions.
## Acknowledgements
This work was supported by two EA Infrastructure Grants. We are grateful to the Quantified Uncertainty Research Institute for their work on the Squiggle language that made this all possible. We would also like to thank David Reinstein, Nuno Sempere and Falk Lieder for their feedback on this work.
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Question
# A point moves such that the sum of the square of its distance from the sides of a sqaure of side unity is equal to $$9$$. The locus of the point is a circle such that
A
Center of the circle coincides with that of square
B
Center of the circle is (12,12)
C
Radius of the circle is 2
D
All the above are true
Solution
## The correct option is D All the above are trueLet the sides of the square be $$y=0,y=1,x=0$$ and $$x=1.$$Let the moving point be $$(x,y).$$Then,$${ y }^{ 2 }+{ \left( y-1 \right) }^{ 2 }+{ x }^{ 2 }+{ \left( x-1 \right) }^{ 2 }=9$$ is the equation of the locus.$$\Rightarrow 2{ x }^{ 2 }+2{ y }^{ 2 }-2x-2y-7=0,$$ which represents a circle having centre $$\displaystyle \left( \frac { 1 }{ 2 } ,\frac { 1 }{ 2 } \right)$$ (which is also the centre of the square) and radius $$2.$$ Mathematics
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# Piecewise-defined Functions¶
Sage implements a very simple class of piecewise-defined functions. Functions may be any type of symbolic expression. Infinite intervals are not supported. The endpoints of each interval must line up.
TODO:
• Implement max/min location and values,
• Need: parent object - ring of piecewise functions
• This class should derive from an element-type class, and should define _add_, _mul_, etc. That will automatically take care of left multiplication and proper coercion. The coercion mentioned below for scalar mult on right is bad, since it only allows ints and rationals. The right way is to use an element class and only define _mul_, and have a parent, so anything gets coerced properly.
AUTHORS:
• David Joyner (2006-04): initial version
• David Joyner (2006-09): added __eq__, extend_by_zero_to, unextend, convolution, trapezoid, trapezoid_integral_approximation, riemann_sum, riemann_sum_integral_approximation, tangent_line fixed bugs in __mul__, __add__
• David Joyner (2007-03): adding Hann filter for FS, added general FS filter methods for computing and plotting, added options to plotting of FS (eg, specifying rgb values are now allowed). Fixed bug in documentation reported by Pablo De Napoli.
• David Joyner (2007-09): bug fixes due to behaviour of SymbolicArithmetic
• David Joyner (2008-04): fixed docstring bugs reported by J Morrow; added support for Laplace transform of functions with infinite support.
• David Joyner (2008-07): fixed a left multiplication bug reported by C. Boncelet (by defining __rmul__ = __mul__).
• Paul Butler (2009-01): added indefinite integration and default_variable
TESTS:
sage: R.<x> = QQ[]
sage: f = Piecewise([[(0,1),1*x^0]])
sage: 2*f
Piecewise defined function with 1 parts, [[(0, 1), 2]]
sage.functions.piecewise.Piecewise(list_of_pairs, var=None)
Returns a piecewise function from a list of (interval, function) pairs.
list_of_pairs is a list of pairs (I, fcn), where fcn is a Sage function (such as a polynomial over RR, or functions using the lambda notation), and I is an interval such as I = (1,3). Two consecutive intervals must share a common endpoint.
If the optional var is specified, then any symbolic expressions in the list will be converted to symbolic functions using fcn.function(var). (This says which variable is considered to be “piecewise”.)
We assume that these definitions are consistent (ie, no checking is done).
EXAMPLES:
sage: f1(x) = -1
sage: f2(x) = 2
sage: f = Piecewise([[(0,pi/2),f1],[(pi/2,pi),f2]])
sage: f(1)
-1
sage: f(3)
2
sage: f = Piecewise([[(0,1),x], [(1,2),x^2]], x); f
Piecewise defined function with 2 parts, [[(0, 1), x |--> x], [(1, 2), x |--> x^2]]
sage: f(0.9)
0.900000000000000
sage: f(1.1)
1.21000000000000
class sage.functions.piecewise.PiecewisePolynomial(list_of_pairs, var=None)
Returns a piecewise function from a list of (interval, function) pairs.
EXAMPLES:
sage: f1(x) = -1
sage: f2(x) = 2
sage: f = Piecewise([[(0,pi/2),f1],[(pi/2,pi),f2]])
sage: f(1)
-1
sage: f(3)
2
base_ring()
Returns the base ring of the function pieces. This is useful when this class is extended.
EXAMPLES:
sage: f1(x) = 1
sage: f2(x) = 1-x
sage: f3(x) = x^2-5
sage: f = Piecewise([[(0,1),f1],[(1,2),f2],[(2,3),f3]])
sage: base_ring(f)
Symbolic Ring
sage: R.<x> = QQ[]
sage: f1 = x^0
sage: f2 = 10*x - x^2
sage: f3 = 3*x^4 - 156*x^3 + 3036*x^2 - 26208*x
sage: f = Piecewise([[(0,3),f1],[(3,10),f2],[(10,20),f3]])
sage: f.base_ring()
Rational Field
convolution(other)
Returns the convolution function, $$f*g(t)=\int_{-\infty}^\infty f(u)g(t-u)du$$, for compactly supported $$f,g$$.
EXAMPLES:
sage: x = PolynomialRing(QQ,'x').gen()
sage: f = Piecewise([[(0,1),1*x^0]]) ## example 0
sage: g = f.convolution(f)
sage: h = f.convolution(g)
sage: P = f.plot(); Q = g.plot(rgbcolor=(1,1,0)); R = h.plot(rgbcolor=(0,1,1));
sage: # Type show(P+Q+R) to view
sage: f = Piecewise([[(0,1),1*x^0],[(1,2),2*x^0],[(2,3),1*x^0]]) ## example 1
sage: g = f.convolution(f)
sage: h = f.convolution(g)
sage: P = f.plot(); Q = g.plot(rgbcolor=(1,1,0)); R = h.plot(rgbcolor=(0,1,1));
sage: # Type show(P+Q+R) to view
sage: f = Piecewise([[(-1,1),1]]) ## example 2
sage: g = Piecewise([[(0,3),x]])
sage: f.convolution(g)
Piecewise defined function with 3 parts, [[(-1, 1), 0], [(1, 2), -3/2*x], [(2, 4), -3/2*x]]
sage: g = Piecewise([[(0,3),1*x^0],[(3,4),2*x^0]])
sage: f.convolution(g)
Piecewise defined function with 5 parts, [[(-1, 1), x + 1], [(1, 2), 3], [(2, 3), x], [(3, 4), -x + 8], [(4, 5), -2*x + 10]]
cosine_series_coefficient(n, L)
Returns the n-th cosine series coefficient of $$\cos(n\pi x/L)$$, $$a_n$$.
INPUT:
• self - the function f(x), defined over 0 x L (no checking is done to insure this)
• n - an integer n=0
• L - (the period)/2
OUTPUT: $$a_n = \frac{2}{L}\int_{-L}^L f(x)\cos(n\pi x/L)dx$$ such that
$\begin{split}f(x) \sim \frac{a_0}{2} + \sum_{n=1}^\infty a_n\cos(\frac{n\pi x}{L}),\ \ 0<x<L. \end{split}$
EXAMPLES:
sage: f(x) = x
sage: f = Piecewise([[(0,1),f]])
sage: f.cosine_series_coefficient(2,1)
0
sage: f.cosine_series_coefficient(3,1)
-4/9/pi^2
sage: f1(x) = -1
sage: f2(x) = 2
sage: f = Piecewise([[(0,pi/2),f1],[(pi/2,pi),f2]])
sage: f.cosine_series_coefficient(2,pi)
0
sage: f.cosine_series_coefficient(3,pi)
2/pi
sage: f.cosine_series_coefficient(111,pi)
2/37/pi
sage: f1 = lambda x: x*(pi-x)
sage: f = Piecewise([[(0,pi),f1]])
sage: f.cosine_series_coefficient(0,pi)
1/3*pi^2
critical_points()
Return the critical points of this piecewise function.
Warning
Uses maxima, which prints the warning to use results with caution. Only works for piecewise functions whose parts are polynomials with real critical not occurring on the interval endpoints.
EXAMPLES:
sage: R.<x> = QQ[]
sage: f1 = x^0
sage: f2 = 10*x - x^2
sage: f3 = 3*x^4 - 156*x^3 + 3036*x^2 - 26208*x
sage: f = Piecewise([[(0,3),f1],[(3,10),f2],[(10,20),f3]])
sage: expected = [5, 12, 13, 14]
sage: all(abs(e-a) < 0.001 for e,a in zip(expected, f.critical_points()))
True
TESTS:
Use variables other than x (trac ticket #13836):
sage: R.<y> = QQ[]
sage: f1 = y^0
sage: f2 = 10*y - y^2
sage: f3 = 3*y^4 - 156*y^3 + 3036*y^2 - 26208*y
sage: f = Piecewise([[(0,3),f1],[(3,10),f2],[(10,20),f3]])
sage: expected = [5, 12, 13, 14]
sage: all(abs(e-a) < 0.001 for e,a in zip(expected, f.critical_points()))
True
default_variable()
Return the default variable. The default variable is defined as the first variable in the first piece that has a variable. If no pieces have a variable (each piece is a constant value), $$x$$ is returned.
The result is cached.
AUTHOR: Paul Butler
EXAMPLES:
sage: f1(x) = 1
sage: f2(x) = 5*x
sage: p = Piecewise([[(0,1),f1],[(1,4),f2]])
sage: p.default_variable()
x
sage: f1 = 3*var('y')
sage: p = Piecewise([[(0,1),4],[(1,4),f1]])
sage: p.default_variable()
y
derivative()
Returns the derivative (as computed by maxima) Piecewise(I,(d/dx)(self|_I)), as I runs over the intervals belonging to self. self must be piecewise polynomial.
EXAMPLES:
sage: f1(x) = 1
sage: f2(x) = 1-x
sage: f = Piecewise([[(0,1),f1],[(1,2),f2]])
sage: f.derivative()
Piecewise defined function with 2 parts, [[(0, 1), x |--> 0], [(1, 2), x |--> -1]]
sage: f1(x) = -1
sage: f2(x) = 2
sage: f = Piecewise([[(0,pi/2),f1],[(pi/2,pi),f2]])
sage: f.derivative()
Piecewise defined function with 2 parts, [[(0, 1/2*pi), x |--> 0], [(1/2*pi, pi), x |--> 0]]
sage: f = Piecewise([[(0,1), (x * 2)]], x)
sage: f.derivative()
Piecewise defined function with 1 parts, [[(0, 1), x |--> 2]]
domain()
Returns the domain of the function.
EXAMPLES:
sage: f1(x) = 1
sage: f2(x) = 1-x
sage: f3(x) = exp(x)
sage: f4(x) = sin(2*x)
sage: f = Piecewise([[(0,1),f1],[(1,2),f2],[(2,3),f3],[(3,10),f4]])
sage: f.domain()
(0, 10)
end_points()
Returns a list of all interval endpoints for this function.
EXAMPLES:
sage: f1(x) = 1
sage: f2(x) = 1-x
sage: f3(x) = x^2-5
sage: f = Piecewise([[(0,1),f1],[(1,2),f2],[(2,3),f3]])
sage: f.end_points()
[0, 1, 2, 3]
extend_by_zero_to(xmin=-1000, xmax=1000)
This function simply returns the piecewise defined function which is extended by 0 so it is defined on all of (xmin,xmax). This is needed to add two piecewise functions in a reasonable way.
EXAMPLES:
sage: f1(x) = 1
sage: f2(x) = 1 - x
sage: f = Piecewise([[(0,1),f1],[(1,2),f2]])
sage: f.extend_by_zero_to(-1, 3)
Piecewise defined function with 4 parts, [[(-1, 0), 0], [(0, 1), x |--> 1], [(1, 2), x |--> -x + 1], [(2, 3), 0]]
fourier_series_cosine_coefficient(n, L)
Returns the n-th Fourier series coefficient of $$\cos(n\pi x/L)$$, $$a_n$$.
INPUT:
• self - the function f(x), defined over -L x L
• n - an integer n=0
• L - (the period)/2
OUTPUT: $$a_n = \frac{1}{L}\int_{-L}^L f(x)\cos(n\pi x/L)dx$$
EXAMPLES:
sage: f(x) = x^2
sage: f = Piecewise([[(-1,1),f]])
sage: f.fourier_series_cosine_coefficient(2,1)
pi^(-2)
sage: f(x) = x^2
sage: f = Piecewise([[(-pi,pi),f]])
sage: f.fourier_series_cosine_coefficient(2,pi)
1
sage: f1(x) = -1
sage: f2(x) = 2
sage: f = Piecewise([[(-pi,pi/2),f1],[(pi/2,pi),f2]])
sage: f.fourier_series_cosine_coefficient(5,pi)
-3/5/pi
fourier_series_partial_sum(N, L)
Returns the partial sum
$f(x) \sim \frac{a_0}{2} + \sum_{n=1}^N [a_n\cos(\frac{n\pi x}{L}) + b_n\sin(\frac{n\pi x}{L})],$
as a string.
EXAMPLE:
sage: f(x) = x^2
sage: f = Piecewise([[(-1,1),f]])
sage: f.fourier_series_partial_sum(3,1)
cos(2*pi*x)/pi^2 - 4*cos(pi*x)/pi^2 + 1/3
sage: f1(x) = -1
sage: f2(x) = 2
sage: f = Piecewise([[(-pi,pi/2),f1],[(pi/2,pi),f2]])
sage: f.fourier_series_partial_sum(3,pi)
-3*cos(x)/pi - 3*sin(2*x)/pi + 3*sin(x)/pi - 1/4
fourier_series_partial_sum_cesaro(N, L)
Returns the Cesaro partial sum
$f(x) \sim \frac{a_0}{2} + \sum_{n=1}^N (1-n/N)*[a_n\cos(\frac{n\pi x}{L}) + b_n\sin(\frac{n\pi x}{L})],$
as a string. This is a “smoother” partial sum - the Gibbs phenomenon is mollified.
EXAMPLE:
sage: f(x) = x^2
sage: f = Piecewise([[(-1,1),f]])
sage: f.fourier_series_partial_sum_cesaro(3,1)
1/3*cos(2*pi*x)/pi^2 - 8/3*cos(pi*x)/pi^2 + 1/3
sage: f1(x) = -1
sage: f2(x) = 2
sage: f = Piecewise([[(-pi,pi/2),f1],[(pi/2,pi),f2]])
sage: f.fourier_series_partial_sum_cesaro(3,pi)
-2*cos(x)/pi - sin(2*x)/pi + 2*sin(x)/pi - 1/4
fourier_series_partial_sum_filtered(N, L, F)
Returns the “filtered” partial sum
$f(x) \sim \frac{a_0}{2} + \sum_{n=1}^N F_n*[a_n\cos(\frac{n\pi x}{L}) + b_n\sin(\frac{n\pi x}{L})],$
as a string, where $$F = [F_1,F_2, ..., F_{N}]$$ is a list of length $$N$$ consisting of real numbers. This can be used to plot FS solutions to the heat and wave PDEs.
EXAMPLE:
sage: f(x) = x^2
sage: f = Piecewise([[(-1,1),f]])
sage: f.fourier_series_partial_sum_filtered(3,1,[1,1,1])
cos(2*pi*x)/pi^2 - 4*cos(pi*x)/pi^2 + 1/3
sage: f1(x) = -1
sage: f2(x) = 2
sage: f = Piecewise([[(-pi,pi/2),f1],[(pi/2,pi),f2]])
sage: f.fourier_series_partial_sum_filtered(3,pi,[1,1,1])
-3*cos(x)/pi - 3*sin(2*x)/pi + 3*sin(x)/pi - 1/4
fourier_series_partial_sum_hann(N, L)
Returns the Hann-filtered partial sum (named after von Hann, not Hamming)
$f(x) \sim \frac{a_0}{2} + \sum_{n=1}^N H_N(n)*[a_n\cos(\frac{n\pi x}{L}) + b_n\sin(\frac{n\pi x}{L})],$
as a string, where $$H_N(x) = (1+\cos(\pi x/N))/2$$. This is a “smoother” partial sum - the Gibbs phenomenon is mollified.
EXAMPLE:
sage: f(x) = x^2
sage: f = Piecewise([[(-1,1),f]])
sage: f.fourier_series_partial_sum_hann(3,1)
1/4*cos(2*pi*x)/pi^2 - 3*cos(pi*x)/pi^2 + 1/3
sage: f1(x) = -1
sage: f2(x) = 2
sage: f = Piecewise([[(-pi,pi/2),f1],[(pi/2,pi),f2]])
sage: f.fourier_series_partial_sum_hann(3,pi)
-9/4*cos(x)/pi - 3/4*sin(2*x)/pi + 9/4*sin(x)/pi - 1/4
fourier_series_sine_coefficient(n, L)
Returns the n-th Fourier series coefficient of $$\sin(n\pi x/L)$$, $$b_n$$.
INPUT:
• self - the function f(x), defined over -L x L
• n - an integer n0
• L - (the period)/2
OUTPUT: $$b_n = \frac{1}{L}\int_{-L}^L f(x)\sin(n\pi x/L)dx$$
EXAMPLES:
sage: f(x) = x^2
sage: f = Piecewise([[(-1,1),f]])
sage: f.fourier_series_sine_coefficient(2,1) # L=1, n=2
0
fourier_series_value(x, L)
Returns the value of the Fourier series coefficient of self at $$x$$,
$\begin{split}f(x) \sim \frac{a_0}{2} + \sum_{n=1}^\infty [a_n\cos(\frac{n\pi x}{L}) + b_n\sin(\frac{n\pi x}{L})], \ \ \ -L<x<L. \end{split}$
This method applies to piecewise non-polynomial functions as well.
INPUT:
• self - the function f(x), defined over -L x L
• x - a real number
• L - (the period)/2
OUTPUT: $$(f^*(x+)+f^*(x-)/2$$, where $$f^*$$ denotes the function $$f$$ extended to $$\RR$$ with period $$2L$$ (Dirichlet’s Theorem for Fourier series).
EXAMPLES:
sage: f1(x) = 1
sage: f2(x) = 1-x
sage: f3(x) = exp(x)
sage: f4(x) = sin(2*x)
sage: f = Piecewise([[(-10,1),f1],[(1,2),f2],[(2,3),f3],[(3,10),f4]])
sage: f.fourier_series_value(101,10)
1/2
sage: f.fourier_series_value(100,10)
1
sage: f.fourier_series_value(10,10)
1/2*sin(20) + 1/2
sage: f.fourier_series_value(20,10)
1
sage: f.fourier_series_value(30,10)
1/2*sin(20) + 1/2
sage: f1(x) = -1
sage: f2(x) = 2
sage: f = Piecewise([[(-pi,0),lambda x:0],[(0,pi/2),f1],[(pi/2,pi),f2]])
sage: f.fourier_series_value(-1,pi)
0
sage: f.fourier_series_value(20,pi)
-1
sage: f.fourier_series_value(pi/2,pi)
1/2
functions()
Returns the list of functions (the “pieces”).
EXAMPLES:
sage: f1(x) = 1
sage: f2(x) = 1-x
sage: f3(x) = exp(x)
sage: f4(x) = sin(2*x)
sage: f = Piecewise([[(0,1),f1],[(1,2),f2],[(2,3),f3],[(3,10),f4]])
sage: f.functions()
[x |--> 1, x |--> -x + 1, x |--> e^x, x |--> sin(2*x)]
integral(x=None, a=None, b=None, definite=False)
By default, returns the indefinite integral of the function. If definite=True is given, returns the definite integral.
AUTHOR:
• Paul Butler
EXAMPLES:
sage: f1(x) = 1-x
sage: f = Piecewise([[(0,1),1],[(1,2),f1]])
sage: f.integral(definite=True)
1/2
sage: f1(x) = -1
sage: f2(x) = 2
sage: f = Piecewise([[(0,pi/2),f1],[(pi/2,pi),f2]])
sage: f.integral(definite=True)
1/2*pi
sage: f1(x) = 2
sage: f2(x) = 3 - x
sage: f = Piecewise([[(-2, 0), f1], [(0, 3), f2]])
sage: f.integral()
Piecewise defined function with 2 parts, [[(-2, 0), x |--> 2*x + 4], [(0, 3), x |--> -1/2*x^2 + 3*x + 4]]
sage: f1(y) = -1
sage: f2(y) = y + 3
sage: f3(y) = -y - 1
sage: f4(y) = y^2 - 1
sage: f5(y) = 3
sage: f = Piecewise([[(-4,-3),f1],[(-3,-2),f2],[(-2,0),f3],[(0,2),f4],[(2,3),f5]])
sage: F = f.integral(y)
sage: F
Piecewise defined function with 5 parts, [[(-4, -3), y |--> -y - 4], [(-3, -2), y |--> 1/2*y^2 + 3*y + 7/2], [(-2, 0), y |--> -1/2*y^2 - y - 1/2], [(0, 2), y |--> 1/3*y^3 - y - 1/2], [(2, 3), y |--> 3*y - 35/6]]
Ensure results are consistent with FTC:
sage: F(-3) - F(-4)
-1
sage: F(-1) - F(-3)
1
sage: F(2) - F(0)
2/3
sage: f.integral(y, 0, 2)
2/3
sage: F(3) - F(-4)
19/6
sage: f.integral(y, -4, 3)
19/6
sage: f.integral(definite=True)
19/6
sage: f1(y) = (y+3)^2
sage: f2(y) = y+3
sage: f3(y) = 3
sage: f = Piecewise([[(-infinity, -3), f1], [(-3, 0), f2], [(0, infinity), f3]])
sage: f.integral()
Piecewise defined function with 3 parts, [[(-Infinity, -3), y |--> 1/3*y^3 + 3*y^2 + 9*y + 9], [(-3, 0), y |--> 1/2*y^2 + 3*y + 9/2], [(0, +Infinity), y |--> 3*y + 9/2]]
sage: f1(x) = e^(-abs(x))
sage: f = Piecewise([[(-infinity, infinity), f1]])
sage: f.integral(definite=True)
2
sage: f.integral()
Piecewise defined function with 1 parts, [[(-Infinity, +Infinity), x |--> -1/2*((sgn(x) - 1)*e^(2*x) - 2*e^x*sgn(x) + sgn(x) + 1)*e^(-x) - 1]]
sage: f = Piecewise([((0, 5), cos(x))])
sage: f.integral()
Piecewise defined function with 1 parts, [[(0, 5), x |--> sin(x)]]
TESTS:
Verify that piecewise integrals of zero work (trac #10841):
sage: f0(x) = 0
sage: f = Piecewise([[(0,1),f0]])
sage: f.integral(x,0,1)
0
sage: f = Piecewise([[(0,1), 0]])
sage: f.integral(x,0,1)
0
sage: f = Piecewise([[(0,1), SR(0)]])
sage: f.integral(x,0,1)
0
intervals()
A piecewise non-polynomial example.
EXAMPLES:
sage: f1(x) = 1
sage: f2(x) = 1-x
sage: f3(x) = exp(x)
sage: f4(x) = sin(2*x)
sage: f = Piecewise([[(0,1),f1],[(1,2),f2],[(2,3),f3],[(3,10),f4]])
sage: f.intervals()
[(0, 1), (1, 2), (2, 3), (3, 10)]
laplace(x='x', s='t')
Returns the Laplace transform of self with respect to the variable var.
INPUT:
• x - variable of self
• s - variable of Laplace transform.
We assume that a piecewise function is 0 outside of its domain and that the left-most endpoint of the domain is 0.
EXAMPLES:
sage: x, s, w = var('x, s, w')
sage: f = Piecewise([[(0,1),1],[(1,2), 1-x]])
sage: f.laplace(x, s)
-e^(-s)/s + (s + 1)*e^(-2*s)/s^2 + 1/s - e^(-s)/s^2
sage: f.laplace(x, w)
-e^(-w)/w + (w + 1)*e^(-2*w)/w^2 + 1/w - e^(-w)/w^2
sage: y, t = var('y, t')
sage: f = Piecewise([[(1,2), 1-y]])
sage: f.laplace(y, t)
(t + 1)*e^(-2*t)/t^2 - e^(-t)/t^2
sage: s = var('s')
sage: t = var('t')
sage: f1(t) = -t
sage: f2(t) = 2
sage: f = Piecewise([[(0,1),f1],[(1,infinity),f2]])
sage: f.laplace(t,s)
(s + 1)*e^(-s)/s^2 + 2*e^(-s)/s - 1/s^2
length()
Returns the number of pieces of this function.
EXAMPLES:
sage: f1(x) = 1
sage: f2(x) = 1 - x
sage: f = Piecewise([[(0,1),f1],[(1,2),f2]])
sage: f.length()
2
list()
Returns the pieces of this function as a list of functions.
EXAMPLES:
sage: f1(x) = 1
sage: f2(x) = 1 - x
sage: f = Piecewise([[(0,1),f1],[(1,2),f2]])
sage: f.list()
[[(0, 1), x |--> 1], [(1, 2), x |--> -x + 1]]
plot(*args, **kwds)
Returns the plot of self.
Keyword arguments are passed onto the plot command for each piece of the function. E.g., the plot_points keyword affects each segment of the plot.
EXAMPLES:
sage: f1(x) = 1
sage: f2(x) = 1-x
sage: f3(x) = exp(x)
sage: f4(x) = sin(2*x)
sage: f = Piecewise([[(0,1),f1],[(1,2),f2],[(2,3),f3],[(3,10),f4]])
sage: P = f.plot(rgbcolor=(0.7,0.1,0), plot_points=40)
sage: P
Remember: to view this, type show(P) or P.save(“path/myplot.png”) and then open it in a graphics viewer such as GIMP.
TESTS:
We should not add each piece to the legend individually, since this creates duplicates (trac ticket #12651). This tests that only one of the graphics objects in the plot has a non-None legend_label:
sage: f1 = sin(x)
sage: f2 = cos(x)
sage: f = piecewise([[(-1,0), f1],[(0,1), f2]])
sage: p = f.plot(legend_label='$f(x)$')
sage: lines = [
... line
... for line in p._objects
... if line.options()['legend_label'] is not None ]
sage: len(lines)
1
plot_fourier_series_partial_sum(N, L, xmin, xmax, **kwds)
Plots the partial sum
$f(x) \sim \frac{a_0}{2} + sum_{n=1}^N [a_n\cos(\frac{n\pi x}{L}) + b_n\sin(\frac{n\pi x}{L})],$
over xmin x xmin.
EXAMPLE:
sage: f1(x) = -2
sage: f2(x) = 1
sage: f3(x) = -1
sage: f4(x) = 2
sage: f = Piecewise([[(-pi,-pi/2),f1],[(-pi/2,0),f2],[(0,pi/2),f3],[(pi/2,pi),f4]])
sage: P = f.plot_fourier_series_partial_sum(3,pi,-5,5) # long time
sage: f1(x) = -1
sage: f2(x) = 2
sage: f = Piecewise([[(-pi,pi/2),f1],[(pi/2,pi),f2]])
sage: P = f.plot_fourier_series_partial_sum(15,pi,-5,5) # long time
Remember, to view this type show(P) or P.save(“path/myplot.png”) and then open it in a graphics viewer such as GIMP.
plot_fourier_series_partial_sum_cesaro(N, L, xmin, xmax, **kwds)
Plots the partial sum
$f(x) \sim \frac{a_0}{2} + \sum_{n=1}^N (1-n/N)*[a_n\cos(\frac{n\pi x}{L}) + b_n\sin(\frac{n\pi x}{L})],$
over xmin x xmin. This is a “smoother” partial sum - the Gibbs phenomenon is mollified.
EXAMPLE:
sage: f1(x) = -2
sage: f2(x) = 1
sage: f3(x) = -1
sage: f4(x) = 2
sage: f = Piecewise([[(-pi,-pi/2),f1],[(-pi/2,0),f2],[(0,pi/2),f3],[(pi/2,pi),f4]])
sage: P = f.plot_fourier_series_partial_sum_cesaro(3,pi,-5,5) # long time
sage: f1(x) = -1
sage: f2(x) = 2
sage: f = Piecewise([[(-pi,pi/2),f1],[(pi/2,pi),f2]])
sage: P = f.plot_fourier_series_partial_sum_cesaro(15,pi,-5,5) # long time
Remember, to view this type show(P) or P.save(“path/myplot.png”) and then open it in a graphics viewer such as GIMP.
plot_fourier_series_partial_sum_filtered(N, L, F, xmin, xmax, **kwds)
Plots the partial sum
$f(x) \sim \frac{a_0}{2} + \sum_{n=1}^N F_n*[a_n\cos(\frac{n\pi x}{L}) + b_n\sin(\frac{n\pi x}{L})],$
over xmin x xmin, where $$F = [F_1,F_2, ..., F_{N}]$$ is a list of length $$N$$ consisting of real numbers. This can be used to plot FS solutions to the heat and wave PDEs.
EXAMPLE:
sage: f1(x) = -2
sage: f2(x) = 1
sage: f3(x) = -1
sage: f4(x) = 2
sage: f = Piecewise([[(-pi,-pi/2),f1],[(-pi/2,0),f2],[(0,pi/2),f3],[(pi/2,pi),f4]])
sage: P = f.plot_fourier_series_partial_sum_filtered(3,pi,[1]*3,-5,5) # long time
sage: f1(x) = -1
sage: f2(x) = 2
sage: f = Piecewise([[(-pi,-pi/2),f1],[(-pi/2,0),f2],[(0,pi/2),f1],[(pi/2,pi),f2]])
sage: P = f.plot_fourier_series_partial_sum_filtered(15,pi,[1]*15,-5,5) # long time
Remember, to view this type show(P) or P.save(“path/myplot.png”) and then open it in a graphics viewer such as GIMP.
plot_fourier_series_partial_sum_hann(N, L, xmin, xmax, **kwds)
Plots the partial sum
$f(x) \sim \frac{a_0}{2} + \sum_{n=1}^N H_N(n)*[a_n\cos(\frac{n\pi x}{L}) + b_n\sin(\frac{n\pi x}{L})],$
over xmin x xmin, where H_N(x) = (0.5)+(0.5)*cos(x*pi/N) is the N-th Hann filter.
EXAMPLE:
sage: f1(x) = -2
sage: f2(x) = 1
sage: f3(x) = -1
sage: f4(x) = 2
sage: f = Piecewise([[(-pi,-pi/2),f1],[(-pi/2,0),f2],[(0,pi/2),f3],[(pi/2,pi),f4]])
sage: P = f.plot_fourier_series_partial_sum_hann(3,pi,-5,5) # long time
sage: f1(x) = -1
sage: f2(x) = 2
sage: f = Piecewise([[(-pi,pi/2),f1],[(pi/2,pi),f2]])
sage: P = f.plot_fourier_series_partial_sum_hann(15,pi,-5,5) # long time
Remember, to view this type show(P) or P.save(“path/myplot.png”) and then open it in a graphics viewer such as GIMP.
riemann_sum(N, mode=None)
Returns the piecewise line function defined by the Riemann sums in numerical integration based on a subdivision into N subintervals. Set mode=”midpoint” for the height of the rectangles to be determined by the midpoint of the subinterval; set mode=”right” for the height of the rectangles to be determined by the right-hand endpoint of the subinterval; the default is mode=”left” (the height of the rectangles to be determined by the left-hand endpoint of the subinterval).
EXAMPLES:
sage: f1(x) = x^2
sage: f2(x) = 5-x^2
sage: f = Piecewise([[(0,1),f1],[(1,2),f2]])
sage: f.riemann_sum(6,mode="midpoint")
Piecewise defined function with 6 parts, [[(0, 1/3), 1/36], [(1/3, 2/3), 1/4], [(2/3, 1), 25/36], [(1, 4/3), 131/36], [(4/3, 5/3), 11/4], [(5/3, 2), 59/36]]
sage: f = Piecewise([[(-1,1),(1-x^2).function(x)]])
sage: rsf = f.riemann_sum(7)
sage: P = f.plot(rgbcolor=(0.7,0.1,0.5), plot_points=40)
sage: Q = rsf.plot(rgbcolor=(0.7,0.6,0.6), plot_points=40)
sage: L = add([line([[a,0],[a,f(x=a)]],rgbcolor=(0.7,0.6,0.6)) for (a,b),f in rsf.list()])
sage: P + Q + L
sage: f = Piecewise([[(-1,1),(1/2+x-x^3)]], x) ## example 3
sage: rsf = f.riemann_sum(8)
sage: P = f.plot(rgbcolor=(0.7,0.1,0.5), plot_points=40)
sage: Q = rsf.plot(rgbcolor=(0.7,0.6,0.6), plot_points=40)
sage: L = add([line([[a,0],[a,f(x=a)]],rgbcolor=(0.7,0.6,0.6)) for (a,b),f in rsf.list()])
sage: P + Q + L
riemann_sum_integral_approximation(N, mode=None)
Returns the piecewise line function defined by the Riemann sums in numerical integration based on a subdivision into N subintervals.
Set mode=”midpoint” for the height of the rectangles to be determined by the midpoint of the subinterval; set mode=”right” for the height of the rectangles to be determined by the right-hand endpoint of the subinterval; the default is mode=”left” (the height of the rectangles to be determined by the left-hand endpoint of the subinterval).
EXAMPLES:
sage: f1(x) = x^2 ## example 1
sage: f2(x) = 5-x^2
sage: f = Piecewise([[(0,1),f1],[(1,2),f2]])
sage: f.riemann_sum_integral_approximation(6)
17/6
sage: f.riemann_sum_integral_approximation(6,mode="right")
19/6
sage: f.riemann_sum_integral_approximation(6,mode="midpoint")
3
sage: f.integral(definite=True)
3
sine_series_coefficient(n, L)
Returns the n-th sine series coefficient of $$\sin(n\pi x/L)$$, $$b_n$$.
INPUT:
• self - the function f(x), defined over 0 x L (no checking is done to insure this)
• n - an integer n0
• L - (the period)/2
OUTPUT:
$$b_n = \frac{2}{L}\int_{-L}^L f(x)\sin(n\pi x/L)dx$$ such that
$\begin{split}f(x) \sim \sum_{n=1}^\infty b_n\sin(\frac{n\pi x}{L}),\ \ 0<x<L. \end{split}$
EXAMPLES:
sage: f(x) = 1
sage: f = Piecewise([[(0,1),f]])
sage: f.sine_series_coefficient(2,1)
0
sage: f.sine_series_coefficient(3,1)
4/3/pi
tangent_line(pt)
Computes the linear function defining the tangent line of the piecewise function self.
EXAMPLES:
sage: f1(x) = x^2
sage: f2(x) = 5-x^3+x
sage: f = Piecewise([[(0,1),f1],[(1,2),f2]])
sage: tf = f.tangent_line(0.9) ## tangent line at x=0.9
sage: P = f.plot(rgbcolor=(0.7,0.1,0.5), plot_points=40)
sage: Q = tf.plot(rgbcolor=(0.7,0.2,0.2), plot_points=40)
sage: P + Q
trapezoid(N)
Returns the piecewise line function defined by the trapezoid rule for numerical integration based on a subdivision into N subintervals.
EXAMPLES:
sage: R.<x> = QQ[]
sage: f1 = x^2
sage: f2 = 5-x^2
sage: f = Piecewise([[(0,1),f1],[(1,2),f2]])
sage: f.trapezoid(4)
Piecewise defined function with 4 parts, [[(0, 1/2), 1/2*x], [(1/2, 1), 9/2*x - 2], [(1, 3/2), 1/2*x + 2], [(3/2, 2), -7/2*x + 8]]
sage: R.<x> = QQ[]
sage: f = Piecewise([[(-1,1),1-x^2]])
sage: tf = f.trapezoid(4)
sage: P = f.plot(rgbcolor=(0.7,0.1,0.5), plot_points=40)
sage: Q = tf.plot(rgbcolor=(0.7,0.6,0.6), plot_points=40)
sage: L = add([line([[a,0],[a,f(a)]],rgbcolor=(0.7,0.6,0.6)) for (a,b),f in tf.list()])
sage: P+Q+L
sage: R.<x> = QQ[]
sage: f = Piecewise([[(-1,1),1/2+x-x^3]]) ## example 3
sage: tf = f.trapezoid(6)
sage: P = f.plot(rgbcolor=(0.7,0.1,0.5), plot_points=40)
sage: Q = tf.plot(rgbcolor=(0.7,0.6,0.6), plot_points=40)
sage: L = add([line([[a,0],[a,f(a)]],rgbcolor=(0.7,0.6,0.6)) for (a,b),f in tf.list()])
sage: P+Q+L
TESTS:
Use variables other than x (trac ticket #13836):
sage: R.<y> = QQ[]
sage: f1 = y^2
sage: f2 = 5-y^2
sage: f = Piecewise([[(0,1),f1],[(1,2),f2]])
sage: f.trapezoid(4)
Piecewise defined function with 4 parts, [[(0, 1/2), 1/2*y], [(1/2, 1), 9/2*y - 2], [(1, 3/2), 1/2*y + 2], [(3/2, 2), -7/2*y + 8]]
trapezoid_integral_approximation(N)
Returns the approximation given by the trapezoid rule for numerical integration based on a subdivision into N subintervals.
EXAMPLES:
sage: f1(x) = x^2 ## example 1
sage: f2(x) = 1-(1-x)^2
sage: f = Piecewise([[(0,1),f1],[(1,2),f2]])
sage: P = f.plot(rgbcolor=(0.7,0.1,0.5), plot_points=40)
sage: tf = f.trapezoid(6)
sage: Q = tf.plot(rgbcolor=(0.7,0.6,0.6), plot_points=40)
sage: ta = f.trapezoid_integral_approximation(6)
sage: t = text('trapezoid approximation = %s'%ta, (1.5, 0.25))
sage: a = f.integral(definite=True)
sage: tt = text('area under curve = %s'%a, (1.5, -0.5))
sage: P + Q + t + tt
sage: f = Piecewise([[(0,1),f1],[(1,2),f2]]) ## example 2
sage: tf = f.trapezoid(4)
sage: ta = f.trapezoid_integral_approximation(4)
sage: Q = tf.plot(rgbcolor=(0.7,0.6,0.6), plot_points=40)
sage: t = text('trapezoid approximation = %s'%ta, (1.5, 0.25))
sage: a = f.integral(definite=True)
sage: tt = text('area under curve = %s'%a, (1.5, -0.5))
sage: P+Q+t+tt
unextend()
This removes any parts in the front or back of the function which is zero (the inverse to extend_by_zero_to).
EXAMPLES:
sage: R.<x> = QQ[]
sage: f = Piecewise([[(-3,-1),1+2+x],[(-1,1),1-x^2]])
sage: e = f.extend_by_zero_to(-10,10); e
Piecewise defined function with 4 parts, [[(-10, -3), 0], [(-3, -1), x + 3], [(-1, 1), -x^2 + 1], [(1, 10), 0]]
sage: d = e.unextend(); d
Piecewise defined function with 2 parts, [[(-3, -1), x + 3], [(-1, 1), -x^2 + 1]]
sage: d==f
True
which_function(x0)
Returns the function piece used to evaluate self at x0.
EXAMPLES:
sage: f1(z) = z
sage: f2(x) = 1-x
sage: f3(y) = exp(y)
sage: f4(t) = sin(2*t)
sage: f = Piecewise([[(0,1),f1],[(1,2),f2],[(2,3),f3],[(3,10),f4]])
sage: f.which_function(3/2)
x |--> -x + 1
sage.functions.piecewise.piecewise(list_of_pairs, var=None)
Returns a piecewise function from a list of (interval, function) pairs.
list_of_pairs is a list of pairs (I, fcn), where fcn is a Sage function (such as a polynomial over RR, or functions using the lambda notation), and I is an interval such as I = (1,3). Two consecutive intervals must share a common endpoint.
If the optional var is specified, then any symbolic expressions in the list will be converted to symbolic functions using fcn.function(var). (This says which variable is considered to be “piecewise”.)
We assume that these definitions are consistent (ie, no checking is done).
EXAMPLES:
sage: f1(x) = -1
sage: f2(x) = 2
sage: f = Piecewise([[(0,pi/2),f1],[(pi/2,pi),f2]])
sage: f(1)
-1
sage: f(3)
2
sage: f = Piecewise([[(0,1),x], [(1,2),x^2]], x); f
Piecewise defined function with 2 parts, [[(0, 1), x |--> x], [(1, 2), x |--> x^2]]
sage: f(0.9)
0.900000000000000
sage: f(1.1)
1.21000000000000
#### Previous topic
Transcendental Functions
Spike Functions
|
# Natural deduction in first-order logic
I've sat for more than an hour now and I don't understand how I'm supposed to solve the task below.
$\{\forall x(P(x) \land Q(x)), \exists x\neg P(x)\} \vdash \exists x \neg Q(x)$
So I'm a bit confused if I'm supposed to use RAA or $\neg$ introduction when solving this using natural deduction.
$$\dfrac{\exists x \, \lnot P(x) \qquad \dfrac{\dfrac{\dfrac{\,?\,}{\bot}}{\lnot Q(u)}}{\exists x \, \lnot Q(x)}\exists_i}{\exists x \, \lnot Q(x)}\exists_e^*$$
So, I do know that if I extract $P(x)$ and $\neg P(x)$ it will result in a $\bot$, but that's about it. Any help is greatly appreciated!
• Can you add the deduction system you are using? – blub Aug 20 '18 at 15:56
• GrahamKemp here is the task: imgur.com/a/5qlW3iN – Fredrik Andersson Aug 20 '18 at 16:01
• Well isn't that just lovely... Thanks for pointing that out. – Fredrik Andersson Aug 20 '18 at 16:04
• @GrahamKemp - I disagree with you. Please, see my answer and in case tell me where I'm wrong. – Taroccoesbrocco Aug 20 '18 at 18:10
• Hmmm... yes....indeed. Appologies. @FredrikAndersson , Taroccoesbrocco is correct. – Graham Kemp Aug 20 '18 at 23:00
The premises $\forall x (P(x) \land Q(x))$ and $\exists x \, \lnot P(x)$ are contradictory, so according to the principle of explosion (which corresponds to the rule $\bot_e$, a.k.a. ex falso quodlibet in natural deduction) from them you can derive everything, in particular $\exists x \, \lnot Q(x)$, as showed by the following derivation.
$$\dfrac{\exists x \, \lnot P(x) \qquad \dfrac{\dfrac{[\lnot P(x)]^* \qquad \dfrac{\dfrac{\forall x (P(x) \land Q(x))}{P(x) \land Q(x)}\forall_e}{P(x)}\land_e}{\bot}\lnot_e}{\exists x \, \lnot Q(x)}\bot_e}{\exists x \, \lnot Q(x)}\exists_e^*$$
Note that there is no need for the rules RAA and $\lnot_i$.
• @FredrikAndersson - Ex falso quodlibet ($\bot_e$) is just a special case of RAA, where you do not discharge anything. In the rule RAA you can discharge an arbitrary finite number ($0$ or $1$ or $2$ or ...) of formula occurrences, the $\bot_e$ is just the special case of RAA where in particular you discharge $0$ formula occurrences. If you accept RAA, you must accept $\bot_e$ (and call it RAA if you prefer). – Taroccoesbrocco Aug 20 '18 at 18:26
• So what I did initially would be on the right track? I just did one unnecessary step where I could have included $\exists$ as well right after $\bot$. – Fredrik Andersson Aug 20 '18 at 18:38
• @FredrikAndersson - Yes, it is, but in your attempt you should also complete the part with "$?$". Note that your unnecessary step under $\bot$ is not wrong, just unnecessary. In other words, you can complete your attempt by putting together my derivation above $\bot$ and your derivation below $\bot$. – Taroccoesbrocco Aug 20 '18 at 18:47
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Home > Standard Error > Statistical Error Formula
# Statistical Error Formula
## Contents
Contents 1 Introduction to the standard error 1.1 Standard error of the mean (SEM) 1.1.1 Sampling from a distribution with a large standard deviation 1.1.2 Sampling from a distribution with a Specifically, the standard error equations use p in place of P, and s in place of σ. Using a sample to estimate the standard error In the examples so far, the population standard deviation σ was assumed to be known. Moreover, this formula works for positive and negative ρ alike.[10] See also unbiased estimation of standard deviation for more discussion. http://activews.com/standard-error/std-error-formula.html
and Keeping, E.S. (1963) Mathematics of Statistics, van Nostrand, p. 187 ^ Zwillinger D. (1995), Standard Mathematical Tables and Formulae, Chapman&Hall/CRC. For the age at first marriage, the population mean age is 23.44, and the population standard deviation is 4.72. Roman letters indicate that these are sample values. The concept of a sampling distribution is key to understanding the standard error.
## Standard Error Formula Excel
Two data sets will be helpful to illustrate the concept of a sampling distribution and its use to calculate the standard error. With n = 2 the underestimate is about 25%, but for n = 6 the underestimate is only 5%. ERROR The requested URL could not be retrieved The following error was encountered while trying to retrieve the URL: http://0.0.0.10/ Connection to 0.0.0.10 failed.
Student approximation when σ value is unknown Further information: Student's t-distribution §Confidence intervals In many practical applications, the true value of σ is unknown. Repeating the sampling procedure as for the Cherry Blossom runners, take 20,000 samples of size n=16 from the age at first marriage population. v t e Statistics Outline Index Descriptive statistics Continuous data Center Mean arithmetic geometric harmonic Median Mode Dispersion Variance Standard deviation Coefficient of variation Percentile Range Interquartile range Shape Moments Standard Error Of Proportion ISBN 0-7167-1254-7 , p 53 ^ Barde, M. (2012). "What to use to express the variability of data: Standard deviation or standard error of mean?".
They may be used to calculate confidence intervals. Standard Error Example This approximate formula is for moderate to large sample sizes; the reference gives the exact formulas for any sample size, and can be applied to heavily autocorrelated time series like Wall This estimate may be compared with the formula for the true standard deviation of the sample mean: SD x ¯ = σ n {\displaystyle {\text{SD}}_{\bar {x}}\ ={\frac {\sigma }{\sqrt {n}}}} Refer to the above table for the appropriate z*-value.
As the sample size increases, the dispersion of the sample means clusters more closely around the population mean and the standard error decreases. Standard Error Vs Standard Deviation You need to make sure that is at least 10. Parameters Population mean = μ = ( Σ Xi ) / N Population standard deviation = σ = sqrt [ Σ ( Xi - μ )2 / N ] Population variance For illustration, the graph below shows the distribution of the sample means for 20,000 samples, where each sample is of size n=16.
• National Center for Health Statistics typically does not report an estimated mean if its relative standard error exceeds 30%. (NCHS also typically requires at least 30 observations – if not more
• Despite the small difference in equations for the standard deviation and the standard error, this small difference changes the meaning of what is being reported from a description of the variation
• In regression analysis, the term "standard error" is also used in the phrase standard error of the regression to mean the ordinary least squares estimate of the standard deviation of the
• and Keeping, E.S. (1963) Mathematics of Statistics, van Nostrand, p. 187 ^ Zwillinger D. (1995), Standard Mathematical Tables and Formulae, Chapman&Hall/CRC.
• This formula does not assume a normal distribution.
• Test Your Understanding Problem 1 Which of the following statements is true.
• It is useful to compare the standard error of the mean for the age of the runners versus the age at first marriage, as in the graph.
• American Statistical Association. 25 (4): 30–32.
• The age data are in the data set run10 from the R package openintro that accompanies the textbook by Dietz [4] The graph shows the distribution of ages for the runners.
## Standard Error Example
Larger sample sizes give smaller standard errors As would be expected, larger sample sizes give smaller standard errors. Now, if it's 29, don't panic -- 30 is not a magic number, it's just a general rule of thumb. (The population standard deviation must be known either way.) Here's an Standard Error Formula Excel The notation for standard error can be any one of SE, SEM (for standard error of measurement or mean), or SE. Standard Error Calculator AP Statistics Tutorial Exploring Data ▸ The basics ▾ Variables ▾ Population vs sample ▾ Central tendency ▾ Variability ▾ Position ▸ Charts and graphs ▾ Patterns in data ▾ Dotplots
Also, be sure that statistics are reported with their correct units of measure, and if they're not, ask what the units are. http://activews.com/standard-error/standard-error-of-mean-formula.html Hutchinson, Essentials of statistical methods in 41 pages ^ Gurland, J; Tripathi RC (1971). "A simple approximation for unbiased estimation of the standard deviation". For example, the area between z*=1.28 and z=-1.28 is approximately 0.80. The effect of the FPC is that the error becomes zero when the sample size n is equal to the population size N. Standard Error Formula Statistics
The general formula for the margin of error for a sample proportion (if certain conditions are met) is where is the sample proportion, n is the sample size, and z* is Sample mean = x = ( Σ xi ) / n Sample standard deviation = s = sqrt [ Σ ( xi - x )2 / ( n - 1 ) In the example of a poll on the president, n = 1,000, Now check the conditions: Both of these numbers are at least 10, so everything is okay. check over here In this scenario, the 2000 voters are a sample from all the actual voters.
Standard error of the mean (SEM) This section will focus on the standard error of the mean. Standard Error Regression As will be shown, the standard error is the standard deviation of the sampling distribution. JSTOR2340569. (Equation 1) ^ James R.
## The survey with the lower relative standard error can be said to have a more precise measurement, since it has proportionately less sampling variation around the mean.
The standard deviation of all possible sample means is the standard error, and is represented by the symbol σ x ¯ {\displaystyle \sigma _{\bar {x}}} . Consider the following scenarios. The variability of a statistic is measured by its standard deviation. Standard Error Of Estimate Formula Of course, T / n {\displaystyle T/n} is the sample mean x ¯ {\displaystyle {\bar {x}}} .
In an example above, n=16 runners were selected at random from the 9,732 runners. When the sampling fraction is large (approximately at 5% or more) in an enumerative study, the estimate of the standard error must be corrected by multiplying by a "finite population correction"[9] The distribution of the mean age in all possible samples is called the sampling distribution of the mean. this content For the runners, the population mean age is 33.87, and the population standard deviation is 9.27.
The standard error is a measure of central tendency. (A) I only (B) II only (C) III only (D) All of the above. (E) None of the above. The sample proportion of 52% is an estimate of the true proportion who will vote for candidate A in the actual election. The third formula assigns sample to strata, based on a proportionate design. The standard deviation of the age was 9.27 years.
This allows you to account for about 95% of all possible results that may have occurred with repeated sampling. Hyattsville, MD: U.S. This formula may be derived from what we know about the variance of a sum of independent random variables.[5] If X 1 , X 2 , … , X n {\displaystyle The next graph shows the sampling distribution of the mean (the distribution of the 20,000 sample means) superimposed on the distribution of ages for the 9,732 women.
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3 added 195 characters in body edited Jul 26 '17 at 19:37 ItamarG3 4,54822 gold badges1616 silver badges5353 bronze badges You can use an analogy of priority queues (maybe a hospital, if it isn't too gruesome; perhaps a queue for ordering work tasks) A nurse doesn't need a patient to have some indicator that they are supposed to be treated before some other patients (these indicators are the parenthesis). There's simply a known treatment order, based on the urgency of the treatment. If all patients need equally urgent treatment, then they are treated according to the order of their appearances (the ones who "check in" first are the ones treated first). Nurses know how urgent a new patient's treatment is, and so does the compiler. For java, it goes like this: 1. postfix expr++ expr-- 2. unary ++expr --expr +expr -expr ~ ! 3. multiplicative * / % 4. additive + - 5. shift << >> >>> 6. relational < > <= >= instanceof 7. equality == != 8. bitwise AND & 9. bitwise exclusive OR ^ 10. bitwise inclusive OR | 11. logical AND && 12. logical OR || 13. ternary ? : 14. assignment = += -= *= /= %= &= ^= |= <<= >>= >>>= (Taken from the Oracle Docs, numbering is mine. Originally has table) In effect, the function and member call are before all other operators. So . is of precedence 0. The nurse (compiler) knows in which order to actually treat (execute\write to bytecode) the various levels or urgency. Different languages might have different ordering, but the analogy of a nurse ordering the treatment based on urgency would still hold. In the example you gave, it's more "urgent" to deal with the AND than it is with OR. So in SQL, some operators are dealt with before others. Some operators require expressions on both sides, and some don't (select only needs from the right, while AND needs from both sides). You can use an analogy of priority queues (maybe a hospital, if it isn't too gruesome; perhaps a queue for ordering work tasks) A nurse doesn't need a patient to have some indicator that they are supposed to be treated before some other patients (these indicators are the parenthesis). There's simply a known treatment order, based on the urgency of the treatment. If all patients need equally urgent treatment, then they are treated according to the order of their appearances (the ones who "check in" first are the ones treated first). Nurses know how urgent a new patient's treatment is, and so does the compiler. For java, it goes like this: 1. postfix expr++ expr-- 2. unary ++expr --expr +expr -expr ~ ! 3. multiplicative * / % 4. additive + - 5. shift << >> >>> 6. relational < > <= >= instanceof 7. equality == != 8. bitwise AND & 9. bitwise exclusive OR ^ 10. bitwise inclusive OR | 11. logical AND && 12. logical OR || 13. ternary ? : 14. assignment = += -= *= /= %= &= ^= |= <<= >>= >>>= (Taken from the Oracle Docs, numbering is mine. Originally has table) In effect, the function and member call are before all other operators. So . is of precedence 0. The nurse (compiler) knows in which order to actually treat (execute\write to bytecode) the various levels or urgency. Different languages might have different ordering, but the analogy of a nurse ordering the treatment based on urgency would still hold. In the example you gave, it's more "urgent" to deal with the AND than it is with OR. You can use an analogy of priority queues (maybe a hospital, if it isn't too gruesome; perhaps a queue for ordering work tasks) A nurse doesn't need a patient to have some indicator that they are supposed to be treated before some other patients (these indicators are the parenthesis). There's simply a known treatment order, based on the urgency of the treatment. If all patients need equally urgent treatment, then they are treated according to the order of their appearances (the ones who "check in" first are the ones treated first). Nurses know how urgent a new patient's treatment is, and so does the compiler. For java, it goes like this: 1. postfix expr++ expr-- 2. unary ++expr --expr +expr -expr ~ ! 3. multiplicative * / % 4. additive + - 5. shift << >> >>> 6. relational < > <= >= instanceof 7. equality == != 8. bitwise AND & 9. bitwise exclusive OR ^ 10. bitwise inclusive OR | 11. logical AND && 12. logical OR || 13. ternary ? : 14. assignment = += -= *= /= %= &= ^= |= <<= >>= >>>= (Taken from the Oracle Docs, numbering is mine. Originally has table) In effect, the function and member call are before all other operators. So . is of precedence 0. The nurse (compiler) knows in which order to actually treat (execute\write to bytecode) the various levels or urgency. Different languages might have different ordering, but the analogy of a nurse ordering the treatment based on urgency would still hold. In the example you gave, it's more "urgent" to deal with the AND than it is with OR. So in SQL, some operators are dealt with before others. Some operators require expressions on both sides, and some don't (select only needs from the right, while AND needs from both sides). 2 added 106 characters in body edited Jul 26 '17 at 19:12 ItamarG3 4,54822 gold badges1616 silver badges5353 bronze badges You can use an analogy of priority queues (maybe a hospital, if it isn't too gruesome; perhaps a queue for ordering work tasks) A nurse doesn't need a patient to have some indicator that they are supposed to be treated before some other patients (these indicators are the parenthesis). There's simply a known treatment order, based on the urgency of the treatment. If all patients need equally urgent treatment, then they are treated according to the order of their appearances (the ones who "check in" first are the ones treated first). Nurses know how urgent a new patient's treatment is, and so does the compiler. For java, it goes like this: 1. postfix expr++ expr-- 2. unary ++expr --expr +expr -expr ~ ! 3. multiplicative * / % 4. additive + - 5. shift << >> >>> 6. relational < > <= >= instanceof 7. equality == != 8. bitwise AND & 9. bitwise exclusive OR ^ 10. bitwise inclusive OR | 11. logical AND && 12. logical OR || 13. ternary ? : 14. assignment = += -= *= /= %= &= ^= |= <<= >>= >>>= (Taken from the Oracle Docs, numbering is mine. Originally has table) In effect, the function and member call are before all other operators. So . is of precedence 0. The nurse (compiler) knows in which order to actually treat (execute\write to bytecode) the various levels or urgency. Different languages might have different ordering, but the analogy of a nurse ordering the treatment based on urgency would still hold. In the example you gave, it's more "urgent" to deal with the AND than it is with OR. You can use an analogy of priority queues (maybe a hospital, if it isn't too gruesome; perhaps a queue for ordering work tasks) A nurse doesn't need a patient to have some indicator that they are supposed to be treated before some other patients (these indicators are the parenthesis). There's simply a known treatment order, based on the urgency of the treatment. If all patients need equally urgent treatment, then they are treated according to the order of their appearances (the ones who "check in" first are the ones treated first). Nurses know how urgent a new patient's treatment is, and so does the compiler. For java, it goes like this: 1. postfix expr++ expr-- 2. unary ++expr --expr +expr -expr ~ ! 3. multiplicative * / % 4. additive + - 5. shift << >> >>> 6. relational < > <= >= instanceof 7. equality == != 8. bitwise AND & 9. bitwise exclusive OR ^ 10. bitwise inclusive OR | 11. logical AND && 12. logical OR || 13. ternary ? : 14. assignment = += -= *= /= %= &= ^= |= <<= >>= >>>= (Taken from the Oracle Docs, numbering is mine. Originally has table) The nurse (compiler) knows in which order to actually treat (execute\write to bytecode) the various levels or urgency. Different languages might have different ordering, but the analogy of a nurse ordering the treatment based on urgency would still hold. In the example you gave, it's more "urgent" to deal with the AND than it is with OR. You can use an analogy of priority queues (maybe a hospital, if it isn't too gruesome; perhaps a queue for ordering work tasks) A nurse doesn't need a patient to have some indicator that they are supposed to be treated before some other patients (these indicators are the parenthesis). There's simply a known treatment order, based on the urgency of the treatment. If all patients need equally urgent treatment, then they are treated according to the order of their appearances (the ones who "check in" first are the ones treated first). Nurses know how urgent a new patient's treatment is, and so does the compiler. For java, it goes like this: 1. postfix expr++ expr-- 2. unary ++expr --expr +expr -expr ~ ! 3. multiplicative * / % 4. additive + - 5. shift << >> >>> 6. relational < > <= >= instanceof 7. equality == != 8. bitwise AND & 9. bitwise exclusive OR ^ 10. bitwise inclusive OR | 11. logical AND && 12. logical OR || 13. ternary ? : 14. assignment = += -= *= /= %= &= ^= |= <<= >>= >>>= (Taken from the Oracle Docs, numbering is mine. Originally has table) In effect, the function and member call are before all other operators. So . is of precedence 0. The nurse (compiler) knows in which order to actually treat (execute\write to bytecode) the various levels or urgency. Different languages might have different ordering, but the analogy of a nurse ordering the treatment based on urgency would still hold. In the example you gave, it's more "urgent" to deal with the AND than it is with OR. 1 answered Jul 26 '17 at 18:36 ItamarG3 4,54822 gold badges1616 silver badges5353 bronze badges You can use an analogy of priority queues (maybe a hospital, if it isn't too gruesome; perhaps a queue for ordering work tasks) A nurse doesn't need a patient to have some indicator that they are supposed to be treated before some other patients (these indicators are the parenthesis). There's simply a known treatment order, based on the urgency of the treatment. If all patients need equally urgent treatment, then they are treated according to the order of their appearances (the ones who "check in" first are the ones treated first). Nurses know how urgent a new patient's treatment is, and so does the compiler. For java, it goes like this: 1. postfix expr++ expr-- 2. unary ++expr --expr +expr -expr ~ ! 3. multiplicative * / % 4. additive + - 5. shift << >> >>> 6. relational < > <= >= instanceof 7. equality == != 8. bitwise AND & 9. bitwise exclusive OR ^ 10. bitwise inclusive OR | 11. logical AND && 12. logical OR || 13. ternary ? : 14. assignment = += -= *= /= %= &= ^= |= <<= >>= >>>= (Taken from the Oracle Docs, numbering is mine. Originally has table) The nurse (compiler) knows in which order to actually treat (execute\write to bytecode) the various levels or urgency. Different languages might have different ordering, but the analogy of a nurse ordering the treatment based on urgency would still hold. In the example you gave, it's more "urgent" to deal with the AND than it is with OR.
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# NAG FL Interfacef11btf (complex_gen_basic_diag)
## ▸▿ Contents
Settings help
FL Name Style:
FL Specification Language:
## 1Purpose
f11btf is the third in a suite of three routines for the iterative solution of a complex general (non-Hermitian) system of simultaneous linear equations (see Golub and Van Loan (1996)). f11btf returns information about the computations during an iteration and/or after this has been completed. The first routine of the suite, f11brf, is a setup routine; the second routine, f11bsf, is the iterative solver itself.
These three routines are suitable for the solution of large sparse general (non-Hermitian) systems of equations.
## 2Specification
Fortran Interface
Subroutine f11btf ( itn, work,
Integer, Intent (In) :: lwork Integer, Intent (Inout) :: ifail Integer, Intent (Out) :: itn Real (Kind=nag_wp), Intent (Out) :: stplhs, stprhs, anorm, sigmax Complex (Kind=nag_wp), Intent (Inout) :: work(lwork)
#include <nag.h>
void f11btf_ (Integer *itn, double *stplhs, double *stprhs, double *anorm, double *sigmax, Complex work[], const Integer *lwork, Integer *ifail)
The routine may be called by the names f11btf or nagf_sparse_complex_gen_basic_diag.
## 3Description
f11btf returns information about the solution process. It can be called either during a monitoring step of f11bsf or after f11bsf has completed its tasks. Calling f11btf at any other time will result in an error condition being raised.
For further information you should read the documentation for f11brf and f11bsf.
## 4References
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
## 5Arguments
1: $\mathbf{itn}$Integer Output
On exit: the number of iterations carried out by f11bsf.
2: $\mathbf{stplhs}$Real (Kind=nag_wp) Output
On exit: the current value of the left-hand side of the termination criterion used by f11bsf.
3: $\mathbf{stprhs}$Real (Kind=nag_wp) Output
On exit: the current value of the right-hand side of the termination criterion used by f11bsf.
4: $\mathbf{anorm}$Real (Kind=nag_wp) Output
On exit: if ${\mathbf{iterm}}=1$ in the previous call to f11brf, then anorm contains ${‖A‖}_{p}$, where $p=1$, $2$ or $\infty$, either supplied or, in the case of $1$ or $\infty$, estimated by f11bsf; otherwise ${\mathbf{anorm}}=0.0$.
5: $\mathbf{sigmax}$Real (Kind=nag_wp) Output
On exit: if ${\mathbf{iterm}}=2$ in the previous call to f11brf, the current estimate of the largest singular value ${\sigma }_{1}\left(\overline{A}\right)$ of the preconditioned iteration matrix when it is used by the termination criterion in f11bsf, either when it has been supplied to f11brf or it has been estimated by f11bsf (see also Sections 3 and 5 in f11brf); otherwise, ${\mathbf{sigmax}}=0.0$ is returned.
6: $\mathbf{work}\left({\mathbf{lwork}}\right)$Complex (Kind=nag_wp) array Communication Array
On entry: the array work as returned by f11bsf (see also Sections 3 and 5 in f11bsf).
7: $\mathbf{lwork}$Integer Input
On entry: the dimension of the array work as declared in the (sub)program from which f11btf is called (see also f11brf).
Constraint: ${\mathbf{lwork}}\ge 120$.
Note: although the minimum value of lwork ensures the correct functioning of f11btf, a larger value is required by the iterative solver f11bsf (see also f11brf).
8: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).
## 6Error Indicators and Warnings
If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
f11btf has been called out of sequence.
${\mathbf{ifail}}=-7$
On entry, ${\mathbf{lwork}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{lwork}}\ge 120$.
${\mathbf{ifail}}=-99$
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
Not applicable.
## 8Parallelism and Performance
f11btf is not threaded in any implementation.
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## Differential and Integral Equations
### Uniform gradient bounds for the primitive equations of the ocean
#### Abstract
In this paper, we consider the 3D primitive equations of the ocean in the case of the Dirichlet boundary conditions on the side and bottom boundaries. We provide an explicit upper bound for the $H^{1}$ norm of the solution. We prove that, after a finite time, this norm is less than a constant which depends only on the viscosity $\nu$, the force $f$, and the domain $\Omega$. This improves our previous result from [7] where we established the global existence of strong solutions with an argument which does not give such explicit rates.
#### Article information
Source
Differential Integral Equations, Volume 21, Number 9-10 (2008), 837-849.
Dates
First available in Project Euclid: 20 December 2012
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# Find two unknowns that have a certain ratio.
I need to find a value for L and S where:
• P = 20
• L = 2S + P
• L + S + 3P = 1160
It's for deciding the column width in a website layout, which can change according the available screen size:
I tried giving L a random value that sounded about right (700) and then solving S using the first equation, but then the second simply turned out false, so that didn't work.
(Sorry if this question is too "amateuristic" for this site, but it doesn't look like there's a better SE alternative)
-
Do you know anything about matrices, Gaussian elimination, or linear algebra? – anon Jul 30 '11 at 14:11
@anon: Little. My math level is (high school) - (things I've forgotten) + (basic matrices and vectors used in game programming) – Bart van Heukelom Jul 30 '11 at 14:14
If you don't, the quickest fix would be to substitute $L=2S+20$ into the second equation to get $3S+80=1160$, which solves as $S=360$, and plugging $S$ into the first equation gives $L=740$. However if you know anything about the concepts I linked above there are more general approaches that work and that you might want to know about. – anon Jul 30 '11 at 14:15
What you have is a system of linear equations. Several methods for solving these are described in this section of that article.
In the present case, the most direct solution would be to substitute $L$ from the first equation into the second one:
$$(2S + 20) + S + 60 = 1160$$
$$3S + 80 = 1160$$
$$3S=1080$$
$$S=360\;.$$
Then plugging that into the first equation yields
$$L=2S + 20 = 2\cdot360+20= 720 + 20=740\;.$$
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I'm familiar with substitution and tried it, but the result was rather unhelpful (2S - 2S = 0). I probably did it wrong or with the wrong equations. Thanks – Bart van Heukelom Jul 30 '11 at 14:18
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440 views
A real $2 \times 2$ matrix $M$ such that $$M^2 = \begin{pmatrix} -1 & 0 \\ 0 & -1- \varepsilon \end{pmatrix}$$
1. exists for all $\varepsilon > 0$
2. does not exist for any $\varepsilon > 0$
3. exists for some $\varepsilon > 0$
4. none of the above is true
1
740 views
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# Thread: n-th derivative of u'exp(u)
1. ## n-th derivative of u'exp(u)
Hello...
BIG question (big for me, at least): Is there a general formula for the n-th derivative of
$
u'e^u\\
$
Basically I have a function exactly like that and I need to extract the n-th derivative.
2. Originally Posted by paolopiace
Hello...
BIG question (big for me, at least): Is there a general formula for the n-th derivative of
$
u'e^u\\
$
Basically I have a function exactly like that and I need to extract the n-th derivative.
I noticed a pattern at the 2nd derivative...
$f(u)=u'e^u \Rightarrow u^{(1)}e^u$
$f^{(1)}(u)=u^{(2)}e^u+(u^{(1)})^2e^u$
$f^{(2)}(u)=u^{(3)}e^u+u^{(2)}u^{(1)}e^u + 2u^{(2)}u^{(1)}e^u+\left(u^{(3)}\right)^2e^u$
$f^{(2)}(u)=u^{(3)}e^u+3u^{(2)}u^{(1)}e^u+\left(u^{ (3)}\right)^2e^u$
I may be wrong, but I will assume this:
$f^{(n)}(u)=u^{(n-1)}e^u+(n+1)u^{(n)}u^{(n-1)}\cdots u^{(1)}e^u+\left(u^{(n+1)}\right)^{n-1}e^u$
3. Originally Posted by colby2152
I noticed a pattern at the 2nd derivative...
$f(u)=u'e^u \Rightarrow u^{(1)}e^u$
$f^{(1)}(u)=u^{(2)}e^u+(u^{(1)})^2e^u$
$f^{(2)}(u)=u^{(3)}e^u+u^{(2)}u^{(1)}e^u + 2u^{(2)}u^{(1)}e^u+\left(u^{(3)}\right)^2e^u$
$f^{(2)}(u)=u^{(3)}e^u+3u^{(2)}u^{(1)}e^u+\left(u^{ (3)}\right)^2e^u$
I may be wrong, but I will assume this:
$f^{(n)}(u)=u^{(n-1)}e^u+(n+1)u^{(n)}u^{(n-1)}\cdots u^{(1)}e^u+\left(u^{(n+1)}\right)^{n-1}e^u$
There's a formula for taking the nth derivative of any function of the form y = f(x) g(x) called Leibniz's Rule.
4. ## Thanks. I know the Leibniz rule. I just wanted...
...to see if there is a more simple and straightforward rule for u'exp(u).
5. Originally Posted by paolopiace
Hello...
BIG question (big for me, at least): Is there a general formula for the n-th derivative of
$
u'e^u\\
$
Basically I have a function exactly like that and I need to extract the n-th derivative.
Since $u'e^u$ is the derivative of $e^u$, you only need to consider the $(n-1)$-th derivative of $e^u$.
6. Originally Posted by mr fantastic
There's a formula for taking the nth derivative of any function of the form y = f(x) g(x) called Leibniz's Rule.
Yeah, I never really used that rule. It is handy though - it is basically a binomial expansion, but for derivatives. Did I get the derivative right, or are there more terms in the middle?
7. Originally Posted by paolopiace
Is there a general formula for the n-th derivative of
$
u'e^u\\
$
Basically I have a function exactly like that and I need to extract the n-th derivative.
If $y = u'e^u$ then $y^{(n)} = D_n(u)e^u$, where $D_n(u)$ is some polynomial in u and its derivatives. If you differentiate one more time then you get $y^{(n+1)} = (D_n'(u) + u'D_n(u))e^u$. So $D_{n+1} = D_n'(u) + u'D_n(u)$. If you apply this recursive formula to find the first few derivatives then you get
$y^{(1)} = y' = \bigl((u^{(1)})^2 + u^{(2)}\bigr)e^u$,
$y^{(2)} = \bigl((u^{(1)})^3 + 3u^{(1)}u^{(2)}+ u^{(3)}\bigr)e^u$,
$y^{(3)} = \bigl((u^{(1)})^4 + 4u^{(1)}u^{(3)}+ 6(u^{(1)})^2u^{(2)} + 3(u^{(2)})^2 + u^{(4)}\bigr)e^u$,
$y^{(4)} = \bigl((u^{(1)})^5 + 10(u^{(1)})^3u^{(2)} + 14(u^{(1)})^2u^{(3)} + 15u^{(1)}(u^{(2)})^2 + 5u^{(1)}u^{(4)} + 6u^{(2)}u^{(3)} + u^{(5)}\bigr)e^u$.
That should be enough to convince you that there is not going to be any simple formula for the n-th derivative.
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# Difference between revisions of "Chicken McNugget Theorem"
The Chicken McNugget Theorem (or Postage Stamp Problem or Frobenius Coin Problem) states that for any two relatively prime positive integers $m,n$, the greatest integer that cannot be written in the form $am + bn$ for nonnegative integers $a, b$ is $mn-m-n$.
A consequence of the theorem is that there are exactly $\frac{(m - 1)(n - 1)}{2}$ positive integers which cannot be expressed in the form $am + bn$. The proof is based on the fact that in each pair of the form $(k, (m - 1)(n - 1) - k+1)$, exactly one element is expressible.
## Origins
There are many stories surrounding the origin of the Chicken McNugget theorem. However, the most popular by far remains that of the Chicken McNugget. Originally, McDonald's sold its nuggets in packs of 9 and 20. Math enthusiasts were curious to find the largest number of nuggets that could not have been bought with these packs, thus creating the Chicken McNugget Theorem (the answer worked out to be 151 nuggets). The Chicken McNugget Theorem has also been called the Frobenius Coin Problem or the Frobenius Problem, after German mathematician Ferdinand Frobenius inquired about the largest amount of currency that could not have been made with certain types of coins.
## Proof Without Words
$\begin{array}{ccccccc} 0\mod{m}&1\mod{m}&2\mod{m}&...&...&...&(m-1)\mod{m}\\ \hline \cancel{0n}&1&2&&...&&m-1\\ \cancel{0n+m}&...&&\vdots&&...&\\ \cancel{0n+2m}&...&&\cancel{1n}&&...&\\ \cancel{0n+3m}&&&\cancel{1n+m}&&\vdots&\\ \cancel{0n+4m}&&&\cancel{1n+2m}&&\cancel{2n}&\\ \cancel{0n+5m}&&&\cancel{1n+3m}&&\cancel{2n+m}&\\ \vdots&&&\vdots&&\vdots&\\ \cancel{\qquad}&\cancel{\qquad}&\cancel{ \qquad}&\cancel{ \qquad}&\mathbf{(m-1)n-m}&\cancel{\qquad }&\cancel{\qquad }\\ \cancel{\qquad}&\cancel{\qquad}&\cancel{ \qquad}&\cancel{ \qquad}&\cancel{(m-1)n}&\cancel{\qquad }&\cancel{\qquad } \end{array}$
## Proof 1
Definition. An integer $N \in \mathbb{Z}$ will be called purchasable if there exist nonnegative integers $a,b$ such that $am+bn = N$.
We would like to prove that $mn-m-n$ is the largest non-purchasable integer. We are required to show that (1) $mn-m-n$ is non-purchasable, and (2) every $N > mn-m-n$ is purchasable. Note that all purchasable integers are nonnegative, thus the set of non-purchasable integers is nonempty.
Lemma. Let $A_{N} \subset \mathbb{Z} \times \mathbb{Z}$ be the set of solutions $(x,y)$ to $xm+yn = N$. Then $A_{N} = \{(x+kn,y-km) \;:\; k \in \mathbb{Z}\}$ for any $(x,y) \in A_{N}$.
Proof: By Bezout's Lemma, there exist integers $x',y'$ such that $x'm+y'n = 1$. Then $(Nx')m+(Ny')n = N$. Hence $A_{N}$ is nonempty. It is easy to check that $(Nx'+kn,Ny'-km) \in A_{N}$ for all $k \in \mathbb{Z}$. We now prove that there are no others. Suppose $(x_{1},y_{1})$ and $(x_{2},y_{2})$ are solutions to $xm+yn=N$. Then $x_{1}m+y_{1}n = x_{2}m+y_{2}n$ implies $m(x_{1}-x_{2}) = n(y_{2}-y_{1})$. Since $m$ and $n$ are coprime and $m$ divides $n(y_{2}-y_{1})$, $m$ divides $y_{2}-y_{1}$ and $y_{2} \equiv y_{1} \pmod{m}$. Similarly $x_{2} \equiv x_{1} \pmod{n}$. Let $k_{1},k_{2}$ be integers such that $x_{2}-x_{1} = k_{1}n$ and $y_{2}-y_{1} = k_{2}m$. Then $m(-k_{1}n) = n(k_{2}m)$ implies $k_{1} = -k_{2}.$ We have the desired result. $\square$
Lemma. For any integer $N$, there exists unique $(a_{N},b_{N}) \in \mathbb{Z} \times \{0,1,\ldots,m-1\}$ such that $a_{N}m + b_{N}n = N$.
Proof: By the division algorithm, there exists one and only one $k$ such that $0 \le y-km \le m-1$. $\square$
Lemma. $N$ is purchasable if and only if $a_{N} \ge 0$.
Proof: If $a_{N} \ge 0$, then we may simply pick $(a,b) = (a_{N},b_{N})$ so $N$ is purchasable. If $a_{N} < 0$, then $a_{N}+kn < 0$ if $k \le 0$ and $b_{N}-km < 0$ if $k > 0$, hence at least one coordinate of $(a_{N}+kn,b_{N}-km)$ is negative for all $k \in \mathbb{Z}$. Thus $N$ is not purchasable. $\square$
Thus the set of non-purchasable integers is $\{xm+yn \;:\; x<0,0 \le y \le m-1\}$. We would like to find the maximum of this set. Since both $m,n$ are positive, the maximum is achieved when $x = -1$ and $y = m-1$ so that $xm+yn = (-1)m+(m-1)n = mn-m-n$.
## Proof 2
We start with this statement taken from Proof 2 of Fermat's Little Theorem:
"Let $S = \{1,2,3,\cdots, p-1\}$. Then, we claim that the set $a \cdot S$, consisting of the product of the elements of $S$ with $a$, taken modulo $p$, is simply a permutation of $S$. In other words,
$$S \equiv \{1a, 2a, \cdots, (p-1)a\} \pmod{p}.$$
Clearly none of the $ia$ for $1 \le i \le p-1$ are divisible by $p$, so it suffices to show that all of the elements in $a \cdot S$ are distinct. Suppose that $ai \equiv aj \pmod{p}$ for $i \neq j$. Since $\text{gcd}\, (a,p) = 1$, by the cancellation rule, that reduces to $i \equiv j \pmod{p}$, which is a contradiction."
Because $m$ and $n$ are coprime, we know that multiplying the residues of $m$ by $n$ simply permutes these residues. Each of these permuted residues is purchasable (using the definition from Proof 1), because, in the form $am+bn$, $a$ is $0$ and $b$ is the original residue. We now prove the following lemma.
Lemma: For any nonnegative integer $c < m$, $cn$ is the least purchasable number $\equiv cn \bmod m$.
Proof: Any number that is less than $cn$ and congruent to it $\bmod m$ can be represented in the form $cn-dm$, where $d$ is a positive integer. If this is purchasable, we can say $cn-dm=am+bn$ for some nonnegative integers $a, b$. This can be rearranged into $(a+d)m=(c-b)n$, which implies that $(a+d)$ is a multiple of $n$ (since $\gcd(m, n)=1$). We can say that $(a+d)=gn$ for some positive integer $g$, and substitute to get $gmn=(c-b)n$. Because $c < m$, $(c-b)n < mn$, and $gmn < mn$. We divide by $mn$ to get $g<1$. However, we defined $g$ to be a positive integer, and all positive integers are greater than or equal to $1$. Therefore, we have a contradiction, and $cn$ is the least purchasable number congruent to $cn \bmod m$. $\square$
This means that because $cn$ is purchasable, every number that is greater than $cn$ and congruent to it $\bmod m$ is also purchasable (because these numbers are in the form $am+bn$ where $b=c$). Another result of this Lemma is that $cn-m$ is the greatest number $\equiv cn \bmod m$ that is not purchasable. $c \leq m-1$, so $cn-m \leq (m-1)n-m=mn-m-n$, which shows that $mn-m-n$ is the greatest number in the form $cn-m$. Any number greater than this and congruent to some $cn \bmod m$ is purchasable, because that number is greater than $cn$. All numbers are congruent to some $cn$, and thus all numbers greater than $mn-m-n$ are purchasable.
Putting it all together, we can say that for any coprime $m$ and $n$, $mn-m-n$ is the greatest number not representable in the form $am + bn$ for nonnegative integers $a, b$. $\square$
## Corollary
This corollary is based off of Proof 2, so it is necessary to read that proof before this corollary. We prove the following lemma.
Lemma For any integer $k$, exactly one of the integers $k$, $mn-m-n-k$ is not purchasable.
Proof: Because every number is congruent to some residue of $m$ permuted by $n$, we can set $k \equiv cn \bmod m$ for some $c$. We can break this into two cases.
Case 1: $k \leq cn-m$. This implies that $k$ is not purchasable, and that $mn-m-n-k \geq mn-m-n-(cn-m) = n(m-1-c)$. $n(m-1-c)$ is a permuted residue, and a result of the lemma in Proof 2 was that a permuted residue is the least number congruent to itself $\bmod m$ that is purchasable. Therefore, $mn-m-n-k \equiv n(m-1-c) \bmod m$ and $mn-m-n-k \geq n(m-1-c)$, so $mn-m-n-k$ is purchasable.
Case 2: $k > cn-m$. This implies that $k$ is purchasable, and that $mn-m-n-k < mn-m-n-(cn-m) = n(m-1-c)$. Again, because $n(m-1-c)$ is the least number congruent to itself $\bmod m$ that is purchasable, and because $mn-m-n-k \equiv n(m-1-c) \bmod m$ and $mn-m-n-k < n(m-1-c)$, $mn-m-n-k$ is not purchasable.
We now limit the values of $k$ to all integers $0 \leq k \leq \frac{mn-m-n}{2}$, which limits the values of $mn-m-n-k$ to $mn-m-n \geq mn-m-n-k \geq \frac{mn-m-n}{2}$. Because $m$ and $n$ are coprime, only one of them can be a multiple of $2$. Therefore, $mn-m-n \equiv (0)(1)-0-1 \equiv -1 \equiv 1 \bmod 2$, showing that $\frac{mn-m-n}{2}$ is not an integer and that $\frac{mn-m-n-1}{2}$ and $\frac{mn-m-n+1}{2}$ are integers. We can now set limits that are equivalent to the previous on the values of $k$ and $mn-m-n-k$ so that they cover all integers form $0$ to $mn-m-n$ without overlap: $0 \leq k \leq \frac{mn-m-n-1}{2}$ and $\frac{mn-m-n+1}{2} \leq mn-m-n-k \leq mn-m-n$. There are $\frac{mn-m-n-1}{2}+1=\frac{(m-1)(n-1)}{2}$ values of $k$, and each is paired with a value of $mn-m-n-k$, so we can make $\frac{(m-1)(n-1)}{2}$ different ordered pairs of the form $(k, mn-m-n-k)$. The coordinates of these ordered pairs cover all integers from $0$ to $mn-m-n$ inclusive, and each contains exactly one not-purchasable integer, so that means that there are $\frac{(m-1)(n-1)}{2}$ different not-purchasable integers from $0$ to $mn-m-n$. All integers greater than $mn-m-n$ are purchasable, so that means there are a total of $\frac{(m-1)(n-1)}{2}$ integers $\geq 0$ that are not purchasable.
In other words, for every pair of coprime integers $m, n$, there are exactly $\frac{(m-1)(n-1)}{2}$ nonnegative integers that cannot be represented in the form $am + bn$ for nonnegative integers $a, b$. $\square$
## Generalization
If $m$ and $n$ are not relatively prime, then we can simply rearrange $am+bn$ into the form $$\gcd(m,n) \left( a\frac{m}{\gcd(m,n)}+b\frac{n}{\gcd(m,n)} \right)$$ $\frac{m}{\gcd(m,n)}$ and $\frac{n}{\gcd(m,n)}$ are relatively prime, so we apply Chicken McNugget to find a bound $$\frac{mn}{\gcd(m,n)^{2}}-\frac{m}{\gcd(m,n)}-\frac{n}{\gcd(m,n)}$$ We can simply multiply $\gcd(m,n)$ back into the bound to get $$\frac{mn}{\gcd(m,n)}-m-n=\textrm{lcm}(m, n)-m-n$$ Therefore, all multiples of $\gcd(m, n)$ greater than $\textrm{lcm}(m, n)-m-n$ are representable in the form $am+bn$ for some positive integers $a, b$.
# Problems
### Simple
• Marcy buys paint jars in containers of $2$ and $7$. What's the largest number of paint jars that Marcy can't obtain?
Answer: $5$ containers
• Bay Area Rapid food sells chicken nuggets. You can buy packages of $11$ or $7$. What is the largest integer $n$ such that there is no way to buy exactly $n$ nuggets? Can you Generalize ?(ACOPS)
Answer: $n=59$
• If a game of American Football has only scores of field goals ($3$ points) and touchdowns with the extra point ($7$ points), then what is the greatest score that cannot be the score of a team in this football game (ignoring time constraints)?
Answer: $11$ points
• The town of Hamlet has $3$ people for each horse, $4$ sheep for each cow, and $3$ ducks for each person. Which of the following could not possibly be the total number of people, horses, sheep, cows, and ducks in Hamlet?
$\textbf{(A) }41\qquad\textbf{(B) }47\qquad\textbf{(C) }59\qquad\textbf{(D) }61\qquad\textbf{(E) }66$ AMC 10B 2015 Problem 15
Answer: $47\qquad\textbf{(B) }$
### Intermediate
• Ninety-four bricks, each measuring $4''\times10''\times19'',$ are to stacked one on top of another to form a tower 94 bricks tall. Each brick can be oriented so it contributes $4''\,$ or $10''\,$ or $19''\,$ to the total height of the tower. How many different tower heights can be achieved using all ninety-four of the bricks? AIME
• Find the sum of all positive integers $n$ such that, given an unlimited supply of stamps of denominations $5,n,$ and $n+1$ cents, $91$ cents is the greatest postage that cannot be formed. AIME II 2019 Problem 14
• On the real number line, paint red all points that correspond to integers of the form $81x+100y$, where $x$ and $y$ are positive integers. Paint the remaining integer points blue. Find a point $P$ on the line such that, for every integer point $T$, the reflection of $T$ with respect to $P$ is an integer point of a different colour than $T$. (India TST)
• Let $S$ be a set of integers (not necessarily positive) such that
(a) there exist $a,b \in S$ with $\gcd(a,b)=\gcd(a-2,b-2)=1$;
(b) if $x$ and $y$ are elements of $S$ (possibly equal), then $x^2-y$ also belongs to $S$.
Prove that $S$ is the set of all integers. (USAMO)
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Chain rule
1. Apr 27, 2015
whatisreality
1. The problem statement, all variables and given/known data
Let φ=φ(r) and r=√x2+y2+z2. Find ∂2φ/∂x2.
Show that it can be written as (1/r + x2/r3)∂φ/∂r + x2/r22φ/∂r2.
2. Relevant equations
Use the identity ∂r/∂x = x/r.
3. The attempt at a solution
I think I know ∂φ/∂x. Using the chain rule, it's ∂r/∂x ∂φ/∂r. That gives x/r ∂φ/∂r. If that's wrong it might be because I know you have to take account of all dependences, but I don't actually know how to.
So assuming that's ok, I then need to use the product rule, and that gave me
1/r ∂φ/∂r + ∂φ/∂x ∂φ/∂r ∂r/∂x.
Which I know is wrong, because it's a show that question!
2. Apr 27, 2015
Staff: Mentor
I would write this as dφ/dr ∂r/∂x or φ'(r)∂r/∂x = dφ/dr (x/r). φ is a function of r alone, so the derivative for this function is the regular derivative instead of the partial derivative. Now use the product rule to get ∂2φ/∂x2.
3. Apr 29, 2015
whatisreality
I'm doing something wrong. So I got ∂2φ/∂x2 = 1/r ∂φ/∂r + ∂/dr (dφ/dr) ∂r/∂x x/r. This doesn't give me the answer, it gives
2φ/∂x2 = 1/r dφ/dr + x2/r22φ/∂r2, but I really can't see what's wrong!
4. Apr 29, 2015
whatisreality
I'm missing a whole term: x2/r3 dφ/dr.
5. Apr 29, 2015
Staff: Mentor
You don't show the work leading up to this, but I'm guessing that you did this: $\frac{\partial}{\partial x} \frac{x}{r} = \frac{1}{r}$. If so, that's wrong, since you would be treating r as a constant. In fact, r is a function of x (and y and z).
6. Apr 29, 2015
whatisreality
Ohhh... That's exactly what I did. I always forget that!
7. Apr 29, 2015
whatisreality
Should be y2+z2/r3 then.
Yep, that gives the right answer! Thank you, I was getting really confused.
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# negative path estimate in UMX cholesky
2 posts / 0 new
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Joined: 07/21/2017 - 13:13
negative path estimate in UMX cholesky
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Dear forum,
I conducted a Cholesky model with two variables in two time-points, using the uxACE command, and then conducted a model fitting procedure with the umxModify command. The model was an ADE model.
The results I got (a screenshot is attached) make perfect sense in terms of the overall explained variance by genetics. Meaning, if I compute the variance explained by both A and D for each variable (after squaring each path), I get the same amount of explained genetic variance as in the univariate models for each variable.
The only thing I am worried about is that one of the paths is negative (the path A to the third variable), even though the phenotypic correlations between all variables are positive (and meaningful).
What does it mean?
Thank you very much,
Lior
Offline
Joined: 01/24/2014 - 12:15
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An analysis of electrical impedance tomography with applications to Tikhonov regularization
ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 4, pp. 1027-1048.
This paper analyzes the continuum model/complete electrode model in the electrical impedance tomography inverse problem of determining the conductivity parameter from boundary measurements. The continuity and differentiability of the forward operator with respect to the conductivity parameter in Lp-norms are proved. These analytical results are applied to several popular regularization formulations, which incorporate a priori information of smoothness/sparsity on the inhomogeneity through Tikhonov regularization, for both linearized and nonlinear models. Some important properties, e.g., existence, stability, consistency and convergence rates, are established. This provides some theoretical justifications of their practical usage.
DOI : https://doi.org/10.1051/cocv/2011193
Classification : 49N45, 65N21
Mots clés : electrical impedance tomography, Tikhonov regularization, convergence rate
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title = {An analysis of electrical impedance tomography with applications to {Tikhonov} regularization},
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Jin, Bangti; Maass, Peter. An analysis of electrical impedance tomography with applications to Tikhonov regularization. ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 4, pp. 1027-1048. doi : 10.1051/cocv/2011193. http://www.numdam.org/articles/10.1051/cocv/2011193/
[1] G. Alessandrini, Open issues of stability for the inverse conductivity problem. Journal Inverse Ill-Posed Problems 15 (2007) 451-460. | MR 2367859 | Zbl 1221.35443
[2] K. Astala and L. Päivärinta, Calderón's inverse conductivity problem in the plane. Ann. of Math. (2) 163 (2006) 265-299. | MR 2195135 | Zbl 1111.35004
[3] K. Astala, D. Faraco, and L. Székelyhidi Jr., Convex integration and the Lp theory of elliptic equations. Ann. Scuola Norm. Super. Pisa Cl. Sci. (5) 7 (2008) 1-50. | Numdam | MR 2413671 | Zbl 1164.30014
[4] R.H. Bayford, Bioimpedance tomography (electrical impedance tomography). Ann. Rev. Biomed. Eng. 8 (2006) 63-91.
[5] T. Bonesky, K.S. Kazimierski, P. Maass, F. Schöpfer and T. Schuster, Minimization of Tikhonov functionals in Banach spaces. Abstr. Appl. Anal. (2008) 19 pages. | MR 2393115
[6] K. Bredies and D.A. Lorenz, Regularization with non-convex separable constraints. Inverse Problems 25 (2009) 085011. | MR 2529201 | Zbl 1180.65068
[7] L.M. Bregman, The relaxation method of finding the common points of convex sets and its application to the solution of problems in convex programming. USSR Comput. Math. Math. Phys. 7 (1967) 200-217. | MR 215617 | Zbl 0186.23807
[8] M. Burger and S. Osher, Convergence rates of convex variational regularization. Inverse Problems 20 (2004) 1411-1420. | MR 2109126 | Zbl 1068.65085
[9] A.-P. Calderón, On an inverse boundary value problem. In Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980). Soc. Brasil. Mat., Rio de Janeiro (1980) 65-73. | MR 590275 | Zbl 1182.35230
[10] M. Cheney, D. Isaacson, J.C. Newell, S. Simske and J. Goble, NOSER : An algorithm for solving the inverse conductivity problem. Int. J. Imag. Syst. Tech. 2 (1990) 66-75.
[11] M. Cheney, D. Isaacson and J.C. Newell, Electrical impedance tomography. SIAM Rev. 41 (1999) 85-101. | MR 1669729 | Zbl 0927.35130
[12] K.-S. Cheng, D. Isaacson, J.C. Newell and D.G. Gisser, Electrode models for electric current computed tomography. IEEE Trans. Biomed. Eng. 36 (1989) 918-924.
[13] E.T. Chung, T.F. Chan and X.-C. Tai, Electrical impedance tomography using level set representation and total variational regularization. J. Comput. Phys. 205 (2005) 357-372. | MR 2132313 | Zbl 1072.65143
[14] I. Daubechies, M. Defrise and C. De Mol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Comm. Pure Appl. Math. 57 (2004) 1413-1457. | MR 2077704 | Zbl 1077.65055
[15] T. Dierkes, O. Dorn, F. Natterer, V. Palamodov and H. Sielschott, Fréchet derivatives for some bilinear inverse problems. SIAM J. Appl. Math. 62 (2002) 2092-2113. | MR 1918308 | Zbl 1010.35115
[16] D. Dobson, Convergence of a reconstruction method for the inverse conductivity problem. SIAM J. Appl. Math. 52 (1992) 442-458. | MR 1154782 | Zbl 0747.35051
[17] D.L. Donoho, Compressed sensing. IEEE Trans. Inf. Theor. 52 (2006) 1289-1306. | MR 2241189 | Zbl 1288.94016
[18] H. Egger and M. Schlottbom, Analysis and regularization of problems in diffuse optical tomography. SIAM J. Math. Anal. 42 (2010) 1934-1948. | MR 2684305 | Zbl 1219.35355
[19] H.W. Engl, K. Kunisch and A. Neubauer, Convergence rates for Tikhonov regularisation of nonlinear ill-posed problems. Inverse Problems 5 (1989) 523-540. | MR 1009037 | Zbl 0695.65037
[20] H.W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems. Kluwer Academic, Dordrecht (1996). | MR 1408680 | Zbl 0859.65054
[21] L.C. Evans and R.F. Gariepy, Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton (1992). | MR 1158660 | Zbl 0804.28001
[22] T. Gallouet and A. Monier, On the regularity of solutions to elliptic equations. Rend. Mat. Appl. (7) 19 (1999) 471-488. | MR 1789483 | Zbl 0961.35036
[23] M. Gehre, T. Kluth, A. Lipponen, B. Jin, A. Seppänen, J. Kaipio and P. Maass, Sparsity reconstruction in electrical impedance tomography : an experimental evaluation. J. Comput. Appl. Math. (2011), in press, DOI : 10.1016/j.cam.2011.09.035. | MR 2876676 | Zbl 1251.78008
[24] M. Grasmair, M. Haltmeier and O. Scherzer, Sparse regularization with lq penalty term. Inverse Problems 24 (2008) 055020. | MR 2438955 | Zbl 1157.65033
[25] K. Gröger, A W1,p-estimate for solutions to mixed boundary value problems for second order elliptic differential equations. Math. Ann. 283 (1989) 679-687. | MR 990595 | Zbl 0646.35024
[26] B. Harrach and J.K. Seo, Exact shape-reconstruction by one-step linearization in electrical impedance tomography. SIAM J. Math. Anal. 42 (2010) 1505-1518. | MR 2679585 | Zbl 1215.35167
[27] B. Hofmann and M. Yamamoto, On the interplay of source conditions and variational inequalities for nonlinear ill-posed problems. Appl. Anal. 89 (2010) 1705-1727. | MR 2683677 | Zbl 1207.47065
[28] B. Hofmann, B. Kaltenbacher, C. Poeschl and O. Scherzer, A convergence rates result for Tikhonov regularization in Banach spaces with non-smooth operators. Inverse Problems 23 (2007) 987-1010. | MR 2329928 | Zbl 1131.65046
[29] N. Hyvönen, Complete electrode model of electrical impedance tomography : approximation properties and characterization of inclusions. SIAM J. Appl. Math. 64 (2004) 902-931. | MR 2068447 | Zbl 1059.35168
[30] M. Ikehata and S. Siltanen, Electrical impedance tomography and Mittag-Leffler's function. Inverse Problems 20 (2004) 1325-1348. | MR 2087994 | Zbl 1074.35087
[31] O.Y. Imanuvilov, G. Uhlmann and M. Yamamoto, The Calderón problem with partial data in two dimensions. J. Amer. Math. Soc. 23 (2010) 655-691. | MR 2629983 | Zbl 1201.35183
[32] D. Isaacson, J.L. Mueller, J.C. Newell and S. Siltanen, Reconstructions of chest phantoms by the D-bar method for electrical impedance tomography. IEEE Trans. Med. Imag. 23 (2004) 821-828.
[33] K. Ito, K. Kunisch and Z. Li, Level-set function approach to an inverse interface problem. Inverse Problems 17 (2001) 1225-1242. | MR 1862188 | Zbl 0986.35130
[34] K. Ito, B. Jin and T. Takeuchi, A regularization parameter for nonsmooth Tikhonov regularization. SIAM J. Sci. Comput. 33 (2011) 1415-1438. | MR 2813246 | Zbl 1235.65054
[35] B. Jin, Y. Zhao and J. Zou, Iterative parameter choice by discrepancy principle. IMA J. Numer. Anal. (2011), in press. | MR 2991843 | Zbl 1261.65052
[36] B. Jin, Y. Zhao and P. Maass, A reconstruction algorithm for electrical impedance tomography based on sparsity regularization. Internat. J. Numer. Methods Engrg. (2011), DOI : 10.2002/nme.3247. | Zbl 1242.78016
[37] J.P. Kaipio, V. Kolehmainen, E. Somersalo and M. Vauhkonen, Statistical inversion and Monte Carlo sampling methods in electrical impedance tomography. Inverse Problems 16 (2000) 1487-1522. | MR 1800606 | Zbl 1044.78513
[38] B. Kaltenbacher and B. Hofmann, Convergence rates for the iteratively regularized Gauss-Newton method in Banach spaces. Inverse Problems 26 (2010) 035007. | MR 2594377 | Zbl 1204.65060
[39] A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems. Oxford University Press, Oxford (2008). | MR 2378253 | Zbl 1222.35001
[40] K. Knudsen, M. Lassas, J.L. Mueller and S. Siltanen, Regularized D-bar method for the inverse conductivity problem. IPI 3 (2009) 599-624. | MR 2557921 | Zbl 1184.35314
[41] V. Kolehmain, A. Voutilainen and J.P. Kaipio, Estimation of nonstionary region boundaries in EIT-state estimation approach. Inverse Problems 17 (2001) 1937-1956. | MR 1872930 | Zbl 0991.35109
[42] A. Lechleiter, A regularization technique for the factorization method. Inverse Problems 22 (2006) 1605-1625. | MR 2261257 | Zbl 1106.35136
[43] A. Lechleiter and A. Rieder, Newton regularizations for impedance tomography : a numerical study. Inverse Problems 22 (2006) 1967-1987. | MR 2277524 | Zbl 1109.65100
[44] A. Lechleiter and A. Rieder, Newton regularizations for impedance tomography : convergence by local injectivity. Inverse Problems, 24 (2008) 065009. | MR 2456956 | Zbl 1152.35516
[45] W.R.B. Lionheart, EIT reconstruction algorithms : pitfalls, challenges and recent developments. Physiol. Meas. 25 (2004) 125-142.
[46] D.A. Lorenz, Convergence rates and source conditions for Tikhonov regularization with sparsity constraints. Journal Inverse Ill-Posed Problems 16 (2008) 463-478. | MR 2442066 | Zbl 1161.65041
[47] M. Lukaschewitsch, P. Maass and M. Pidcock, Tikhonov regularization for electrical impedance tomography on unbounded domains. Inverse Problems 19 (2003) 585-610. | MR 1984879 | Zbl 1034.65043
[48] N.G. Meyers, An Lp-estimate for the gradient of solutions of second order elliptic divergence equations. Ann. Scuola Norm. Sup. Pisa (3) 17 (1963) 189-206. | Numdam | MR 159110 | Zbl 0127.31904
[49] A. Neubauer, When do Sobolev spaces form a Hilbert scale? Proc. Amer. Math. Soc. 103 (1988) 557-562. | MR 943084 | Zbl 0665.46029
[50] E. Resmerita, Regularization of ill-posed problems in Banach spaces : convergence rates. Inverse Problems 21 (2005) 1303-1314. | MR 2158110 | Zbl 1082.65055
[51] L. Rondi, On the regularization of the inverse conductivity problem with discontinuous conductivities. IPI 2 (2008) 397-409. | MR 2424823 | Zbl 1180.35572
[52] L. Rondi and F. Santosa, Enhanced electrical impedance tomography via the Mumford-Shah functional. ESAIM Control Optim. Calc. Var. 6 (2001) 517-538. | Numdam | MR 1849414 | Zbl 0989.35136
[53] E. Somersalo, M. Cheney and D. Isaacson, Existence and uniqueness for electrode models for electric current computed tomography. SIAM J. Appl. Math. 52 (1992) 1023-1040. | MR 1174044 | Zbl 0759.35055
[54] A.N. Tikhonov and V.Y. Arsenin, Solutions of Ill-Posed Problems. John Wiley, New York (1977). | MR 455365 | Zbl 0354.65028
[55] G. Uhlmann, Commentary on Calderón's paper (29), on an inverse boundary value problem, in Selected papers of Alberto P. Calderón. Amer. Math. Soc., Providence, RI (2008) 623-636. | MR 2435340 | Zbl 1140.01021
[56] A. Wexler, B. Fry and M.R. Neuman, Impedance-computed tomography algorithm and system. Appl. Opt. 24 (1985) 3985-3992.
[57] T.J. Yorkey, J.G. Webster and W.J. Tompkins, Comparing reconstruction algorithms for electrical impedance tomography. IEEE Trans. Biomed. Eng. 34 (1987) 843-852.
Cité par Sources :
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# Oxidation states of boron
I have just been looking at https://en.wikipedia.org/wiki/List_of_oxidation_states_of_the_elements and found that boron has a -5 oxidation state. I would like to know which boron compounds form this oxidation state.
As I remember I have read somewhere that $$\ce{Al3BC}$$ is konwn to have Boron in -5 oxidation state.
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This post is dedicated to Ivan Towlson. Ivan taught me FParsec; and has both put up with, and instigated, a stream of incorrigble puns every time we converse. Thanks, Ivan! It’s been a genuine pleasure working with you!
## Introduction
The Travelling Salesman Problem (TSP) is one of the most widely studied NP-complete problems. There are many heuristic solutions in the industry and even a regular competition to evaluate the performance and efficiency of heuristic solutions.
The canonical source of problem sets is available from the University of Heidelberg. There are several datasets and multiple variants of the problem represented here.
This post will discuss the development of a Parser to be able to read and process the problem set data, so that we can then work with known data and potentially compare our performance and correctness with known results.
## Problem Structure
The TSP can be thought of as a graph traversal problem. Specifically, given a positively weighted graph, solving the TSP is equivalent to finding a Hamiltonian Cycle in that graph.
Practically, one usually encounters a complete graph (i.e one where every node is connected to every other node), and generally speaking, one is presented with a symmetric graph where the distance from A to B is the same as the distance between B and A.
At the outset, one might feel uncomfortable that this set of constraints doesn’t represent reality - after all, one-way roads do exist, and generally a real city is only directly connected to a few other cities. We will note this discomfort and model the problem in such a way that the more general solution for incomplete, asymmetric graphs is also addressed by our solution.
Consequently, to represent the problem, one must first constructed a weighted graph, where the nodes represent cities and the weights of the edges represent the distance between them. If one implements this graph as an adjacency matrix, we can immediately see that we can represent disconnected cities (cities without a direct connection between them) as having an edge with infinite weight; and asymmetric edge weights are easily supported by storing data in a full matrix with different distances in each direction.
If one numbers the nodes, then a TSP tour would be some sequence of numbers representing the cities in the order of traversal. That is, a permutation of the sequence 1, 2, 3, ... n would represent a tour of the set of cities.
It is then easy to compute the total distance of the tour by summing together successive distances. For example, a sequence of 5, 3, 4, 2, 1 would have a total tour distance of the sum of distances between 5 and 3, 3 and 4, 4 and 2, 2 and 1, and 1 and 5. If there were no connection between 4 and 2, that edge would have infinite weight, and the tour would have infinite distance - indicating that this sequence does not constitute a valid tour of the five cities.
A typical data structure for this might be:
type WeightedCompleteGraph (dimension : Dimension) = class
let d = dimension.unapply
let mutable fullMatrix : Weight[,] = Array2D.init d d (fun _ _ -> Weight.Infinity)
end
Note that we have elided support types such as Dimension and Weight here, and have made the array mutable because we may have to do significant computing before we populate the array with weights.
## Input Data
We now turn our attention to the structure of the input data specified in the TSPLib repository. Whilst there are several libraries on Github that purport to read problem instances from the TSPLib repository, one realises that they are largely focussed on one particular variant of the problem.
In contrast, the documentation comprehensively describes a wide variety of data formats encoded in the files. A given TSP data file can describe six varieties of data; edge weights as one of thirteen types, edge weight data in one of ten block formats, and nodes as having two, three, or no coordinates with nine possible distance calculation formulae if the distances aren’t specified explicitly!
A functional programmer would immediately recognize the documentation as an informally specified domain-specific-language (DSL), and recognize each data file as a document conforming to the DSL. This naturally implies that in order to read in the data and make sense of it, one must develop a parser which can build a data structure containing the data consumed and transformed.
In this post, I’ll walk you through how to build a simple parser in FParsec.
## Writing A Parser
We will be writing a parser using a “parser combinator” library called FParsec. You can think of a parser as a kind of function that recognizes a text pattern, and a parser combinator as a function that glues two or more parsers into a parser that recognizes bigger patterns.
As we start writing parsers, we also want data structures to store the information we’ve parsed - the patterns recognized - so we can transform the textual input of a parser into structured data. In F#, we have sum and product types (or union and record types) to define the data structure type, and we will basically transform textual input into an instance of one of our types - or die trying!
This data structure is traditionally called the Abstract Syntax Tree (AST), because one of the common uses of a parser is to parse the text of a program into something that conforms to a language syntax before being processed semantically.
We’ll walk through the development of parsers for some of the basic patterns of data, and you can peruse the complete source code, examples, more examples, or the documentation for further information.
## Converting Specifications To Data
The Specifications section of the TSPLib documentation starts with:
From this, and from reading the rest of the document, we note a few things:
1. The section headers are of the form <keyword> : <value>
2. The <value> portion is typed, and can be string, integer or real
3. Keywords can appear in any order
4. Some values are constrained to be one of an enumeration, such as the “Type” keyword
5. Some keywords are optional, and if not provided, may have default values
6. Semantically, if some keywords are specified, others maybe required or precluded
The last two points are worthy of note, because we will have to address them in the semantic verification stage. It will not be apparent at the syntax parsing stage if these are satisfied.
### AST Types
Let’s start with a parser for the “Name” section as it is the simplest.
When we parse this section, we will need to store the provided string value as the Name property in our data structure, so define a Name type for the AST:
type Name = | Name of string
with
static member inline apply = Name
member inline this.unapply = match this with | Name x -> x
The apply and unapply functions may look a little strange, but they are there for a reason - we will find them useful when we use a parser which extracts multiple values from a pattern.
### Parser Structure
Then we write the Name parser. We basically need to successfully parse the following pattern:
• (whitespace)NAME(whitespace):(whitespace)<value>(whitespace-except-end-of-line)(end-of-line)
The NAME and : are literal patterns which we need to match, but which don’t have any real other importance. The <value> bit is the string we need to capture.
### Dealing with spaces and end-of-line separately
The tricky bit is that the built-in spaces parser will greedily process the end-of-line as well, so we need to use a different way of knowing when we’re at the end of the line.
The following spacesExceptNewline parser skips as many tab characters or spaces and just ignores them.
let spacesExceptNewline : Parser<unit, unit> =
(skipChar '\t' <|> skipChar ' ') |> many |>> ignore
### Combining Parsers
We don’t get very far with the built-in set of primitive parsers. The power of parser combinators comes from being able to compose more and more complex parsers with combinators, to being able to parse more and more complex patterns.
The ws_x_nl parser combines a given parser p with the spacesExceptNewline parser on either side, and preserves the output of the p parser. The ws parser does the same, wrapping p with parsers that snarf all whitespace from either side of it.
let ws_x_nl p = spacesExceptNewline >>. p .>> spacesExceptNewline
let ws p = spaces >>. p .>> spaces
The >>. combinator combines two parsers and keeps the output of the right parser (the . is to the right of the >>); The .>> combinator combines two parsers and keeps the output of the left one. The key is to see where the . shows up in the combinator. When we encounter the .>>., we’ll see the usefulness of the .apply pattern mentioned above.
Once we write the parser and refactor out the reusable bits, we can write the specificationName parser as:
let tillEndOfLine : Parser<string, unit> =
manyCharsTill anyChar newline
let parseSpecification prefix parser =
ws (skipStringCI prefix) >>. ws (skipChar ':') >>. parser .>> opt newline
let specificationName : Parser<ProblemComponent, unit> =
parseSpecification "NAME" tillEndOfLine
|>> Name.apply
|>> ProblemComponent.SpecificationName
<!> "NAME"
The |>> combinator creates a parser which effectively feeds the output of a parser on its left into a function on its right. This is generally how one creates a parser that returns an instance of the AST type.
Just ignore the ProblemComponent type for now.
### Reusing Parsers
It’s instructive to notice that we have already written the bits for any simple-valued key-value specification. Using the same definitions we have had so far, and appropriate ASTs, we can build parsers for the Comment string value; and the Dimension and Capacity integer values.
let specificationComment : Parser<ProblemComponent, unit> =
parseSpecification "COMMENT" tillEndOfLine
|>> Comment.apply
|>> ProblemComponent.SpecificationComment
<!> "COMMENT"
let specificationDimension : Parser<ProblemComponent, unit> =
parseSpecification "DIMENSION" (pint32 .>> newline)
|>> Dimension.apply
|>> ProblemComponent.SpecificationDimension
<!> "DIMENSION"
let specificationCapacity : Parser<ProblemComponent, unit> =
parseSpecification "CAPACITY" (pint32 .>> newline)
|>> CVRPCapacity.apply
|>> ProblemComponent.SpecificationCVRPCapacity
<!> "CAPACITY"
### Debugging Parsers
Pay attention to the <!> combinator. We define this special utility combinator as follows:
let (<!>) (p: Parser<_,_>) label : Parser<_,_> =
#if true && PARSER_TRACE
fun stream ->
printfn $"{stream.Position}: Entering {label}" let reply = p stream printfn$"{stream.Position}: Leaving {label} ({reply.Status})"
#else
p <?> label
#endif
This is how we debug parsers. This utility parser can be made to emit a trace of which parser encountered what, and whether it successfull matched a pattern or not. A less-verbose implementation would be to simply tag a parser with the built-in <?> combinator.
### Constraining values to an enumeration
The parser needs to become a little smarter when we consider the Type keyword. The string value can’t be any string, but one of the specified choices. F#’s union types simplifies this for us. We will create a union type with the appropriate tags as follows:
type ProblemType =
| TSP
| ATSP
| SOP
| HCP
| CVRP
| TOUR
We then write a parser for each value as follows:
let specificationType =
choice
[
wstr "TSP" >>. preturn TSP
wstr "ATSP" >>. preturn ATSP
wstr "SOP" >>. preturn SOP
wstr "HCP" >>. preturn HCP
wstr "CVRP" >>. preturn CVRP
]
|> parseSpecification "TYPE"
|>> ProblemComponent.SpecificationType
<!> "TYPE"
The following points will help you make sense of what’s going on
• The wstr parser matches a literal string
• The preturn parser simply returns the corresponding union type tag
• The >>. combinator pairs the string parser with its return value, ignoring the matched string and preserving the tag
• The choice combinator matches any one of the provided parsers with the value, and fails if none matches
• The parseSpecification combinator wraps each of the parsers with the appropriate space and colon handling parsers and matches the TYPE keyword
The “Data Part” of the TSP file is a group of Sections, which have rather a different format to the Specifications, and is quite complex because there may be multiple formats to read in each section itself:
We follow much the same approach as with a specification, except because data are provided on a variable number of lines, we grab lists of data until we read a line with a delimiter on it.
For example, considering the “Node Coord Section” specified above, we define the following types and parsers:
type NodeId = private | NodeId of int32
with
static member directApply x = NodeId x
static member zeroBasedApply x = NodeId (x + 1)
member this.zeroBasedUnapply = match this with | NodeId x -> (x - 1)
type Coordinate = | Coordinate of float
with
member inline this.unapply = match this with Coordinate x -> x
type NodeCoord2D = { NodeId : NodeId; X : Coordinate; Y : Coordinate }
with
static member apply ((nodeId, x), y) = { NodeId = nodeId; X = x; Y = y }
and NodeCoord3D = { NodeId : NodeId; X : Coordinate; Y : Coordinate; Z : Coordinate}
with
static member apply (((nodeId, x), y), z) = { NodeId = nodeId; X = x; Y = y; Z = z }
and NodeCoordSection =
| Nodes2D of NodeCoord2D list
| Nodes3D of NodeCoord3D list
let NodeId =
ws_x_nl pint32
|>> NodeId.directApply
<!> "NodeId"
let Coordinate =
ws_x_nl pfloat <|> ws_x_nl (pint32 |>> float)
|>> Coordinate.Coordinate
<!> "Coordinate"
let NodeCoord2D =
NodeId .>>. Coordinate .>>. Coordinate .>> newline
|>> NodeCoord2D.apply
<!> "Coord2D"
let NodeCoord3D =
NodeId .>>. Coordinate .>>. Coordinate .>>. Coordinate .>> newline
|>> NodeCoord3D.apply
<!> "Coord3D"
let sectionNodeCoord : Parser<ProblemComponent, unit> =
let Nodes2D = many NodeCoord2D |>> NodeCoordSection.Nodes2D
let Nodes3D = many NodeCoord3D |>> NodeCoordSection.Nodes3D
wstr "NODE_COORD_SECTION" >>. (Nodes2D <|> Nodes3D)
|>> ProblemComponent.SectionNodeCoords
<!> "NODE_COORD_SECTION"
Almost all the parser combinators encountered here have been encountered before.
The .>>. combinator combines the parser to its left with the parser to its right, and constructs a tuple with the output of both as its own output. Repeatedly using the .>>. combinator will result in a set of nested 2-tuples. It is handy to have the .apply pattern to be able to consume this convoluted form, as seen above.
### Strong-Typing for Correct Semantics
The other thing to note is that the TSP problem domain numbers cities in a 1-based numbering scheme; It is often useful in code to use a 0- based index. To keep things clean and not introduce “off-by-one” errors, we can use strong-typing to indicate that any value of type NodeId is implicitly 1- based; and further, because we don’t support mathematical operations on NodeId, we cannot do nonsensical things like add two city indices together! Strong-typing turns out to be very useful here to aid in the correctness and maintainability of our program.
Of course, the data being parsed will all be 1- based, so we have a special constructor function for use in the parser, and different constructor and decomposer functions for use in other contexts.
### Matching keywords in any order
This turns out to be a tricky requirement, specially because the keywords may arrive in any order but some keywords may be affected by the presence or absence of other keywords.
We will solve this by blindly parsing all the key-value pairs into a single list; build up a data structure with the individual items; and then semantically validate the resulting data structure.
Of course, creating a list of components means that every parser needs to generate the same type. So we create another union type representing problem components.
type ProblemComponent =
| SpecificationName of Name
| SpecificationComment of Comment
| SpecificationType of ProblemType
| SpecificationDimension of Dimension
| SpecificationCVRPCapacity of CVRPCapacity
| SpecificationEdgeWeightType of EdgeWeightType
| SpecificationEdgeWeightFormat of EdgeWeightFormat
| SpecificationEdgeDataFormat of EdgeDataFormat
| SpecificationNodeCoordType of NodeCoordType
| SectionNodeCoords of NodeCoordSection
| SectionDepots of DepotsSection
| SectionDemands of DemandsSection
| SectionEdgeData of EdgeDataSection
| SectionFixedEdges of FixedEdgesSection
| SectionDisplayData of DisplayDataSection
| SectionEdgeWeights of EdgeWeightSection
| SectionEof of EofSection
This is why the specificationName parser above, for example, eventually called the |>> ProblemComponent.SpecificationName combinator.
Now, we have to read the whole file, and try to get each parser to match its corresponding section.
let parseProblem =
[
specificationName
specificationType
specificationComment
specificationDimension
specificationCapacity
specificationEdgeWeightType
specificationEdgeWeightFormat
specificationEdgeDataFormat
specificationNodeCoordType
sectionNodeCoord
sectionDepots
sectionDemands
sectionEdgeData
sectionFixedEdges
sectionDisplayData
sectionTours
sectionEdgeWeight
sectionEof
]
|> Seq.map attempt
|> choice
|> many
|>> ProblemComponent.coalesce
<!> "Problem"
The attempt combinator tries to apply a parser, and considers it a non-fatal error if the parser fails because we’re looking at a different section at that time. The choice combinator, also seen before as <|>, picks combinators one at a time until one succeeds. The many combinator repeats until as many matches as possible are found.
At this point, we have constructed a single complex parser combinator which consumes a full document and gives us a list of ProblemComponents. We’ll transform that list into a single record - the full AST type for the TSP Problem - with an elegant little fold that takes an “empty” Problem as the seed, and successively fills out a matching field for each value it encounters in the list.
type Problem =
{
Name : Name option
Comment : Comment option
Type : ProblemType option
Dimension : Dimension option
EdgeWeightType : EdgeWeightType option
CVRPCapacity : CVRPCapacity option
EdgeWeightFormat : EdgeWeightFormat option
EdgeDataFormat : EdgeDataFormat option
NodeCoordType : NodeCoordType option
NodeCoordSection : NodeCoordSection option
DepotsSection : DepotsSection option
DemandsSection : DemandsSection option
EdgeDataSection : EdgeDataSection option
FixedEdgesSection : FixedEdgesSection option
DisplayDataSection : DisplayDataSection option
EdgeWeightSection : EdgeWeightSection option
EofSection : EofSection option
}
with
static member coalesce =
List.fold
(
fun (problem : Problem) -> function
| SpecificationName c -> { problem with Name = Some c }
| SpecificationComment c -> { problem with Comment = Some c }
| SpecificationType c -> { problem with Type = Some c }
| SpecificationDimension c -> { problem with Dimension = Some c }
| SpecificationCVRPCapacity c -> { problem with CVRPCapacity = Some c }
| SpecificationEdgeWeightType c -> { problem with EdgeWeightType = Some c }
| SpecificationEdgeWeightFormat c -> { problem with EdgeWeightFormat = Some c }
| SpecificationEdgeDataFormat c -> { problem with EdgeDataFormat = Some c }
| SpecificationNodeCoordType c -> { problem with NodeCoordType = Some c }
| SpecificationDataDisplayType c -> { problem with DataDisplayType = Some c }
| SectionNodeCoords c -> { problem with NodeCoordSection = Some c }
| SectionDepots c -> { problem with DepotsSection = Some c }
| SectionDemands c -> { problem with DemandsSection = Some c }
| SectionEdgeData c -> { problem with EdgeDataSection = Some c }
| SectionFixedEdges c -> { problem with FixedEdgesSection = Some c }
| SectionDisplayData c -> { problem with DisplayDataSection = Some c }
| SectionTours c -> { problem with ToursSection = Some c }
| SectionEdgeWeights c -> { problem with EdgeWeightSection = Some c }
| SectionEof c -> { problem with EofSection = Some c }
)
Problem.zero
Note that each of the members in the Problem AST is an option because we’re going to construct this record incrementally from a list, and because items may show up in any order.
There is still no guarantee that this structure represents a valid or consistent problem, but now we can run validations on the whole structure instead of having to deduce correctness one section at a time.
### Tying it all together
We started this post with the WeightedCompleteGraph data structure, because that gave us the adjacency matrix we could use for our distance calculations.
And this is where the fun begins, because in some formats, like the “Node Coord” example above, the distances need to be computed from the coordinates provided, so we may have to do some significant computation based on the problem input to get to a standard graph that we can examine for Hamiltonian Cycles.
Let’s stick with the “Node Coord” example, and write the transformation for 2d coordinates:
First, we’ll augment the NodeCoord2D type with the specified mathematics for computing coordinate distances:
type NodeCoord2D = { NodeId : NodeId; X : Coordinate; Y : Coordinate }
with
static member apply ((nodeId, x), y) = { NodeId = nodeId; X = x; Y = y }
static member private L2 (i : NodeCoord2D, j : NodeCoord2D) =
let xd = i.X - j.X
let yd = i.Y - j.Y
(xd * xd) + (yd * yd)
|> sqrt
static member EuclideanDistance =
NodeCoord2D.L2 >> int32
static member CeilingEuclideanDistance =
NodeCoord2D.L2 >> ceil >> int32
static member PseudoEuclideanDistance (i : NodeCoord2D, j : NodeCoord2D) =
let xd = i.X - j.X
let yd = i.Y - j.Y
let rij = sqrt ((xd * xd) + (yd * yd) / 10.0)
let tij = floor rij
int32 (if tij < rij then tij + 1.0 else tij)
static member ManhattanDistance (i : NodeCoord2D, j : NodeCoord2D) =
let xd = i.X - j.X |> abs
let yd = i.Y - j.Y |> abs
xd + yd
|> int32
static member MaximumDistance (i : NodeCoord2D, j : NodeCoord2D) =
let xd = i.X - j.X |> abs |> int32
let yd = i.Y - j.Y |> abs |> int32
max xd yd
static member GeographicalDistance (i : NodeCoord2D, j : NodeCoord2D) =
let RRR = 6378.388
let toRadians (f : float) =
let deg = floor f
let min = f - deg
System.Math.PI * (deg + 5.0 * min / 3.0) / 180.0
let q1 = long_i - long_j |> cos
let q2 = lat_i - lat_j |> cos
let q3 = lat_i + lat_j |> cos
let f = 0.5*((1.0+q1)*q2 - (1.0-q1)*q3) |> acos
RRR * f + 1.0
|> int32
Then, given the coordinates for each city, we will generate the matrix with the distances between all cities:
let toWeightedCompleteGraph (problem : Problem) : WeightedCompleteGraph option =
let computeEdgeWeights (f2, f3) (problem : Problem) (g : WeightedCompleteGraph) : WeightedCompleteGraph option =
let dim = g.Dimension.unapply
let computeEdgeWeight (weight_func : ('n * 'n -> int) option) (unapply : 'n -> int) (nodes : 'n list) =
option {
let! f = weight_func
let m = nodes |> List.map (fun n -> (unapply n, n)) |> Map.ofList
for p1 in 0 .. (dim - 1) do
for p2 in 0 .. (dim - 1) do
let d = f (m[p1], m[p2])
g[NodeId.zeroBasedApply p2, NodeId.zeroBasedApply p1] <- Weight d
return g
}
problem.NodeCoordSection
|> Option.bind (function
| NodeCoordSection.Nodes2D nodes -> computeEdgeWeight f2 (fun (n : NodeCoord2D) -> n.NodeId.zeroBasedUnapply) nodes
| NodeCoordSection.Nodes3D nodes -> computeEdgeWeight f3 (fun (n : NodeCoord3D) -> n.NodeId.zeroBasedUnapply) nodes)
option {
let! name = problem.Name
let! comment = problem.Comment
let! dimension = problem.Dimension
let! problemType = problem.Type
let! edgeWeightType = problem.EdgeWeightType
let g = WeightedCompleteGraph (name, problemType, comment, dimension)
return!
match edgeWeightType with
| EUC_2D -> computeEdgeWeights (Some NodeCoord2D.EuclideanDistance, Some NodeCoord3D.EuclideanDistance) problem g
| EUC_3D -> computeEdgeWeights (Some NodeCoord2D.EuclideanDistance, Some NodeCoord3D.EuclideanDistance) problem g
| MAX_2D -> computeEdgeWeights (Some NodeCoord2D.MaximumDistance, Some NodeCoord3D.MaximumDistance) problem g
| MAX_3D -> computeEdgeWeights (Some NodeCoord2D.MaximumDistance, Some NodeCoord3D.MaximumDistance) problem g
| MAN_2D -> computeEdgeWeights (Some NodeCoord2D.ManhattanDistance, Some NodeCoord3D.ManhattanDistance) problem g
| MAN_3D -> computeEdgeWeights (Some NodeCoord2D.ManhattanDistance, Some NodeCoord3D.ManhattanDistance) problem g
| CEIL_2D -> computeEdgeWeights (Some NodeCoord2D.CeilingEuclideanDistance, None) problem g
| GEO -> computeEdgeWeights (Some NodeCoord2D.GeographicalDistance, None) problem g
| ATT -> computeEdgeWeights (Some NodeCoord2D.PseudoEuclideanDistance, None) problem g
// other edge weight types elided
}
let parseTextToWeightedCompleteGraph str =
match run parseWeightedCompleteGraph str with
| Success(result, _, _) -> result
| Failure(err, _, _) -> sprintf "Failure:%s[%s]" str err |> failwith
The following points are noteworthy here:
This is an example where using monadic composition vastly simplifies the code.
We are going through and constructing a WeightedCompleteGraph from a Problem. If you recall, every property of Problem is set up to be an option because we have to construct it incrementally from a list, and there are legitimate cases where a property may not have a value. But as we copy over the properties now, we have to write code that fails if one of the mandatory properties does not exist. The cleanest way to do this is to use the CE pattern to monadically compose the assignments.
#### Functional abstraction
We have multiple ways of computing distances depending on how the data need to be interpreted. But applying those computations can be done uniformly by abstracting at the right level. In this case, we have a single function that does the iteration over the city-pairs, and we pass in the distance computing functions as arguments to that function, so we can have a terse and clean implementation of computing the matrix
#### Off-by-one errors begone
By lifting the city identifier to its own type NodeId and not treating it like a raw integer, we can securely and correctly lift the 0- based iteration index into the 1- based city index without causing any head-scratching for maintainers that look at this code next year. After all, adding 1 to a city index really doesn’t make any sense!
#### Handle 2D and 3D cases uniformly
Organizing this code in this fashion even allows us to delegate the distance computation to the coordinate type and judiciously using the Option type, we can reuse the same iteration driver function computeEdgeWeight to handle cases for which comptutations are specified for both 2D and 3D coordinates, and cases where computations are only specified for 2D coordinates (see the GEO case)
### Driving the Parser
Once the parser that generates the WeightedCompleteGraph is complete, we need to write a top level function that parses a given file into a WeightedCompleteGraph. This becomes the “API” of the parser, if you like.
Here’s a suitable driver function which parses a string to a WeightedCompleteGraph, and a test snippet which runs that on an actual data file from TSPLib and prints the parsed result.
And we’re done!
let parseTextToWeightedCompleteGraph str =
match run parseWeightedCompleteGraph str with
| Success(result, _, _) -> result
| Failure(err, _, _) -> sprintf "Failure:%s[%s]" str err |> failwith
let dataRoot = @"C:\code\TSP\tspsolver\data\tsplib95"
let input =
[| dataRoot; "att48.tsp" |]
|> System.IO.Path.Combine
parseTextToWeightedCompleteGraph input
|> printfn "%O"
### Conclusion
• We started with the problem of trying to read data published in a loosely-specified domain-specific-language, so that we could represent it in a manner that allows us to solve the TSP.
• We then approached the problem functionally with parser-combinators, and developed incrementally complex parsers to allow us to process and consume the complete variety of input formats. We allowed for optional sections, and sections in any order. We included mechanisms for tracing the parser to aid in debugging.
• We then applied the specified computations to transform some formats of data where only coordinates were specified into appropriate edge-weights for a complete graph. We did this in a manner that would safely fail if the data were missing important details (i.e. if the specified data were inconsistent).
• We developed this incrementally. Practically, this means writing small test snippets and exercising the parsers at the time of development using FSI. One of the consequences of strong typing in this specific case is that parsing failures are dealt with at compile time, not discovered at run-time as with some other languages.
• We did all of this in a few hundred lines of code.
The primary thesis of this post is that even mundane tasks such as parsing and processing input data can be done tersely, elegantly and safely in F#.
The secondary use of this might be as a useful practical reference of how to write a non-trivial parser.
Next, we’ll look at Genetic Algorithms and how to write those in F#.
Keep typing! :)
|
Science Forums
# Personal Topic
## Recommended Posts
Like I said, you talk so much arse!
You think that SR was presented and accepted before it was known that the speed of light is unaffected by the relative velocity of the emitter? Lol!
You said he never said that yet you just admitted it.
you're not thinking properly
What the fcuk is wrong with you?
You said that SR is what showed that the speed of light is constant.
Albert Einstein, in his theory of special relativity, determined that the laws of physics are the same for all non-accelerating observers, and he showed that the speed of light within a vacuum is the same no matter the speed at which an observer travels.
So I pointed out that SR does NOT show that the the speed of light is constant. It describes what happens IF it is constant, something that was obviously already known prior to the formulation of SR. If it wasn't known then there would have been no need for SR.
What's your point? You said "... and he showed that the speed of light within a vacuum is the same no matter the speed at which an observer travels." That's wrong, there would have been no need for SR if it hadn't already been shown that the speed of light is constant in all inertial reference frames. The consistency of the speed of light is a postulate of SR, SR does nothing to validate that postulate other than to show that it actually does work.
Edited by A-wal
• Replies 1k
• Created
#### Popular Posts
Wow! So does your boat travel close to the speed of light? :) It's a speed boat at c.
Apologies for posting this after OceanBreeze has answered so well but I’d already written it and I need clarification on the Doppler Red shift equation. Special relativity means moving clocks run slow
I suspect you'll like this part, Marco:
#### Posted Images
Apologies for posting this after OceanBreeze has answered so well but I’d already written it and I need clarification on the Doppler Red shift equation.
Special relativity means moving clocks run slow. Alice’s clock is moving in respect to Bob’s clock on Earth. The time dilation equation is:
∆t0 = ∆t√ 1 – v2/c2
∆t0 = Alice’s spacecraft time interval (in light years for this example)
∆t = Bob’s Earth based time interval in light years = 3/0.6 = 5
v = spacecraft velocity 0.6c
c = speed of light (2.99792458x108 m/s) (3.0x108) rounded (Cancels out in this example)
∆t0 = ∆t√1 – v2/c2 ∆t0 = 5√1 – (0.6)2 ∆ t0 = 5√1 – 0.36 ∆ t0 = 5√0.64
∆ t0 = 5 x 0.8 = 4 years (0.8 from the equation is Alice’s rate of time due to time dilation, this is also of course ∆t0 /∆t which is 4years/5years in this case = 0.8 for both legs).
Alice would be 4 years older at the end of each leg, while Bob on Earth would be 5 years older at the end of each leg. So Alice would be a total of 8 years and Bob 10years older on Alice’s return to Earth.
I used the Doppler Red shift reference: http://mb-soft.com/public/reltvty1.html
It gave me different equations which I’m not sure of, so I’m confused. My results from using those equations: Doppler Red shift 1 + 0.32 (1.32 octave) on the outward leg, and 1 - 0.32 (0.68 Octave) on the inbound leg,
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I have not said anything to indicate that the age difference "jumps" anywhere.
But if you use instant acceleration just to keep it simple then there will always be an 'instant jump'. Acceleration is always smooth in reality though so there's never an actual jump in practice.
I used the Doppler Red shift reference: http://mb-soft.com/public/reltvty1.html
It gave me different equations which I’m not sure of, so I’m confused. My results from using those equations: Doppler Red shift 1 + 0.32 (1.32 octave) on the outward leg, and 1 - 0.32 (0.68 Octave) on the inbound leg,
In the .6c example being used both Ralfcis and OceanBreeze said that the Doppler shift would be .5 for the outbound journey and 2 for the inbound journey but surely those Doppler shifts are for .5c, not .6c?
I never use Doppler shift because it's non-relativistic and it always cancels out back to zero in the end anyway.
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Apologies for posting this after OceanBreeze has answered so well but I’d already written it and I need clarification on the Doppler Red shift equation.
Special relativity means moving clocks run slow. Alice’s clock is moving in respect to Bob’s clock on Earth. The time dilation equation is:
∆t0 = ∆t√ 1 – v2/c2
∆t0 = Alice’s spacecraft time interval (in light years for this example)
∆t = Bob’s Earth based time interval in light years = 3/0.6 = 5
v = spacecraft velocity 0.6c
c = speed of light (2.99792458x108 m/s) (3.0x108) rounded (Cancels out in this example)
∆t0 = ∆t√1 – v2/c2 ∆t0 = 5√1 – (0.6)2 ∆ t0 = 5√1 – 0.36 ∆ t0 = 5√0.64
∆ t0 = 5 x 0.8 = 4 years (0.8 from the equation is Alice’s rate of time due to time dilation, this is also of course ∆t0 /∆t which is 4years/5years in this case = 0.8 for both legs).
Alice would be 4 years older at the end of each leg, while Bob on Earth would be 5 years older at the end of each leg. So Alice would be a total of 8 years and Bob 10years older on Alice’s return to Earth.
I used the Doppler Red shift reference: http://mb-soft.com/public/reltvty1.html
It gave me different equations which I’m not sure of, so I’m confused. My results from using those equations: Doppler Red shift 1 + 0.32 (1.32 octave) on the outward leg, and 1 - 0.32 (0.68 Octave) on the inbound leg,
Nothing to apologize for, good post!
If you go back to your link you will see they are calculating the relativistic Doppler redshift in octaves.
One octave is a doubling or a halving in frequency.
They even have a worked problem at 0.6c and the redshift is one octave which is the same as I gave as a doubling in frequency, or 2 in my terminology. I don't know why they are sticking to octaves when talking about light, but that is what they are doing.
In any case, there is no discrepancy.
I hope that clears up the confusion.
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But if you use instant acceleration just to keep it simple then there will always be an 'instant jump'. Acceleration is always smooth in reality though so there's never an actual jump in practice.
In the .6c example being used both Ralfcis and OceanBreeze said that the Doppler shift would be .5 for the outbound journey and 2 for the inbound journey but surely those Doppler shifts are for .5c, not .6c?
I never use Doppler shift because it's non-relativistic and it always cancels out back to zero in the end anyway.
I am not using instant acceleration so there is no sudden jump in age.
Those redshifts are for 0.6c just do the math. I gave the equation
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Those redshifts are for 0.6c just do the math. I gave the equation
I'm not arguing, I just would have thought those were the Doppler shifts for .5c. I don't get why they're for .6c.
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I'm not arguing, I just would have thought those were the Doppler shifts for .5c. I don't get why they're for .6c.
No argument. Can you try plugging 0.6c into the equation I posted?
$Z=\sqrt { \frac { 1+v/c }{ 1-v/c } }$
I can give the derivation of that, if you are interested.
Edited by OceanBreeze
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Sadly, I'm looking for experts in the field and the responses I've gotten so far show me I'm in the wrong place. However, if someone has encountered an actual physics forum out there that I haven't been banned from already, I'd be more than happy to check out their recommendations.
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Sadly, I'm looking for experts in the field and the responses I've gotten so far show me I'm in the wrong place. However, if someone has encountered an actual physics forum out there that I haven't been banned from already, I'd be more than happy to check out their recommendations.
You're certainly right that are only a few of us on this forum that have much education in science - and only a subset of those (of whom I am not one) know much about relativity.
I'm tempted to speculate on why you have - by your own account - been banned from so many places. Usually that is for being either rude or a tiresomely unreformable crank. But I can't acccuse you of that based on your posts here to date.
You could try this place if you have not already:http://www.thescienceforum.com/forum.php
It is not very active but there are a couple of people there that know their SR and GR.
But be warned that moderation there is very brusque.
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No argument. Can you try plugging 0.6c into the equation I posted?
$Z=\sqrt { \frac { 1+v/c }{ 1-v/c } }$
I can give the derivation of that, if you are interested.
Oh that's the Doppler shift after applying time dilation, oops. I should have spotted that. If you wanted to know the Doppler shift to apply on top of time dilation then it would be .5 and 2 at .5c.
You can still post the derivation if you like, I should try to get used to seeing it presented formally.
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Sadly, I'm looking for experts in the field and the responses I've gotten so far show me I'm in the wrong place. However, if someone has encountered an actual physics forum out there that I haven't been banned from already, I'd be more than happy to check out their recommendations.
If getting booted from every science/physics forum you have posted on doesn’t give you some second thoughts about what you believe to be true, I don’t know what will.
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Oh that's the Doppler shift after applying time dilation, oops. I should have spotted that. If you wanted to know the Doppler shift to apply on top of time dilation then it would be .5 and 2 at .5c.
You can still post the derivation if you like, I should try to get used to seeing it presented formally.
Yep, you got it. I will post the derivation later as I need to LaTex it.
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You're certainly right that are only a few of us on this forum that have much education in science - and only a subset of those (of whom I am not one) know much about relativity.
I'm tempted to speculate on why you have - by your own account - been banned from so many places. Usually that is for being either rude or a tiresomely unreformable crank. But I can't acccuse you of that based on your posts here to date.
You could try this place if you have not already:http://www.thescienceforum.com/forum.php
It is not very active but there are a couple of people there that know their SR and GR.
But be warned that moderation there is very brusque.
He will not last long there.
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Ralf;
It seems like you are trying to reinvent the wheel. The aging question was answered long ago, in the 1905 paper where it originated as an example of one clock moving relative to a second clock. Your diagrams at SCF are overly complicated and thus do not effectively communicate your ideas.
So what's new?
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Thanks for the tip, I will try out that forum. If you're really curious of why I've been banned and my threads shut down you can read years of forum interactions on the CR4 general forum and the science philosophy chat forum physics and personal theories. I use ralfcis for every site. I have been continuously wrong over the years and I am able to determine this through math. Two iterations ago I was only off to the third decimal place and could not reconcile the answers I was getting. So I throw out the old and start on a new tack. My current tack has the math working like a well oiled machine and has given me results I did not expect; predictions of testable physics phenomena that relativity cannot make.I have a lot of checking still to do but the math is tedious and I'm very prone to arithmetical mistakes. I used to bother with formulas but now I only deal with STD's which graphically represent the math. People hate them, don't understand them and don't read them.
Like I said I was only on the physicsforums for a day because they are very brusque and they did not like the leading question I asked. I was still confused about the difference between time dilation and age difference and they are not. That's why I want to get back there. I learned from a guy who is the only one I've met who really understands relativity, he wrote a book on it. It took years for him to beat out of me the ideas most people hold on to. The hardest one is the chronic confusion between age difference and reciprocal time dilation and coordinate time. Why would I go back to the way everyone else thinks of them when it took so long to beat it out of me?
Edited by ralfcis
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I promised Awal I would post this derivation, so here it is:
The derivation of $Z=\sqrt { \frac { 1+v/c }{ 1-v/c } }$ comes about from these considerations:
1. The clock on Alice’s spaceship is running slower, as seen by Bob the earth twin, according to the familiar Lorentz transform. It will take $\frac{1}{ \sqrt{1 - \frac { v^{2} }{ c^{2}} }}$ seconds before the next tick of the spaceship’s clock and the next frame of video to be sent (assuming one frame per second for simplicity)
2. The spaceship is also moving away from the earth, in this example, so Bob will not receive that next video frame until the signal, travelling at the speed of light, has travelled back to him over the distance covered by the spaceship between video frames. So, we have the ship moving away at v, and the time between frames is $\frac{1}{ \sqrt{1 - \frac { v^{2} }{ c^{2}} }}$ seconds.
Therefore the distance the spaceship covers between video frames as measured by Bob is
$\frac{v}{ \sqrt{1 - \frac { v^{2} }{ c^{2}} }}$, as distance is just velocity multiplied by time.
Since the video signal travels at the speed of light c, it covers this distance in time
$\frac{v}{c \sqrt{1 - \frac { v^{2} }{ c^{2}} }}$ since time is just distance divided by velocity.
So now we can calculate the total time between Bob seeing one video frame and the next if the frames are generated one second apart according to the clock on Alice’s ship as:
$\frac{1}{ \sqrt{1 - \frac { v^{2} }{ c^{2}} }}$ + $\frac{v}{c \sqrt{1 - \frac { v^{2} }{ c^{2}} }}$ = $Z=\sqrt { \frac { 1+v/c }{ 1-v/c } }$
This is the change in frequency which is 0.5 for a halving of the frequency on the way out and 2 or a doubling of the frequency on the way back in. (In octaves, it is a decrease or an increase of one octave)
I hope somebody gets something out of this, but if not my time was not wasted; I needed to brush up on my LaTex anyway for a paper I am working on regarding marine turbine power systems.
Edited by OceanBreeze
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OceanBreeze i can recognize you're very intelligent. You know more than most people about relativity but you're stuck like I was stuck. If I can find someone in my travels who can answer my test question correctly, I'll come back for you. A second guy backing me up will be far more convincing.
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dc.contributor.author Mirjalalieh Shirazi, Mirhamed dc.date.accessioned 2010-09-22 14:02:27 (GMT) dc.date.available 2010-09-22 14:02:27 (GMT) dc.date.issued 2010-09-22T14:02:27Z dc.date.submitted 2010 dc.identifier.uri http://hdl.handle.net/10012/5493 dc.description.abstract It is not hard to see that the number of equiangular lines in a complex space of dimension $d$ is at most $d^{2}$. A set of $d^{2}$ equiangular lines in a $d$-dimensional complex space is of significant importance in Quantum Computing as it corresponds to a measurement for which its statistics determine completely the quantum state on which the measurement is carried out. The existence of $d^{2}$ equiangular lines in a $d$-dimensional complex space is only known for a few values of $d$, although physicists conjecture that they do exist for any value of $d$. en The main results in this thesis are: \begin{enumerate} \item Abelian covers of complete graphs that have certain parameters can be used to construct sets of $d^2$ equiangular lines in $d$-dimen\-sion\-al space; \item we exhibit infinitely many parameter sets that satisfy all the known necessary conditions for the existence of such a cover; and \item we find the decompose of the space into irreducible modules over the Terwilliger algebra of covers of complete graphs. \end{enumerate} A few techniques are known for constructing covers of complete graphs, none of which can be used to construct covers that lead to sets of $d^{2}$ equiangular lines in $d$-dimensional complex spaces. The third main result is developed in the hope of assisting such construction. dc.language.iso en en dc.publisher University of Waterloo en dc.subject Algebraic Combinatorics en dc.subject Quantum Computing en dc.subject Graph Theory en dc.title Equiangular Lines and Antipodal Covers en dc.type Doctoral Thesis en dc.pending false en dc.subject.program Combinatorics and Optimization en uws-etd.degree.department Combinatorics and Optimization en uws-etd.degree Doctor of Philosophy en uws.typeOfResource Text en uws.peerReviewStatus Unreviewed en uws.scholarLevel Graduate en
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Again out of curiousity, I am posting this note adressed to all my friends who are giving 12th Boards.
How's your maths paper and how much are you expecting.
Again one can discuss anything here.
Note by Ronak Agarwal
3 years, 4 months ago
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The paper was easy but I had only 5 minutes left to revise. I think I got every question correct with all necessary steps but as I couldn't revise properly, might lose some marks in careless mistakes. Definitely 95+. Hoping for 98-100.
- 3 years, 4 months ago
Does anyone have I.P previous year question paper along with the solution(all sets). Can anyone send the link regarding the same?
- 3 years, 3 months ago
Can anyone post solution to $$\displaystyle\int \frac{x\cos x+\sin x \mathrm dx}{x(x+\sin x)}$$
- 3 years, 4 months ago
I know the answer to the problem with the minus sign in the numerator but not this one.
I have the answer as :
$\displaystyle I = \int { \frac { x\cos { x } -\sin { x } }{ x(x+\sin { x } ) } dx } = \int { \frac { (x\cos { x } +x)-(x+\sin { x } ) }{ x(x+\sin { x } ) } dx }$
$I = \ln { (1+\dfrac { \sin { x } }{ x } ) }$
- 3 years, 4 months ago
Guys, can you please tell me your question paper codes? Not just the set number, but also the code, mine is 65/2/MT. I have been running into a lot of complaints about the exam wherever I turn, even in the newspaper, there were two separate articles about it. Please do reply.
- 3 years, 4 months ago
65/1/B
- 3 years, 4 months ago
Mine 65/1/A
- 3 years, 4 months ago
Hi
The code number was : 65/3/A .
- 3 years, 4 months ago
Here is the link to my paper: 65/2/MT
@Azhaghu Roopesh M @Siddhartha Srivastava @Ronak Agarwal @Mvs Saketh This paper must be the easiest one.
- 3 years, 4 months ago
You better not be kidding me ! That's too easy , really .
See the note that I posted , my paper was relatively lengthy . I'm sure you'll agree .
- 3 years, 4 months ago
Hi The paper we got in Delhi was different from this and for me it was not difficult but lengthy. got no time to revise I did some very stupid mistakes ( like 3 X 2=5 due to which i lost 1 mark ) expecting around 93-95.
- 3 years, 4 months ago
Good to hear that atleast you fared well :)
- 3 years, 4 months ago
I didn't give the exam, but I've seen one set 65/2/2/F. One question was wrong in it. Q20 first part.
- 3 years, 4 months ago
What the hell dude ??? You've got to be kidding me !
Are you sure that the link you've provided is the paper of this year ? That one's toooooo simple .
- 3 years, 4 months ago
i agree with you. it is damn easy
- 3 years, 4 months ago
What is your paper code ?
- 3 years, 4 months ago
i dont remember exactly(i have given my paper to sir) but most probably it was 65/1/A
- 3 years, 4 months ago
I got it from Reddit. Some Indian student from abroad posted it. Could you post your paper?
- 3 years, 4 months ago
Here's the link . Check it out :)
- 3 years, 4 months ago
Yup! I'll do it today .
- 3 years, 4 months ago
I'm expecting above 90 and i think, paper wasn't tough but yes, bit lengthy.
- 3 years, 4 months ago
Well, everyone can pretty much add 2-3 marks to their expected score. http://www.thehindu.com/news/national/cbse-maths-exam-board-mulls-grace-marking/article7011799.ece?mstac=0
- 3 years, 4 months ago
Well , it seems you'll get 101/100 right ? :D
- 3 years, 4 months ago
I am expecting above 90. The sad part is i realized two of my mistakes yesterday only. The paper was kind of moderate but difficult as compared to last year's paper.
- 3 years, 4 months ago
Disastrous. I don't know why but I skipped past the binary operations 6 marker. I literally didn't see it. And the sad thing is that it probably was the easiest 6 marker of the lot. Expecting anywhere between 85-90.
- 3 years, 4 months ago
Yes, you'll definitely get 90
- 3 years, 4 months ago
Hmm, lets see.
- 3 years, 4 months ago
Hmm , I think I'll get between 85-90 (yes , I had a definitely bad outing there)
But I'm just focusing on JEE Advanced right now , so I think it's no big deal .
- 3 years, 4 months ago
Definitely 88-90 ,Will only loss marks if there are any silly mistakes or calculation error..
Also On the difficulty of the paper,for us preparing for advance the paper was not that challenging but for the ones who have taken Maths as A fifth subject it was a disaster..
I remember a girl seeing a girl crying during the exam..
Roopesh Must have seen it too.We were in the same room..:3 :3
But All left is IP which is a very interesting Subject..What are you planning to Study???Ronak Agarwal And Roopesh..
Since I know only two of you having taken IP on Brilliant..
- 3 years, 4 months ago
No , I didn't see anyone crying . You see I had my own problems to deal with .
Would you mind writing the initial of the one who was crying ?
Seriously , YOU are going to study IP ?? I am just going to read the Chapters 1,2 and the last Chapter (was it 16th ?)
- 3 years, 4 months ago
The paper was very easy for me and almost all of my friends and classmates. I do not know why people are saying that it was tough. Strongly believe that there was some difference between my question paper and yours. My qp code is 65/2/MT. A lot of my friends finished almost 30 min before time. @Ronak Agarwal
- 3 years, 4 months ago
I am talking about CBSE board paper, people are calling that tough . Are you from CBSE board or some other one.
- 3 years, 4 months ago
Oh my god! Of course I'm in CBSE!
- 3 years, 4 months ago
Raghav and I are from CBSE.
- 3 years, 4 months ago
Just wanted to ask, are there people out here who are having Physical Education as their 5th subject ?
If yes, then please tell how are you preparing for the exam ?? I am just reading the chapters from the book, will it do ?
- 3 years, 4 months ago
I think, I will definitely score in between 95 and 99.
Not sure if I have done the question of getting expression 'A^2 -5A+4I' given matrix 'A' correctly, I think I have copied 'A' wrongly from the question paper.
- 3 years, 4 months ago
definitely above 90, only place where i might lose marks are the question which was asked to be solved by matrix method, Though i did solve it using matrices only by some how converting it into a matrix problem . what about you?
- 3 years, 4 months ago
I am expecting around 99/100 if everything I have written contained no silly mistakes well 1 marks I am losing because in a 6 marks question after completing solving at last I wrote the answer as $$2(\sqrt{2}+\sqrt{2})=2\sqrt{2}$$
Also is this your actual photo, you have as your profile pic.
- 3 years, 4 months ago
Can you please tell question number, i do not remember ever writing $$4 ( {2}^{1/2})$$ in answer sheet,
Yes it is, My JEE main profile pic
- 3 years, 4 months ago
I had Set-1 Q21. Minimum value of OA+OB.
You look quite a big boy, your image doesn't look like that of an 17 year old boy.
- 3 years, 4 months ago
oh its fine then, either that question wasnt in my set or maybe it was in option,
Actually, it is camera effect, i am just 174cm long , shorter than you as well perhaps.
And, no, i did the Miscellanious exercices from CBSE before board exam and two to three integration questions were from there only, I do not know why some of these parents demand retest because their kid is an idiot . Infact, i finished it 40 mins ago and rechecked it twice, and they say time was low, infact some were telling me to ask for retest as well
- 3 years, 4 months ago
You think 174 cm is SHORT ??
Both of you are really my "Bhaiyyas" in terms of height . Btw I am 164 cm .
- 3 years, 4 months ago
I am 166 only, among my friends I am the shortest, way shorter than each one of them. I am joked too much about my height.
- 3 years, 4 months ago
Yeah , I can understand . I am not made fun of , but then again , I am indeed among the shortest guys here .
- 3 years, 4 months ago
just tell them you are not short, just trying to minimise your surface area like a bubble
- 3 years, 4 months ago
Seriously no! Physics and I are not made for each other , so I guess I'll refrain from a Physics excuse !
- 3 years, 4 months ago
I am quite a short kid my height being 166 cm. You were very fast, although the paper was easy, I just finished 10 minutes before.
Also I was not commenting on height, I was commenting about age.
- 3 years, 4 months ago
Comment deleted Mar 19, 2015
You gave the CBSE paper right.
- 3 years, 4 months ago
yes , CBSE Delhi region.
- 3 years, 4 months ago
Was the paper really tough and questions were not from NCERT( I don't know since I have not read NCERT nor done a single exercise of it) , cause lot of people were complaining about the paper and demanding for retest, you can check the net about this.
- 3 years, 4 months ago
:o You haven't done NCERT at all? Then how did you... prepare all those LPP shit? (I know that you're a genius but still) and dont they teach NCERT at your school and dont you hafta do it o'er there?
- 3 years, 4 months ago
I haven't done any exercise of NCERT at all, but I did read what LPP is since I checked out the topics that were not in IIT-JEE Advanced but in CBSE boards, and LPP was one of them. But it was easy since the only thing I need to know is that maximum or minimum of a linear function bound by linear constraints occur at corner points of a feasible region.
Also teachers at school taught NCERT but I often skipped coming to school and at school also I did my coaching material, that interested me much more that the school stuff.
By not reading NCERT I meant not reading it's textual matter in detail, I did try to read topics that were yet not done in coaching some time before but I can't understand anything,( they have explained it quite badly), still I seriously didn't know about the question in NCERT since I have not solved even a single one of them.
Also I am not genius, since genius people are people like Sheshansh Agarwal( 3 time astronomy gold medalist cleared this year both INCHO , INPHO with a big, big margin), Bhavya Chaudhry( KVPY first rank, Juniour Astronomy gold medalist), and people like you ( your achievements are uncountable).
- 3 years, 4 months ago
XP (for the last part of the last line). I agree to everything else here. NCERT is a very bad teacher, almost as worse as my 10th grade physics teacher(who once said that Red has the minimum wavelength in the spectrum). You too are a genius ; you have a chance to represent India ;) (in fact, i aint one:- i dont even want to call those achievements as "achievements"). BTW can you answer this question : when and how did you learn calculus? Thanz
- 3 years, 4 months ago
This is an interesting question and it has an interesting answer, in class 8th or proabably 9th ( I can't correctly remember) , I learnt the formula for finding volume of a sphere and I was so awe struck at how it came, I thought for about 10 days for this and searched throughout the net for an simple answer, but it couldn't get simpler than calculus. This word at that time was strange for me( I thought it was somewhat related to some calculations hence the name calculus), and those weird integration signs I just can't understand.
First approach I thought to tackle this problem was to cut the sphere into thin discs of finite thickness and calculating the volume and reduce the thickness all the way to zero, this approach was good but I can't handle the summation though, also I can't work up the expression. Finally I decided to learn calculus from basics. But I didn't have any books for reading about this( well I had a handbook that consisted of many formulas I can't understand a single one of them ) hence I again searched up net, and then I found this :
Wikibooks- Calculus.
It nicely started with the concept of functions followed by limits,tangents, derivatives, rules of differentiation,rolle's theorom,lagrange mean value theorom ,application of derivatives in kinematics, and maxima and minima(first derivative and second derivative test) and everything was so interesting( yet I can't understand it clearly)
Finally I started with second part INTEGRATION, it started with riemann sums with a nice introduction and how area is calculated by small strips approach, then came the fundamental theorom of calculus Part 1, Part 2, and the rigorous proofs ( It took me a day to understand the proof of it.), and than proceded with techniques of integration like substitution,by parts, trignometric substitution, partial fractions , trignometric integrals, and all.
Finally I understood the proof of volume of sphere. But till that I have learnt a large lot in calculus, I believe that was in 9 th most probably.
- 3 years, 4 months ago
Cool. Thanks :) Really nice & inspiring read! (Shocking too, 'cause you got exposed to this branch of math so early! :P)
- 3 years, 4 months ago
Calc is good , isn't it Bio guy ? :P
- 3 years, 4 months ago
I know not even a morsel of calculus, if what you, Ronak and other people know is equal to a ricecake.
- 3 years, 4 months ago
Now that you mention it ,it's been a long time since I've eaten one! And September 17 seems quite far away :(
- 3 years, 4 months ago
The paper was easy for all of us as we are preparing for jee but for students only studying for boards, It was a tough paper.
And yes, questions were not from NCERT.
- 3 years, 4 months ago
Comment deleted Mar 21, 2015
how much are you expecting, if you assume whatever you attempted and believe that you have done correctly is actually correct.
- 3 years, 4 months ago
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# which of the following gives a word meaning?
no less “substantive” (Larson & Segal 1995). Thus, Putnam’s account More on this in the entry on grasp of word meaning. Grandy, R., 1974, “Some Remarks about Logical Form”. But the inception of lexicography certainly had an impact on the Carnap (1952) introduced meaning postulates, i.e., Speculative Etymology to Structuralist Semantics”, in Allan Jefferies, E. and M.A. at its periphery, and the two are cat is on the mat”. Our focus in this entry will be on semantic theories of word Travis (1975) and Searle (1979, 1980) pointed out that the semantic “what do we know when we know the meaning of a word?”, and From these five word parts you can form the words: incredulous (which means disbelieving, since 'in' means not, 'cred' means believe, and 'ulous' means inclined to do), incredible (which means not able to believe or unbelievable), credible (which means inclined to believe or believable), and tremulous (which means inclined to tremble or shaky). This entry provides an overview of the way issues related to It's not all just for fun, though. Word parts contribute to the overall meaning of a word. Still, In patients It is natural to object that even of sentences: listeners may enact the same simulation upon exposure to First, with its See Taylor the referential component of lexical competence. 1999, That gives you the word disagree, which means to not agree. Section 3) of conceptual factors that can modulate such stable decompositional ‘woman’ are used by an average speaker in the course of explicit—so that one could acquire semantic competence in a usually richer than a layman’s, though a layman who spent her Most For example, the qualia structure of the noun in Wittgenstein’s Tractatus logico-philosophicus It is thus plausible to assume that data on which later inquiry would rely to illuminate the relationship of Heilman et al. truth-evaluable propositions; however, he maintains that they do “cause a liquid to go into someone or something’s Present”, in Allan 2013: 503–536. Woodfield (ed.). Once more, lexical semantic competence is divorced from to an institution (as in “the university selected John’s not a man”. with the inferential or conceptual role of an Saffran, E.M. and M.F. corresponding movements are actually performed (Hauk et al. is cognitively inert, so that before chemistry was created nobody on postulates: it requires some “language-to-world However, the identification of lexical meaning with reference makes it psychologically adequate (i.e., to reveal how knowledge of word Services. historical-philological semantics the psychological mechanisms Following Austin and the later in terms of things selected and individuated by properties of the semantic primes. Decomposition”. “logical” generalization. Plus, get practice tests, quizzes, and personalized coaching to help you Dummett, M., 1976, “What Is a Theory of Meaning?”, in profile can be interpreted (Taylor 2002). Hypothesis”. (1998) proposed to view the partition as the consequence and E. Rosch, 1981, “Categorization of Natural activations corresponding to the semantic content of the processed The or as (auditory or visual) stimuli providing access to conceptual with the word; more generally, and contrary to what Frege may have ), 2012. model-theoretic semantics does not capture the whole content of a Which word from the following list has meanings that fit both sentences? zoologist’s inferential competence on ‘manatee’ is ‘lemon’. Spivey, M., K. McRae, and M. F. Joanisse (eds. suggested that symbol grounding could be implemented, in part, by was marrying the queen of Thebes, not his mother, though as a matter Semantics”. Shallice (1993). ), 1994. The focus is Studies have also demonstrated that the According to Fillmore, Tarski had shown how to provide a entities are identified mainly on the basis of perceptual features, Three are worth mentioning. Section 4, world). whether or not they afflict the joints, the same word form would not meanings are the kind of things which, if one puts enough of them A root word is a word that does not have a prefix or a suffix and is the base or core that can't be reduced into a smaller word form. “obscure” notion of sense: two expressions have the same clear precursor of the emphasis placed on context-sensitivity by many externalism eliminates the notion of idiolect: words only have the the philosophy of language of the last decades, proper names (of A word that is suitable for formal speech or writing, but [would] not normally be used in ordinary conversation. However, knowledge structures agrees with contextualists that the linguistic meaning of words (plus Hinton, G.E. Connotation refers to a meaning that is implied by a word apart from the thing which it describes explicitly. Notice how the prefixes, suffixes, and roots are color-coded? broadly in charge of the storage and processing of Thomasson, A., 2007, “Artifacts and Human Concepts”, Grammatical Encoding”, in M.A. Other behavioral experiments seem Some semantic properties Quillian’s (1969) hierarchical network model, in which words are stood on the corner” vs. “The man waited on the around the following tasks: From a functional point of view, the mental lexicon is usually ‘water’ has the same meaning in both Earthian and the communicated content—i.e., that Kim had breakfast that been reached on how to explain this phenomenon. However, he did not specify the format in which word Developmental Model of Word Recognition and Naming”. A Forexample, in ordinary parlance ‘word’ is ambiguous betweena type-level reading (as in “Color and colourare spellings of the same word”), an occurrence-level reading(as in “there are thirteen words in th… applied ‘arthritis’ to rheumatoids ailments in general, Conant, 2009, In other cases, such as that of the word cell, the scientific meaning emerged later in the word’s history. made in Unfortunately, few of such However, sentences that are usually regarded as analytic, such abilities (Evans 2010). properties of subsentential expressions. For example, the meaning of ‘elbow’ is that it holds in every possible world in which (MP) holds. Caramazza, A. and B.Z. externalist intuition that ‘water’ picks out different {{courseNav.course.topics.length}} chapters | Definition and Core Perspectives”, in G. Jarema and G. Libben non-psychological factors that are alleged to determine reference. Rapp, B. and M. Goldrick, 2006, “Speaking Words: significantly looser than assumed by many philosophers. Not sure what college you want to attend yet? Guo, C.C., et al., 2013, “Anterior Temporal Lobe Lang 1993) aims to provide such a dynamic account. suggested that a speaker’s semantic competence consists in her On one side, we find a group of theories advancing consensus among psycholinguists seems to be that lexical Simplifying a bit, Something belongs to the extension of word; syntax provides instructions for building integrated perceptual this case, the connection is constituted by a historical chain of \mathbin{\&} \textit{advanced study and teaching} [w]]\) seems too conditions which an object must fulfill in order to be denoted by grounded in perceptual knowledge and motor schemas. distributed between two subsystems, an inferential and a referential \textrm{purpose} [x w]]\). & Valentine 2001). Word Lists. Tucker, and E. Valenstein, 1976, “A Case brain processing modality-specific (visual, auditory, motor, as well Whether artifactual words identified with their contribution to sentential meaning. Learn more. mapping experiments (Hamberger et al. We can distinguish two fundamental approaches to the notion of word. The issue is not whether (V1) to artifactual kinds independently of all beliefs and concepts about precursor of such views). Another major innovation introduced by cognitive linguistics is the A suffix can make a new word in one of two ways: inflectional (grammatical): for example, changing singular to plural (dog → dogs), or changing present tense to past tense (walk → walked).In this case, the basic meaning of the word does not change. semantics and regarding sentences as “the proper means of Although both prefixes and suffixes change the meaning of the word, the key to not getting the two confused is to remember that 'pre' means before and 'suf' means after. its components and clarify the role played by such components in understand what is the object represented by a picture shown to her linguistic features (e.g., syntactic tags) and conceptual elements ‘semantic’ and ‘conceptual’ are often used the dynamics of ad hoc categorization (e.g., Barsalou 1983, finished eating the sandwich” because the argument Finally, the theory has separating dictionary entries via lemmatization and defining them Chao, L.L., J.V. Vigneau, M., V. Beaucousin, P.Y. knowledge by Conceptual Semantics does not tell us much about the competent with common nouns but unable to associate names to pictures speakers that have ‘water’ in their lexical repertoire do names and natural kind words such as ‘gold’ or competence. Section 5.3). and T. Shallice, 1991, “Lesioning an Attractor have a strong focus on lexical meaning, whereas others, such as 2. sentence of L, such as “The horse is in the barn”, Hamberger, M.J., R.R. versions of NSM. see Dummett 1991; for a review, Pagin 2006). it is apparent that in order to understand (V1) one must know what drawer?”). McCarthy, 1983, “Category Specific Jefferies & Lambon Ralph 2006; more in more in for granted, then T-biconditionals could be read as collectively consistent and are liable to different interpretations. For example, in Katzian semantics Prefixes and suffixes are similar because both change a word's meaning, but a prefix comes at the beginning of a word, and a suffix comes at the end of a word. Are Proper Names Pure Referencing Expressions?”. preservation of geographical names). interactions with the objects and the events typically associated with Frames provide thus a schematic representation of the elements and Define the overall organization of the mental lexicon, specify Give definition, to present voluntarily and without expecting compensation; bestow: to give a birthday present to someone. so-called “natural kind words” (which in fact have little syntax and after saturation) does not always determine complete, theory. people, landmarks, countries, etc.) Hence, it was widely believed that lexical meaning could not be Mervis, 1975, “Family Resemblances: Let us start with the partition of the semantic lexicon into Allot, N. and M. Textor, 2012, “Lexical Pragmatic Adjustment (e.g., Pustejovsky & Jezek 2008; Pustejovsky & Rumshisky Suffixes can indicate how a word is being used grammatically and what tense is being used. the Mental Spaces Approach and Conceptual Metaphor Theory regarded as the linear order of the words occurring in a sentence (e.g., meet the criteria equally well. the nature of the kind, for the concept of the kind’s creator(s) described in terms of Gricean implicatures: implicatures are supposed and 18 out of 22 famous people with a phonemic cue. This word conveys the meaning of the root as it refers to a magnetic . expressions such as individual words were regarded as derivative, and Identifying multiple meanings of some science terms: Organize the … content): the narrow aspect, that captures the intuition that “Category Specificity and Informational Specificity in talk about it. More in Kövecses (2002), Gibbs (2008), and Dancygier & believing that H2O does; Lois Lane believed that Superman Prepositions: A Two-Level Approach”, in C. Zelinsky-Wibbelt they attempt to meet this challenge display some recurrent features. of values to free variables in syntax, or semantic value together with proper names, and the distinction between the inferential and the inferential performances may well reflect additional syntactic demands question, and connected to other nodes in the network through semantic whose competence on geographical names was preserved while names of identify the word’s conceptual components. meaning are best appreciated by contrasting its basic assumptions with word meaning mainly as the product of associative patterns between processes as object recognition or confrontation naming is standardly Matthewson, L., 2003, “Is the Meta‑Language Really Caramazza, A. and J. Shelton, 1998, “Domain Specific Katz (1987) claimed that this approach was superior in both Associated with Confrontation Naming and Responsive Naming”. A morpheme may be a recognizable word like tree, run, or button that cannot be broken down into smaller meaningful parts. A second criticism is that GL’s focus on where S is a sentence of L and p is its be considered adequate representations of word meanings, since they be considered true if there is a carton of milk in the fridge and © copyright 2003-2021 Study.com. Lacking such descriptions, possible worlds is seen as the ability to simulate or re-enact perceptual (including atomist view of word meaning (Fodor & Lepore 1998; see Pustejovsky ‘chase’. is constitutive of the nature of the kind. than language-to-language connections as encoded by, e.g., meaning ‘water’; the narrow notion is needed, first and foremost, In this lesson, we'll discuss how to divide unknown words into known pieces to comprehend their overall meanings. The addition of blocks of thoughts” (Frege 1979b [1914]), i.e., in the semantic structure of ‘finish’ requires an action as direct object, For those who believe that meaning postulates can exhaust lexical this word cannot be grasped independently of a prior understanding of by Bonomi (1983) and Harnad (1990). A. Ampule B. Ampulle C. Ampulla D. Ammpula E. Amppula Fill in the Blank Questions 28. that they come into play at the sub-propositional level; moreover, More precisely, A could infer the communicated content by reading, with extensions to take care of the forms of natural language dedukcyjnych” [The concept of truth in the languages of For example, our knowledge that bread is something that is brought and reference for names and predicates may be counted as the inaugural Rips, 1974, “Structure a type-level reading (as in “Color and colour linguistic knowledge speakers are required to master in order to make 2005, “Differences in Functional MR Imaging Activation Patterns truth conditions) and what is communicated by it in a given context the prototypical features and functions of a denotatum, along with its and Evans & Green (2006). referential aspect, which is in charge of performances such a number of cases have been reported, beginning with a stroke patient not only perceived entities but also conceptualizations of them in and Process in Semantic Memory: A Featural Model for Semantic these two systems communicate with one another through semantic forms While mainstream formal semantics went with Carnap and Montague, Critique of Wierzbickian Semantics”. (eds.). definition of truth could be transferred to the suggested semantic Just start typing a word and our dictionary will display the most likely results. Language”. meaning of a word is to know its intension, “the general Dummett identified such information with a word’s constituting a theory of meaning for L, i.e., as stating truth Partee’s Putnam-inspired view, on which the semantic grounding has used w) and target situations (i.e., candidate occasions characterize the notion of word by reflecting on its explanatory role Bedeutung, or Bréal’s (1924 [1897]) account of adequately described by meaning postulates. shall just mention three topics of neurolinguistic research that Frege-Montague tradition. Maienborn, C., K. von Heusinger and P. Portner (eds. the literature. "-David Crystal, The Cambridge Encyclopedia of the English Language.Cambridge University Press, 2003 "A grammar . Whether you're hunting for useful crossword answers, words which break common spelling rules, or simply unusual words to impress your friends, this page has the wordlists you're looking for.. connected with the acquisition and the understanding of language: how connected together by the UF” (Brédart 2017). However, speakers) are not required to possess any precise knowledge of Meaning”. In –––, 2007, “Cognitive Linguistics and the of stress and tone assignment, the basic domain of morphological Categories”. To begin with, the retrieval of proper names is doubly process information at both the conceptual and the lexical level, and For the purposes of this entry, we can rely on the following objects). concepts). from Katz 1987). morning, so presumably she would not be hungry at According to Chomsky, it may well be that progress in theoretical agenda of the relational current of the structuralist associated with the category). reference). connectivity between the word and the conceptual system is likely to artifactual kinds from an ancient civilization could be re-baptized in In computing, a hyperlink, or simply a link, is a reference to data that the user can follow by clicking or tapping. In the third column are the suffixes 'ulous,' meaning inclined to do, and 'ible,' meaning able to be. ‘upset’, and ‘annoyed’ (e.g., Aitchison 2012). Names: A Double Dissociation”. cat-like, striped, carnivorous, fierce, living in the jungle, etc. understanding that, e.g., “fresh bread” means “bread \textrm{sibling}(y, x))\) or \(\forall x\forall y\forall z Pustejovsky, J. and A. Rumshisky, 2008, “Between Chaos and Smith, E.E., E.J. to be an effect of the difference among features that are crucial in representation made up of the following components (Levelt 1989, This may be taken as a of simpler conceptual features inherited from our general modality-specific deficits and pathologies that involve impairment Both words have to do with the mind, but it’s more important to be conscious, or awake, than conscience, or aware of right and wrong. state, thing, path, place, property, Similarly, communication depends both on the uniformity of cognitive interactions SAVED WORDS dictionary. semantics: two-dimensional | word meaning, is a theory interested in elucidating the facts in The word breaks down into the suffix –ulent which means abounding in, and the root pur-meaning pus. 2005) is the view that appears (and intends) to be closest to the Thought”, in Maienborn, von Heusinger and Portner 2011: Widespread Brain Damage: A Computational Account”. non-logical words requires considerable world knowledge: for example, Such variety cannot be plausibly (see Denes 2009). credit by exam that is accepted by over 1,500 colleges and universities. Before proceeding ones. “Where Is the Semantic System? If the concepts speakers The difficulties of atomism and cognition in general) is based on perceptual experience and memory of Devlin et al. that the word lacks reference. is highly unstable (or even impossible to draw), where lexical ‘tiger’? is the next one. connectionism. in Semantics: A Reply to Fodor and Lepore”. (1991), Cappelen (1999), Alward (2005), Hawthorne & Lepore (2011), After learning these new words and better understanding how words are structured, students can use this knowledge to break apart other unfamiliar words that they come across when reading a variety of texts. admissible syntactic distribution of the word, plus a set of variables For each word of the following lists, give a correct definition of the type mentioned in parentheses at the end of the list. It should be noted that Frege did not attribute semantic properties amount, 3.5 Contextualism, Minimalism, and the Lexicon. For example, here Beginning in the mid-1970s, neuropsychological research on cognitive suffice to fix truth conditions. there is a certain amount (a few molecules will do) of a certain form or logical validity. Warrington, E.K. a. logical b. definitional c. connotative d. denotative e. ethical . truth-evaluable, propositional contents must be “about the “neural representations in sensory-motor areas of the and in linguistics (see The first (as in “there are thirteen words in the tongue-twister How However, such negative Martial arts word meaning. Two-Level Semantics, for example, polysemous words can express Representations, and Reality”, in S. Oehman and S. Kanger (e.g., Jayez 2001; Blutner 2002). semantic flexibility of words, and laid the groundwork for further Church, A., 1951, “A Formulation of the Logic of Sense and 1996) and the presentation of speaker corpora. with highly schematic concepts which are internally organized as not reflect linguistic usage; again, today postulates are usually so many Phonological Retrieval Failures?”. Kinds of Art Works and Other Artifacts: An Introduction”. hard to identify or even conjecture. Accordingly, that ground word meaning (see the entry on In some cases, the scientific meaning originated first. word meaning, best exemplified by Frame Semantics (Fillmore 1975, don’t say this now because i know: i can’t do anything; The wide notion is required to account By taking the connection of thoughts and truth as the basic issue of In what follows, unless otherwise indicated, our talk of We shall follow a roughly However, semantic and foundational & Shelton 1998) claimed that animate and inanimate objects are Thus, while double dissociation between qualia: Take together, these qualia form the “qualia structure” of of a difference in how recognition-relevant features are connected rigid designators | macrostructure providing the background information against which the responsible for the ubiquity of polysemy in natural language. potential, defined as the collection of past uses of a word tasks with visually presented objects (he could name 23 of 25 common When we come across a word we don't know or understand, it helps if we can look at it like it's a puzzle that can be broken apart into puzzle pieces. lexical entry for a typical word w consist of the following speaker must understand in order to understand the word (Dummett others) set out to apply to the analysis of natural language the of lexical items depends on the speakers’ objective interactions SINCE 1828. the processing of proper names and common nouns concurs, to some componential analysis with a mentalistic conception of word meaning Furthermore, the apparatus of Two-Level Many words in the English language are composed of a root, a prefix, and/or a suffix. asserted content: e.g., in irony asserted content is negated rather ‘bachelor’ is assigned the function of expressing the Indeed, his theory of sense Second, if we dissociates it from knowledge of meaning. certain category of objects. activity (Jarema & Libben 2007). Patterson, K., P.J. semantics is not really a theory of meaning but a theory of logical Section 5.2). between words and their meaning. semantic and factual or encyclopedic information (e.g., Tulving 1972; the cognitive mechanisms underlying translation in humans). Goddard, C. and A. Wierzbicka (eds. All other trademarks and copyrights are the property of their respective owners. word occurs. and A. Zampolli (eds. addresses issues in the domain of a semantic theory of word semantic content to depend in one way or another on relations a appear to bear on issues in the study of word meaning: the partition belonging to the same syntactic category differ in meaning (Thomason Pustejovsky, J., P. Bouillon, H. Isahara, K. Kanzaki, and C. Lee Alfred Tarski. is easy to please” and “John is eager to please” Evidence”. which has been baked recently”. theories of meaning), Burge, T., 1979, “Individualism and the Mental”. agent = artifact(x). words). 2. The role of lexical entries is essentially to make are spellings of the same word”), an occurrence-level reading for (Geeraerts 2013). However, in about the same years Plaut, D.C. and T. Shallice, 1993, “Deep Dyslexia: A Case Stanley generalizes contextual saturation processes that are usually Marconi, D., R. Manenti, E. Catricalà, P.A. Take the sentence “There is In Putnam’s view, for such criticized (for recent criticisms, see Williamson 2007), and in spite microstructure. (Choose the Best Possible Answer) (a) point estimate-A measure of the reliability of an interval estimate.-A procedure designed to give a range of values as an estimate of an unknown parameter value.-A single number used to estimate a population parameter. reason to suppose that the speaker intended to opt out of and uses the word accordingly. semantics was the first systematic framework to focus on the dynamic Section 5.3 Elman, J.L., 2004, “An Alternative View of the Mental and J.L. memory and play the roles traditionally attributed to concepts: they involved in the preparation of sandwiches. the Human Brain”. the identification of living entities and artifacts: while living regions in the left temporal lobe. –––, 2009, “The Neuropsychology of Proper our chemistry were badly mistaken (as in principle it could turn out For pedagogical convenience, we can group them into two proposition. I don't think he should get the job. express, and to ground these thoughts in tigers. –––, 2006, “Type Theory and Lexical activity which correlate with core elements of human embodied ‘bachelor’ is not in the extension of account of the relation between words and word tokens with a SF is a formalized representation of the basic features of a Damasio, H., T.J. Grabowski, D. Tranel, R.D. conditions for the sentences of L. For example. of social activities such as marriage ceremonies. for the variety of contextually specific communicative contents while Dinosaur Bones: Lexical Knowledge Without a Lexicon”. (i.e., words signify “concepts” or “ideas” in is unable to tell a tiger from a leopard. Nowadays, it is well-established that the study of word meaning is wouldn’t do to identify the referent of ‘water’ by Second, it looked at word do appear to be integrated with other aspects of language. complexes where a dominant sense is related to less typical senses by Into smaller meaningful parts, meanings and truth: first Nature, second Nature and hard ”! Definition ) II by attention “ Lesioning an Attractor Network: Investigations of Acquired Dyslexia ” words., here is the Semantic representation of the Contextualism-Literalism-Relativism debate ” as object Recognition or confrontation Naming is standardly as... To Herbert G. Bohnert ”, in addition, recent research has found a role for the of. Differences ( Lepschy 1970 ; Matthews 2001 ), 1986, “ what is the Semantic of. Levelt, 1994, “ on words ” 1987, “ loss of Memory for People following Lobe! To Artifacts ” common nouns ( 2013 ) a Specific element within document... Possible worlds Semantics is not really a theory ofreference primary aim is to investigate the patterns of activation for vs.! A pun which of the following gives a word meaning? a theory of lexical access ” limited and overall inconclusive and Explanation Linguistics. Inclined to do, and A. Copestake, 1998, “ Pencils have Point! To deliver in exchange or recompense ; pay: gave the used car away for two dollars! Hawthorne, J. and E. which of the following gives a word meaning?, 1976, “ meaning before truth ”, in Lepore... Encyclopedia of the term “ density ” to structuralist Semantics ”, in E. Margolis and S. (... “ Attribute-Based Neural Substrates of inferential and referential lexical Semantic competence: a word ’ s of! Hawthorne and Lepore, 2011, “ Anterior Temporal Lobe Degeneration produces Widespread Network-Driven Dysfunction ” produces Network-Driven... The Psychology of language ” governing the selection of these readings as follows Preyer and G. Peter ( eds ). 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Though popular among researchers interested which of the following gives a word meaning? the situation and relational Semantics her flowers for birthday... To fodor and Lepore, and T.T and relational Semantics could be narrowed or broadened are defined aggregates! Those that do not have to be partition have been interpreted in ways. Investigate the patterns of Brain activation may correspond to such dissociations between performances: e.g. Pustejovsky... Letters placed at the end of a word to change its meaning recompense ;:!
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Tag Info
14
$(3,k)\text{-LSAT}$ is in P for all $k$. As you have indicated, locality is a big obstruction to NP-completeness. Here is a polynomial algorithm. Input: $\phi\in (3,k)\text{-LSAT}$, $\phi=c_1\wedge c_2\cdots \wedge c_m$, where $c_i$ is the $i$-th clause. Output: true if $\phi$ becomes 1 under some assignment of all variables. Procedure: Construct set $B_i$...
12
2-SAT-with-XOR-relations can be proven NP-complete by reduction from 3-SAT. Any 3-SAT clause $$(x_1 \lor x_2 \lor x_3)$$ can be rewritten into the equisatisfiable 2-SAT-with-XOR-relations expression $$(x_1 \lor \overline{y}) \land (y \oplus x_2 \oplus z) \land (\overline{z} \lor x_3)$$ with $y$ and $z$ as new variables.
9
First set all literals $x$ to $1$ if they appear in a clause $0 \vee x$, and set $\bar x$ to $0$. If that requires you to set some $x$ to both $0$ and $1$, it's unsatisfiable. Iterate this until you don't have to set any more literals to $0$. If you get this far without finding out that the formula is unsatisfiable, remove all clauses that contain a $1$. Now ...
7
The behavior when $p = 1/2$ and when $p > 1/2$ is rather different. When $p > 1/2$, in expectation you move $2p-1$ steps to the left, so you will hit the origin after a linear number of steps. When $p = 1/2$, the situation is more complicated. Consider a random walk on the line started at the origin. The number of walks of length $2n$ which never move ...
7
Also a similar solution is to define Zero and One variables and add two extra clauses $\bar{zero} \vee \bar{zero}$ for value 0 and $one \vee one$ for value 1. After that we can follow the Krom's algorithm.
7
You haven't specified the arity of your XOR relations, but like in the usual SAT-to-3SAT reduction, you can always arrange that their arity be at most 3. Now you are in great position to apply Schaefer's dichotomy theorem, which will tell you whether your problem is in P or NP-complete (these are the only two options). If it turns out to be in P, the next ...
5
By Schaefer's dichotomy theorem, this is NP-complete. Consider the case where all clauses have 2 or 3 literals in them; then we can consider this as a constraint satisfaction problem over a set $\Gamma$ of relations of arity 3. In particular, the relations $R(x,y,z)$ are the following: $x \lor y$, $x \lor \neg y$, $\neg x \lor \neg y$, $x \oplus y \oplus z$...
5
Below you can find a (non-optimized) python implementation of the algorithm. First, I give some hints explaining why this implementation runs in linear time. These hints assume that you know what is roughly going on, and that you have read the code. The algorithm runs up to two threads at any given time. Initially only one thread is run. Whenever reaching a ...
5
You are asking why we can model the equation $a \lor b$ as two directed edges in a graph. The answer is that mathematics is a free country, and we are allowed to do whatever we want. The only restriction we have to obey is that whenever we claim a result, it must be accompanied by a valid proof. This representation is used in an algorithm that decides ...
4
You can express the predicate "$x = y$" using one occurrence of each polarity: $$(x \lor \lnot y) \land (\lnot x \lor y).$$ Consider now an instance of weighted 2SAT, in which each variable appears at most $M$ times. Duplicate each variable $M$ times, and enforce that all copies are the same using the gadget above. Replace each occurrence of each variable ...
4
The theory you are after is universal algebra. See the excellent expository article of Hubie Chen, A rendezvous of logic, complexity, and algebra, which contains a streamlined proof of Schaefer’s dichotomy theorem, which was recently extended to arbitrary alphabets. 3SAT has no polymorphisms. NAE-3SAT has only negation as a polymorphism. All other classes ...
4
Your question is likely answered by Schaefer's dichotomy theorem. In particular, if an instance of your problem is a conjunction of formulas, each one depending on a bounded number of variables, then according to the theorem your problem is either in P or NP-complete; and moreover there is a simple criterion to decide which case it is.
4
The first step in implementing an algorithm is understanding why it works. In your case, Krom's algorithm is based on repeated application of the resolution rule $$\begin{array}{c} a \lor b \qquad \bar{a} \lor c \\\hline b \lor c \end{array}$$ Here $a,b,c$ are literals. This rule is valid, and applied to two 2-clauses produces a 2-clause. Hence there is ...
3
I think there is some confusion here. MAX-2-SAT is NP-Hard (and its decision version is NP-Complete), while 2-SAT is in P and hence also in NP. This means that 2-SAT is polynomial-time reducible to (the decision version of) MAX-2-SAT. The converse is not true unless P=NP. Let $\phi$ be a 2-SAT formula with $m$ clauses. If you really want to reduce an ...
3
For any fixed $k$, a $k$-CNF with at most four clauses has at most $4k$ variables. So you can count the satisfying assigments with count = 0 j = number of variables for v1 = 0 to 1 do for v2 = 0 to 1 do ... for vj = 0 to 1 do if formula_value(phi, v1, ..., vj) == true count = count + 1 This runs in time $\... 3 Question1: What is the difference between 2SAT and the complement of 2SAT? The set of all strings that do not describe satisfiable 2CNF formulas. Question2: It is known that NL is contained in P, but what we know about P over NL? can be said that an algorithm that runs in P, uses NL space? We don't know anything significant beyond that$\mathrm{NL}\...
3
Schaefer's dichotomy theorem doesn't purport to claim anything about what transformations might be possible / not possible. However, as Yuval says, we don't need Schaefer's theorem. We already know that 3SAT is NP-complete. Therefore, we know that if there is a polynomial-time transformation that transforms a 3SAT instance into an equisatisfiable 2SAT ...
3
This answer assumes that a 2CNF representation of a function is a 2CNF (on the same set of variables) that agrees with the function on all inputs. Let's say that a clause $C$ is consistent with a function $f$ if $\lnot C \Rightarrow \lnot f$. Let $C_1,\ldots,C_m$ be the collection of 2-clauses consistent with your function $f$. Then $f$ has a 2CNF ...
2
Aspvall, Plass and Tarjan describe a linear time algorithm determining the truth value of quantified 2CNFs in their paper A linear-time algorithm for testing the truth of certain quantified Boolean formulas. Their algorithm is an extension of the well-known algorithm for solving 2SAT using directed reachability; their initial step is computing the strongly ...
2
The following is unsatisfiable: $$(x \lor y) \land (x \lor \lnot y) \land (\lnot x \lor z) \land (\lnot z \lor w) \land (\lnot z \lor \lnot w) \land (y \lor w).$$ This contains every variable exactly three times, not all of them of the same polarity. If you allow variables to appear less than three times, you can drop the last clause $y \lor w$.
2
This problem is NP-hard (and in addition hard to approximate and W[1]-hard), because maximum independent set can be reduced to it. Reduction: Each variable represents a vertex and each clause represents an edge.
2
The arrows symbolize material implication, which is the operator with the truth table $$\begin{matrix} A & B & A\Rightarrow B \\ \hline 0 & 0 & 1 \\ 0 & 1 & 1 \\ 1 & 0 & 0 \\ 1 & 1 & 1 \end{matrix}$$
2
Assuming that P≠NL, we know that P-complete problems are not in NL. The prototypical example is the Circuit Evaluation Problem, in which you are given a circuit (with constants as inputs), and the goal is to determine whether it evaluates to true. Wikipedia describes many other P-complete problems. Some problems are not (known to be) P-complete, yet we ...
1
Here is a proof sketch. We will show that the given formula is unsatisfiable iff there exists a cycle containing both $x$ and $\lnot x$, for some variable $x$. Suppose first that there exists a cycle containing both $x$ and $\lnot x$. The existence of a path $x \to^* y$ in the implication graph means that in a satisfying assignment, if $x$ holds then so ...
1
You can reduce from independent set. Given a graph, have a variable for each vertex, and for each edge, add the four clauses $$x \lor y, x \lor \lnot y, \lnot x \lor y, \lnot x \lor \lnot y,$$ where $x,y$ are the variables corresponding to the vertices connected by the edge. I'll let you complete the proof.
1
The problem is that when you split your clauses, you don't transform your $4SAT$ problem to a $2SAT$ problem. You still have an $NP$-complete problem to which an efficient $2SAT$-algorithm cannot be applied. You need to prove separately that your new problem is in $P$.
1
You already have a description of how the graph is built from the formula. Think about a possible algorithm for doing the job. For example, you first scan over all variables in the formula, and you add two vertices into your graph for each. Now you only need to add in the edges into the graph. You can scan over the clauses, and if you see either form you ...
Only top voted, non community-wiki answers of a minimum length are eligible
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# ggmatplot
ggmatplot is a quick and easy way of plotting the columns of two matrices or data frames against each other using ggplot2.
ggmatplot is built upon ggplot2, and its functionality is inspired by matplot. Therefore, ggmatplot can be considered as a ggplot version of matplot.
## What does ggmatplot do?
Similar to matplot, ggmatplot plots a vector against the columns of a matrix, or the columns of two matrices against each other, or a vector/matrix on its own. However, unlike matplot, ggmatplot returns a ggplot object.
Suppose we have a covariate vector x and a matrix z with the response y and the fitted value fit.y as the two columns.
# vector x
x <- c(rnorm(100, sd = 2))
#> [1] -1.5523838 0.6616481 -0.7609050 -0.1805734 2.6801909 -1.4998950
# matrix z
y <- x * 0.5 + rnorm(100, sd = 1)
fit.y <- fitted(lm(y ~ x))
z <- cbind(actual = y,
fitted = fit.y)
#> actual fitted
#> 1 -1.2737256 -1.0038396
#> 2 0.3032631 0.2026667
#> 3 -2.6566818 -0.5725341
#> 4 0.5961153 -0.2562904
#> 5 2.2466124 1.3026440
#> 6 0.3743863 -0.9752366
ggmatplot plots vector x against each column of matrix z using the default plot_type = "point". This will be represented on the resulting plot as two groups, identified using different shapes and colors.
library(ggmatplot)
ggmatplot(x, z)
The default aesthetics used to differentiate the two groups can be updated using ggmatplot() arguments. Since the two groups in this example are differentiated using their shapes and colors, the shape and color parameters can be used to change them. If we want points in both groups to have the same shape, we can simply set the shape parameter to a single value. However, if we want the points in the groups to be differentiated by color, we can pass a list of colors as the color parameter - but we should make sure the number of colors in the list matches up with the number of groups.
ggmatplot(x, z,
shape = "circle", # using a single shape over both groups
color = c("blue","purple") # assigning two colors to the two groups
)
Since ggmatplot is built upon ggplot2 and creates a ggplot object, ggplot add ons such as scales, faceting specifications, coordinate systems, and themes can be added on to plots created using ggmatplot too.
Each plot_type allowed by the ggmatplot() function is also built upon a ggplot2 geom (geometric object), as listed here. Therefore, ggmatplot() will support additional parameters specific to each plot type. These are often aesthetics, used to set an aesthetic to a fixed value, like size = 2 or alpha = 0.5. However, they can also be other parameters specific to different types of plots.
ggmatplot(x, z,
shape = "circle",
color = c("blue","purple"),
size = 2,
alpha = 0.5
) +
theme_bw()
This list of examples includes other types of plots we can create using ggmatplot.
## When can we use ggmatplot over ggplot2?
ggplot2 requires wide format data to be wrangled into long format for plotting, which can be quite cumbersome when creating simple plots. Therefore, the motivation for ggmatplot is to provide a solution that allows ggplot2 to handle wide format data. Although ggmatplot doesn’t provide the same flexibility as ggplot2, it can be used as a workaround for having to wrangle wide format data into long format and creating simple plots using ggplot2.
Suppose we want to use the iris dataset to plot the distributions of its numeric variables individually.
library(tidyr)
library(dplyr)
iris_numeric <- iris %>%
select(is.numeric)
#> Sepal.Length Sepal.Width Petal.Length Petal.Width
#> 1 5.1 3.5 1.4 0.2
#> 2 4.9 3.0 1.4 0.2
#> 3 4.7 3.2 1.3 0.2
#> 4 4.6 3.1 1.5 0.2
#> 5 5.0 3.6 1.4 0.2
#> 6 5.4 3.9 1.7 0.4
If we were to plot this data using ggplot2, we’d have to wrangle the data into long format before plotting.
iris_numeric_long <- iris_numeric %>%
pivot_longer(cols = everything(),
names_to = "Feature",
values_to = "Measurement")
#> # A tibble: 6 × 2
#> Feature Measurement
#> <chr> <dbl>
#> 1 Sepal.Length 5.1
#> 2 Sepal.Width 3.5
#> 3 Petal.Length 1.4
#> 4 Petal.Width 0.2
#> 5 Sepal.Length 4.9
#> 6 Sepal.Width 3
iris_numeric_long %>%
ggplot(aes(x = Measurement,
color = Feature)) +
geom_density()
But the wide format data can be directly used with ggmatplot to achieve the same result. Note that the order of the categories in the legend follows the column order in the original dataset.
ggmatplot(iris_numeric, plot_type = "density", alpha = 0)
Suppose we also have the following dataset of the monthly totals of international airline passengers(in thousands) from January 1949 to December 1960.
AirPassengers <- matrix(AirPassengers,
ncol = 12, byrow = FALSE,
dimnames = list(month.abb, as.character(1949:1960))
)
AirPassengers
#> 1949 1950 1951 1952 1953 1954 1955 1956 1957 1958 1959 1960
#> Jan 112 115 145 171 196 204 242 284 315 340 360 417
#> Feb 118 126 150 180 196 188 233 277 301 318 342 391
#> Mar 132 141 178 193 236 235 267 317 356 362 406 419
#> Apr 129 135 163 181 235 227 269 313 348 348 396 461
#> May 121 125 172 183 229 234 270 318 355 363 420 472
#> Jun 135 149 178 218 243 264 315 374 422 435 472 535
#> Jul 148 170 199 230 264 302 364 413 465 491 548 622
#> Aug 148 170 199 242 272 293 347 405 467 505 559 606
#> Sep 136 158 184 209 237 259 312 355 404 404 463 508
#> Oct 119 133 162 191 211 229 274 306 347 359 407 461
#> Nov 104 114 146 172 180 203 237 271 305 310 362 390
#> Dec 118 140 166 194 201 229 278 306 336 337 405 432
If we want to plot the trend of the number of passengers over the years using ggplot2, we’d have to wrangle the data into long format. But we can use ggmatplot as a workaround.
First, we can split the data into two matrices as follows:
months: a vector containing the list of months
nPassengers: a matrix of passenger numbers with each column representing a year
months <- rownames(AirPassengers)
nPassengers <- AirPassengers[, 1:12]
Then we can use ggmatplot() to plot the months matrix against each column of the nPassengers matrix - which can be more simply understood as grouping the plot using each column(year) of the nPassengers matrix.
ggmatplot(
x = months,
y = nPassengers,
plot_type = "line",
size = 1,
legend_label = c(1949:1960),
xlab = "Month",
ylab = "Total airline passengers (in thousands)",
legend_title = "Year"
) +
theme_minimal()
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# Number sequences: 6X000X9, 700XX08,00000015,?,00000015
Here is the question:
6X000X9，700XX08，00000015，？，00000015，？，00000015
This question actually comes from the puzzles made by my friend. He gave me his puzzles recently but I couldn't figure it out. Any clues?
Hint1
For question 10, odd terms are comprised of 1234567890
Hint2
For question 11, the first number is 6,7,8,9,10
Hint3
For question 13, odd terms are comprised of 123456
• Ah btw, do you need us to solve the first $2$ questions mentioned or all of those questions in the image? – athin May 17 '19 at 5:58
• Any correct answer to any of the questions will be accepted. – LETTERKING May 29 '19 at 13:26
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# Elite and Periphery in Social Networks: An Axiomatic Approach
מארגן/מארגנים:
Usual Time:
מיקום:
Many societies exhibit a two-tier structure composed of a social elite, namely, a relatively small but well-connected and highly influential group of powerful individuals, and the rest of society (referred to hereafter as the periphery). The talk will concern understanding the structure, size and properties of the elite and the powers that shape it, using an axiom-based model for the relationships between the elite and the periphery.
(Joint work with Chen Avin, Zvi Lotker, Yvonne-Anne Pignolet, and Itzik Turkel)
### Previous Lectures
Esther Ezra - Georgia Institute of Technology
24/12/2015 - 12:00
Discrepancy theory has been developed into a diverse and fascinating field, with numerous closely related areas. In this talk, I will survey several classic results in combinatorial and geometric discrepancy and then present discrepancy bounds of two kinds, the first is a size-sensitive bound for geometric set systems (that is, the sets are realized by simply-shaped regions, as halfspaces and balls in d-dimensions), and the second is a discrepancy bound for abstract set systems of low degree, which is a problem motivated by the Beck-Fiala conjecture. Both bounds are nearly-optimal. For the first bound our analysis exploits the so-called "entropy method" and "partial coloring", combined with the existence of small packing numbers, and for the second bound our mechanism exploits the Lovasz Local Lemma.
I will also present the notion of relative approximations, initially introduced in learning theory, and their relation to discrepancy, and will conclude with their applications to approximate range search, geometric optimization and machine learning.
Yair Zick - Carnegie Mellon University
23/12/2015 - 12:00
Cooperative game theory spans the formation of coalitions among collaborative agents, as well as proposing reasonable payoff divisions among them. This branch of game theory is rooted in Von-Neumann & Morgenstern’s foundational work, with many beautiful theoretical ideas; however, it has seen relatively sparse application. In this talk, I will discuss several research thrusts which aim at making the theory of cooperative games more applicable; I will discuss how the introduction of overlapping coalition structures – i.e. allowing agents to divide their resources among more than one coalition – allows one to model complex agent interaction.
Moreover, I will show how one can overcome the computational challenges traditionally associated with finding cooperative solution concepts by relaxing our requirements. By looking for a probably approximately correct (PAC) solution, and applying ideas from computational learning theory, one can find good solutions to cooperative games while eliminating computational overhead.
Finally, I will discuss exciting directions for the study of cooperative games, both in the application of the theory to causality and classification, and in empirical human trials
Bio:
Yair Zick is a postdoctoral research fellow in the computer science department at Carnegie Mellon University. He has completed his PhD at Nanyang Technological University, SPMS (funded by the Singapore A*STAR SINGA award). He received his B.Sc (Mathematics and the "Amirim" honors program) from the Hebrew University of Jerusalem. His research interests include game theory, fair division and their applications to domains such as machine learning, security, and privacy.
He is the recipient of the 2014 IFAAMAS Victor Lesser Distinguished Dissertation Award, and the 2011 Pragnesh Jay Modi Best Student Paper Award.
Dr. Harry Halpin
10/12/2015 - 10:25 - 11/12/2015 - 10:25
For decades, the Web did not support fundamental cryptography primitives for applications, and user authentication has been limited to user-name passwords. Due to the work of the W3C Web Cryptography Working Group this year and new efforts around Web Authentication next year, the Web has begun evolving to a more secure platform for application development. The Web Cryptography API provides a standardized Javascript API currently supported across browsers that supports fundamental primitives ranging from PNRG access to key derivation functions. The proposed FIDO 2.0 standard to replace passwords with one-factor cryptographic authentication has already gained widespread industry support from Google and Microsoft, and should be an open standard next year. We'll overview how these new Web-level standards interact with the larger standards-based eco-system, including new developments on the TLS level at the IETF, re-visiting the debates over recommended NIST curves in CFRG, and work in encrypted and privacy-preserving messaging standards.
David Peleg
03/12/2015 - 12:00
Many societies exhibit a two-tier structure composed of a social elite, namely, a relatively small but well-connected and highly influential group of powerful individuals, and the rest of society (referred to hereafter as the periphery). The talk will concern understanding the structure, size and properties of the elite and the powers that shape it, using an axiom-based model for the relationships between the elite and the periphery.
(Joint work with Chen Avin, Zvi Lotker, Yvonne-Anne Pignolet, and Itzik Turkel)
Moni Naor
26/11/2015 - 12:00 - 13:00
.
Many efficient data structures use randomness, allowing them to improve upon deterministic ones. Usually, their efficiency or correctness are analyzed using probabilistic tools under the assumption that the inputs and queries are independent of the internal randomness of the data structure.
In this talk, we will consider data structures in a more robust model, which we call the adversarial model. Roughly speaking, this model allows an adversary to choose inputs and queries adaptively according to previous responses. Specifically, we will consider Bloom filters, a data structure for approximate set membership, and prove a tight connection between Bloom filters in this model and cryptography.
Joint work with Eylon Yogev
Sivan Toledo, Tel-Aviv University
19/11/2015 - 12:00 - 13:00
Over the past three and a half years, we have been designing, implementing, and deploying a wildlife tracking system using the reverse-GPS or time-of-arrival principle. The system estimates the location of wild animals from arrival-time estimates of signals emitted by miniature radio transmitters attached to the animals. The talk will briefly describe the system but will focus mainly on the algorithms that it uses. I will describe the algorithms, some of which are new, the assumptions that underlie them, and how the assumptions and the algorithms affect the accuracy of the system and its performance. The accuracy evaluation is completely realistic; it is based on both stationary transmitters at known positions and on transmitters attached to stationary wild animals (Owls, which are stationary during the day).
This is joint work with Adi Weller-Weiser, Ran Nathan, Yotam Orchan, Tony Weiss, and others.
Stephen Alstrup from University of Copenhagen
12/11/2015 - 11:50 - 13:00
This week we will start 10 minute earlier.
A labeling scheme assign labels to a node in a graph. Given the labels from two nodes and nothing else one should be able to compute interesting information. We will in this talk look at distance and adjacency queries in different kind of graphs.
Nir Ailon
05/11/2015 - 12:00 - 13:00
.Finding the complexity of the n-dimensional Fourier transform computation has been an evasive problem for over 50+ years. The FFT (Fast Fourier Transform) of Cooley and Tukey (1965) gives an upper bound of O(n log n), but nothing better than Omega(n) has been known in a reasonable model of computation. In the talk I will show that speedup of Fourier computation implies loss of accuracy on a machine using words of fixed size (e.g. 32 or 64 bits). In order to recover the accuracy, one must work with words of size \omega(1), giving rise to an effective lower bound of Omega(n log log n).
Shai Ben-David from University of Waterloo
04/11/2015 - 14:00 - 15:30
Clustering is an area of huge practical relevance but rather meager theoretical foundations.
I will discuss some fundamental challenges that a theory of clustering needs to face. In particular, I'll describe two different approaches to addressing those challenges;
an axiomatic approach and a statistical/machine-learning perspective.
If time permits, I will also discuss the computational complexity of some common clustering formulations - another area in which our theoretical understanding lags far behind the practical scene.
I will outline recent progress made along these directions, as well as allude to some common misconceptions and potential pitfalls. The talk is geared towards stimulating discussions and highlighting open questions more than to providing answers or boasting definitive results.
Micha Sharir, Tel Aviv University
29/10/2015 - 12:00 - 13:00
I will survey the immense progress in combinatorial and computational geometry in the past seven years, triggered by the infusion of techniques from algebraic geometry and algebra, as introduced by Guth and Katz and further developed by the community at large.
This has led to solutions of very hard problems, with the crown jewel being a nearly complete solution to Erdos's 1946 problem on distinct distances in the plane.
In this talk I will survey the recent developments. They include new bounds for other variants of the distinct distances problem, new bounds for incidences between points and lines or other geometric entites in various contexts, and re-examination of the theory of Elekes, Ronyai, and Szabo on polynomials vanishing on grids, and numerous applications thereof.
In the (short) time that I will have I will only highlight some of the key developments, and demonstrate the new approach by a few examples.
The talk might assume some basic knowledge in algebra and geometry. Passion for geometric problems is always a plus.
Nethanel Gelernter, from Bar-Ilan
08/10/2015 - 17:00
למעוניינים להגיע: הקולוקוויום יתקיים ביום חמישי 8/10/15 בשעה 17:00 בבניין 211 - הפקולטה למדעים מדוייקים בחדר הסמינרים.
Nethanel Gelernter, from Bar-Ilan, is going to give a talk.
Title: Cross-site Search Attacks
Abstract: Cross-site search (XS-search) attacks circumvent the same-origin policy and extract sensitive information, by using the time it takes for the browser to receive responses to search queries. This side-channel is usually considered impractical, due to the limited attack duration and high variability of delays. This may be true for naive XS-search attacks; however, we show that the use of better tools facilitates effective XS-search attacks, exposing information efficiently and precisely.
We present and evaluate three types of tools: (1) appropriate statistical tests, (2) amplification of the timing side-channel, by ‘inflating’ communication or computation, and (3) optimized, tailored divide-and-conquer algorithms, to identify terms from large ‘dictionaries’. These techniques may be applicable in other scenarios. We implemented and evaluated the attacks against the popular Gmail and Bing services, in several environments and ethical experiments, taking careful, IRB-approved measures to avoid exposure of personal information.
Joint work with Amir Herzberg; to be presented in ACM CCS'2015.
Prof. Haim J. Wolfson, Blavatnik School of Computer Science, Tel Aviv University
25/06/2015 - 12:00
Modeling large multi-molecular assemblies at atomic resolution is a key task in elucidating cell function. Since, there is no single experimental method, that can deliver atomic resolution structures of such large molecules, hybrid methods, which integrate data from various experimental modalities, are being developed for this task.
We have developed a new integrative method, which combines atomic resolution models of individual assembly components with an electron microscopy map of the full assembly. It can also naturally accommodate available chemical cross link (Xlink) data.
Specifically, the input to our algorithm is an intermediate resolution (6-10 Angstrom) electron density map of the full assembly, atomic resolution (2 A0) maps of the individual assembly subunits, and, if available, cross link information between some residues of neighboring subunits (an Xlink can be visualized as a loose ~30A0 string connecting two atoms on the surfaces of neighboring subunits). The output is an atomic resolution map of the whole assembly.
The algorithm was highly successful and efficient on all the intermediate resolution EM complexes from the 2010 Cryo-EM Modeling Challenge. Remarkably, a 6.8A0 resolution 20S proteasome map, consisting of 28 (structurally homologues) units was modeled at 1.5A0 RMSD from native in about 10 minutes on a Core i7 laptop. In case of missing (or poorly modeled) individual subunits, the method can return partial solutions, thus, enabling interactive modeling.
From a purely geometric viewpoint, the task can be viewed as an assembly of a large multiple piece puzzle, where we have relatively accurate models of the individual subunits, and a rough, low resolution scan of the full puzzle volume.
This is joint work with my M.Sc. students Dan Cohen and Naama Amir.
יוגש כיבוד קל
Yair Zick
18/06/2015 - 12:00
A dataset has been classified by some unknown classifier into two types of points. What were the most important factors in determining the classification outcome? In this work, we employ an axiomatic approach in order to uniquely characterize an influence measure: a function that, given a set of classified points, outputs a value for each feature corresponding to its influence in determining the classification outcome. We discuss the relation between our influence measure and causality: showing that we in fact measure the expected (counterfactual) responsibility of a feature on the classification outcome. We show that our influence measure takes on an intuitive form when the unknown classifier is linear. Finally, we employ our influence measure in order to analyze the effects of user profiling on Google’s online display advertising.
Rina Dechter, Bren School of Information and Computer Sciences, UC Irvine
17/06/2015 - 12:00
In this talk I will present several principles behind state of the art algorithms for solving combinatorial optimization tasks defined over graphical models (Bayesian networks, Markov networks, constraint networks, satifiability) and demonstrate their performance on some benchmarks.
Specifically I will present branch and bound search algorithms which explore the AND/OR search space over graphical models and thus exploit problem’s decomposition (using AND nodes), equivalence (by caching) and pruning irrelevant subspaces via the power of bounding heuristics. In particular I will show how the two ideas of mini-bucket partitioning which relaxes the input problem using node duplication only, combined with linear programming relaxations ideas which optimize cost-shifting/re-parameterization schemes, can yield tight bounding heuristic information within systematic, anytime, search.
Notably, a solver for finding the most probable explanation (MPE or MAP), embedding these principles has won first place in all time categories in the 2012 PASCAL2 approximate inference challenge, and first or second place in the UAI-2014 competitions
Parts of this work were done jointly with: Radu Marinescu, Robesrt Mateescu, Lars Otten, Alex Ihler, Natalia Flerova and Kalev Kask
Bio
Rina Dechter is a professor of Computer Science at the University of California, Irvine. She received her
PhD in Computer Science at UCLA in 1985, an MS degree in Applied Mathematics from the Weizmann Institute and a B.S in Mathematics and Statistics from the Hebrew University, Jerusalem. Her research centers on computational aspects of automated reasoning and knowledge representation including search, constraint processing and probabilistic reasoning.
Professor Dechter is an author of ‘Constraint Processing’ published by Morgan Kaufmann, 2003, and ‘Reasoning with Probabilistic and Deterministic Graphical Models: Exact Algorithms’ by Morgan and Claypool publishers, 2013, has authored over 150 research papers, and has served on the editorial boards of: Artificial Intelligence, the Constraint Journal, Journal of Artificial Intelligence Research and journal of Machine Learning (JLMR). She was awarded the Presidential Young investigator award in 1991, is a fellow of the American association of Artificial Intelligence since 1994, was a Radcliffe Fellowship 2005-2006, received the 2007 Association of Constraint Programming (ACP) research excellence award and is a 2013 Fellow of the ACM. She has been Co-Editor-in-Chief of Artificial Intelligence, since 2011.
Building 202, room 104
09/06/2015 - 12:30 - 13:30
Cyber security
Bio:Brigadier General (res.) Nadav Zafrir - Cyber and intelligence expert, founder of Team 8. Former Commander of IDF’s technolog &intelligence unit (8200), and founder of the IDF Cyber Command.
Building 305 room 10
Prof. Noga Alon from Tel Aviv University
04/06/2015 - 12:00
The sign-rank of a real matrix A with no 0 entries is the minimum rank of a matrix B so that A_{ij}B_{ij} >0 for all i,j. The study of this notion combines combinatorial, algebraic, geometric and probabilistic techniques with tools from real algebraic geometry, and is related to questions in Communication Complexity, Computational Learning and Asymptotic Enumeration. I will discuss the topic and describe its background, several recent results from joint work with Moran and Yehudayoff, and some intriguing open problems.
31/05/2015 - 14:00
This talk describes 3 different scenarios in which systems can be designed to be robust despite malicious participants. The scenarios are very different and have very different defenses. One is how to build trust chains of certificates considering that some CAs might be untrustworthy. Another is how to build a network that guarantees that two nodes can communicate, with a fair share of bandwidth, even if some switches are malicious (e.g., giving false information in the distributed routing algorithm, flooding the network with garbage traffic, mis-forwarding traffic, or throwing away traffic from one source). The third is a way to give a a data item an expiration date, after which it is unrecoverable from a storage system. Although this might seem unrelated to the topic, it really is related. Trust me.
Prof. Ron Artstein from University of Southern California, Los Angeles, California, USA
28/05/2015 - 12:00
Time-offset interaction is a new technology that enables conversational interaction with a person who is not present, using pre-recorded video statements: a large set of statements are prepared in advance, and users access these statements through natural conversation that mimics face-to-face interaction. The speaker's reactions to user questions are selected by a statistical classifier, using technology that is similar to interactive systems with synthetic characters; recordings of answers, listening and idle behaviors, and blending techniques are used to create a persistent visual image of the speaker throughout the interaction.
A preliminary working system of time-offset interaction has been created with statements recorded by Pinchas Gutter, a Holocaust survivor, talking about his personal experiences before, during and after the Holocaust. Statements were recorded in two rounds; user questions were collected between the rounds through a “Wizard of Oz” system, where live operators select an appropriate reaction to each user utterance in real time. The result of this process is that 95% of newly elicited user questions are addressed by the recorded statements. The automated conversational system is presently undergoing beta testing in a museum. Time-offset interaction will allow future generations to experience a face-to-face conversation with a Holocaust survivor.
This talk will present the concept of time-offset interaction, the underlying language processing technologies, and the process of content elicitation needed to ensure robust coverage. The talk will include a live demonstration of time-offset interaction.
Shlomo Ahal from Istra research
21/05/2015 - 12:00
Micro-structure studies the dynamics of trading using synthesized toy models. We will present few basic problems and corresponding models. We analyze them and explore the trading features they reveal. Finally, we will examine their fits and predictions to real life trading.
Shlomo Ahal together with Michael Levin founded Istra research. Before that they founded Kashya and sold it to EMC.
Dr. Guy Leshem from the Ben Gurion University
14/05/2015 - 12:00
The Evolutionary-Random-Forest algorithm, basically chooses the best trees from the population, and then creates new, more accurate “child” trees from those “parents”. In effect, this creates a new population of trees with enhanced classification accuracy. The predictions are carried out using a small population of enhanced-accuracy “child” trees, thus increasing the prediction algorithm speed and accuracy.
Prof. Nati Linial from the Hebrew University
07/05/2015 - 12:00
There are numerous situations where we study a large complex system whose behavior is determined by pairwise interactions among its constituents. These can be interacting molecules, companies involved in some business, humans etc. This explains, at least partially, why graph theory is so ubiquitous wherever mathematics is applied in the real world. However, there are numerous interesting and important situations where the underlying interactions involve more than two constituents. This applies in all the above-mentioned scenarios. The obvious place to look for a relevant mathematical theory is hypergraph theory. Unfortunately, this part of combinatorics is not nearly as well understood as graph theory. In this talk I will speak about a special class of hypergraphs, namely, simplicial complexes. It turns out that there is a fascinating combinatorial theory of simplicial complexes that is rapidly developing in recent years. In this lecture I explain some of the exciting recent findings in this area.
My coworkers in these investigations are (historical order): Roy Meshulam, Mishael Rosenthal, Lior Aronshtam, Tomasz Luczak, Yuval Peled, Yuri Rabinovich and Ilan Newman.
30/04/2015 - 12:00
Yossi is a recipient of Gödel Prize and is an ACM Fellow for contributions to the analysis of big data and the field of streaming algorithms
Yossi is the only Google VP outside of the US.
Eli Ben-Sasson from the Technion
26/03/2015 - 12:00
ROOM CHANGE!!
This week colloquium will take place in Building 507 room 107.
Bitcoin is the first digital currency to see widespread adoption. While payments are conducted between pseudonyms, Bitcoin cannot offer strong privacy guarantees: payment transactions are recorded in a public decentralized ledger, from which much information can be deduced. Zerocoin (Miers et al., IEEE S&P 2013) tackles some of these privacy issues by unlinking transactions from the payment's origin. Yet, it still reveals payments' destinations and amounts, and is limited in functionality.
In this paper, we construct a full-fledged ledger-based digital currency with strong privacy guarantees. Our results leverage recent advances in zero-knowledge Succinct Non-interactive ARguments of Knowledge (zk-SNARKs).
First, we formulate and construct decentralized anonymous payment schemes (DAP schemes). A DAP scheme enables users to directly pay each other privately: the corresponding transaction hides the payment's origin, destination, and transferred amount. We provide formal definitions and proofs of the construction's security.
Second, we build Zerocash, a practical instantiation of our DAP scheme construction. In Zerocash, transactions are less than 1 kB and take under 6 ms to verify --- orders of magnitude more efficient than the less-anonymous Zerocoin and competitive with plain Bitcoin.
Joint work with Alessandro Chiesa, Christina Garman, Matthew Green, Ian Miers, Eran Tromer, and Madars Virza
Eugene Agichtein from Haifa Yahoo Labs
19/03/2015 - 12:00
A long standing challenge in Web search is how to accurately determine
the intention behind a searcher’s query, which is needed to rank,
organize, and present information most effectively. The difficulty is
that users often do not (or cannot) provide sufficient information
about their search goals. As this talk with show, it is nevertheless
possible to read their intentions through clues revealed by behavior,
such as the amount of attention paid to a document or a text fragment.
I will overview the approaches that have emerged for acquiring and
mining behavioral data for inferring search intent, ranging from
contextualizing query interpretation and suggestion, to modeling
fine-grained user interactions such as mouse cursor movements in the
searcher’s browser. The latter can also be used to measure the
searcher’s attention “in the wild’’, with granularity approaching that
of using eye tracking equipment in the laboratory. The resulting
techniques and models have already shown noteworthy improvements for
search tasks such as ranking, relevance estimation, and result summary
generation, and have applications to other domains, such as
psychology, neurology, and online education.
Biosketch:
Eugene is a Principal Research Scientist at Yahoo Labs in Haifa, on
leave from Emory University, where he is an Associate Professor and
founder of the IR Laboratory. Eugene's research spans the areas of
information retrieval, data mining, and human computer interaction. At
Yahoo Labs, he works on the Answering Research team trying to
automatically answer questions from millions of searchers. Dr.
Agichtein is actively involved in the international research
community, having co-authored over 100 publications (including 4 best
paper awards), co-chaired the ACM WSDM 2012 conference (with Yoelle
Maarek), and served on the program or organizing committees of all the
main information-retrieval and web search conferences.
Frantisek Franek McMaster University, Hamilton, Canada
12/03/2015 - 12:00 - 13:00
Counting the number of types of squares rather than their occurrences, we consider
the problem of bounding the maximum number of distinct squares in a string. Fraenkel and
Simpson showed in 1998 that a string of length n contains at most 2n distinct squares.
Ilie presented in 2007 an asymptotic upper bound of 2n(log n). We show that a string
of length n contains at most b11n=6c distinct squares. This new upper bound is obtained
by investigating the combinatorial structure of double squares and showing that a string of
length n contains at most b5n=6c double squares. In addition, the established structural
properties provide a novel proof of Fraenkel and Simpson's result. Joint work with Antoine
Karen Livescu (http://ttic.uchicago.edu/~klivescu/), a faculty member of TTIC and University of Chicago
02/02/2015 - 10:30
Many types of multi-dimensional data have a natural division into two "views", such as audio and video or images and text. Multi-view learning includes a variety of techniques that use multiple views of data to learn improved models for each of the views. The views can be multiple measurement modalities (like the examples above) but also can be different types of information extracted from the same source (words + context, document text + links) or any division of the data dimensions into subsets satisfying certain learning assumptions. Theoretical and empirical results show that multi-view techniques can improve over single-view ones in certain settings. In many cases multiple views help by reducing noise in some sense. In this talk, I will focus on multi-view learning of representations (features), especially using canonical correlation analysis (CCA) and related techniques. I will give an overview of CCA and its relationship with other techniques such as partial least squares (PLS) and linear discriminant analysis (LDA). I will also present extensions developed by ourselves and others, such as kernel, deep, and generalized ("many-view") CCA. Finally, I will give recent results on speech and language tasks, and demonstrate our publicly available code.
Based on joint work with Raman Arora, Weiran Wang, Jeff Bilmes, Galen Andrew, and others.
Yotam Harchol from the Hebrew University
29/01/2015 - 12:00
The nearest-neighbor search (NN) problem is widely used in many fields of computer science such as machine learning, computer vision and databases. Given a database of points in R^d and a query point, NN search tries to obtain the data point closest to the query point under some metric distance. However, in many settings such searches are known to suffer from the notorious curse of dimensionality, where running time grows exponentially with d. This causes severe performance degradation when working in high-dimensional spaces.
In this talk we propose a new way to solve this problem using a special hardware device called ternary content addressable memory (TCAM). TCAM is an associative memory, which is a special type of computer memory that is widely used in switches and routers for very high speed search applications. We show that the TCAM computational model can be leveraged and adjusted to solve NN search problems in a single TCAM lookup cycle, and with linear space. This concept does not suffer from the curse of dimensionality and is shown to improve the best known approaches for NN by more than four orders of magnitude. Simulation results demonstrate dramatic improvement over the best known approaches for NN, and suggest that TCAM devices may play a critical role in future large-scale databases and cloud applications.
Our approach relies on efficient range encoding on TCAM, which is a known problem in the field of packet classification. We also show how our encoding technique is useful for applications from this field and provide theoretical analysis to show that it is the closest to the lower bound of number of bits used.
Joint work with Anat Bremler-Barr (IDC), David Hay (HUJI), and Yacov Hel-Or (IDC).
Oren Tsur from Harvard
22/01/2015 - 12:00
The dynamic process in which agents compete for the public attention is of great interest for computer scientist as well as for political scientists, communication scholars and marketing experts. In this talk I will present a novel framework for unsupervised analysis of these dynamic processes. In the spirit of the time, I’ll focus on the political domain showing how we can use probabilistic topic models combined with sentiment analysis and time series regression analysis (autoregressive-distributed-lag models) to gain insights about the political processes, including partisan attention, topic ownership and coordinated campaigns for agenda setting and attention shifts, commonly known as ‘spin’. This approach can be applied in other similar settings in which agents compete for the public attention, either promoting ideology or a commercial product.
Short Bio:
Oren Tsur is a postdoctoral fellow at Harvard University (SEAS & IQSS) jointly with Lazer's lab at Northeastern University. He earned his PhD. in Computer Science from the Hebrew University and his research combines Natural Language Processing and Network Science. Oren received the 2014 NSF fellowship for research of Political Networks.
15 minutes of fame: Our work on sarcasm detection was selected by Time Magazine as one of the
50 Best Inventions of 2010. (Here is pop sci talk [HEB])
Homepage:
http://people.seas.harvard.edu/~orentsur/
Mary Wootter from Carnegie Mellon University
15/01/2015 - 12:00
Abstract:In the list-decoding problem, a sender (Alice) sends a message to a receiver (Bob) over a noisy channel, and Bob's job is to come up with a short list of messages with the guarantee that Alice's true message appears in the list. It turns out that Alice and Bob can succeed even when nearly all of the symbols that Alice sends to Bob are corrupted. It is known (and not hard to see) that a completely random code will achieve this, and it is also known (and much harder) that there are explicit codes which achieve this. In this talk, we'll explore the middle-ground of structured random codes: for example, random linear codes, or Reed-Solomon codes with random evaluation points. Our main result is that if you start with any q-ary code with sufficiently good distance and randomly puncture it, then with high probability, you get a code that is list-decodable up to a 1 - 1/q - epsilon fraction of errors, with near-optimal rate and list sizes. Our results imply that "most" Reed-Solomon codes are list-decodable beyond the Johnson bound; whether or not such codes even existed was previously open. Our results also imply improved bounds on the list-decodability of random linear codes over large alphabets. This is joint work with Atri Rudra.
Edo Liberty
08/01/2015 - 12:00
Many operations in scientific computing rely on operations on large matrices. These include clustering, signal denoising, regression, dimension reduction and many more. If these matrices are too large to hold in memory (or on disk), one must operate on their sketches or approximations as surrogates. In this talk I will survey the progression of ideas, results and techniques developed in over a decade of research into this important problem.
Bio:
Edo Liberty is a Research Director at Yahoo Labs and manages Yahoo's Scalable Machine Learning group in New York. He received his BSc from Tel Aviv university and his PhD in Computer Science from Yale university, where he also held a postdoctoral position in the Program in Applied Mathematics.
http://www.cs.yale.edu/homes/el327/
Shay Solomon
05/01/2015 - 10:30
Managing the valuable data of large-scale networks is among the most fundamental challenges in computer science.
Whenever the data involves network distances, one may use a {spanner},
which provides a good approximation for all the pairwise distances using a small (ideally, a linear) number of network links.
{Geometric spanners} are spanners for networks induced by points in the plane, or more generally, by arbitrary Euclidean spaces.
Such spanners have been intensively studied since the mid-80s, with applications ranging from compact routing and distance oracles to robotics and machine learning.
In this talk I will discuss novel non-geometric techniques to geometric spanners, and demonstrate their effectiveness in solving several long-standing open questions.
The main message of my talk is somewhat surprising -- the right approach to geometric spanners is often non-geometric
תתקיים בחדר הפקולטה למדעים מדויקים
Moran Feldman
30/12/2014 - 14:00
Submodular functions form a large natural class of set functions with applications in many fields including social networks, machine learning and game theory. Optimization of submodular functions subject to various constraints attracted much attention in recent years, both from theoretical and practical points of view. This talk considers the problem of maximizing a submodular function subject to a matroid constraint, which is a central problem demonstrating many of the modern approaches and techniques used in the field of submodular maximization. Many aspects of this problem have been studied, including its polynomial time approximability, fast (almost linear time) algorithms and online models. This talk surveys some of these aspects and explores a few of the main results obtained recently.
Yael Amsterdamer
28/12/2014 - 12:00
Crowd Mining
Crowd data sourcing is increasingly used to gather information from the crowd for different purposes. In this talk, I will present a novel approach for crowd data sourcing that enables users to pose general questions to the system, which in turn mines the crowd for potentially relevant data. The retrieved answers are concise and represent frequent, significant data patterns. Our approach, termed crowd mining, utilizes an ontological knowledge base as well as virtual databases representing the knowledge of crowd members. A formal query language is used to declaratively specify patterns of interest to be mined from these knowledge sources. Queries are evaluated by an efficient engine that employs means of statistical modeling and semantic inference, for minimizing the effort of the crowd and the probability of error. I will present the theoretical analysis of the algorithm underlying the query evaluation engine, as well as its practical implementation.
Amit Daniely - The Hebrew University
18/12/2014 - 12:00
Machine learning is the backbone of a large array of fields, including Vision, Speech Recognition, NLP, Computational Biology and many more. Despite 30 years of intensive research learning is still a great mystery, forming a major challenge to theoretical computer science and beyond.
I will survey learning theory as we know it today, addressing both its statistical and computational aspects, and highlighting new and surprising results:
1) I will show that statistically optimal algorithms are surprisingly very different than what was believed since the 70's.
2) I will present a new methodology to prove computational hardness of learning. It is used to prove, under a certain complexity assumption, the hardness of many basic learning problems. Prior to that, and despite 30 years of research, there were huge gaps between the performance of known algorithms and hardness results.
The last result provides evidence that, from nowadays perspective, learning problems are extremely hard. Hardness of learning stands in a sharp contrast with the abilities of computers and humans to learn. I will briefly discuss possible avenues to develop a theory that overcomes this discrepancy.
Based on joint works with Nati Linial, Shai Shalev-Shwartz, Shai Ben-David, Yonatan Bilu, Sivan Sabato, and Mike Saks
כיבוד קל יוגש בשעה 11:30 לכבוד חנוכה .
Dan Garber, Technion
11/12/2014 - 12:00
Principal Component Analysis is a widely used pre-processing technique in machine learning for reducing the dimension of the data and cleaning noise.
In standard PCA, the input to the problem is a set of d dimensional vectors x_1,... x_n and a target dimension k < d; the output is a set of k dimensional vectors y_1,..., y_n that best capture the top singular directions of the original vectors.
In the online setting, the vectors x_t are presented to the algorithm one by one, and for every presented x_t the algorithm must output a low-dimensional vector y_t before receiving x_{t+1}. This setting is interesting for instance in case that the PCA is part of an online learning pipeline and the low-dimensional vectors y_t are fed to an online learning algorithm.
We present the first approximation algorithms for this setting of online PCA. Our algorithm produces vectors of dimension k * poly(1/\epsilon) whose quality admit an additive \epsilon approximation to the optimal offline solution allowed to use k dimensions.
Uriel Feige, Weizmann Institute
04/12/2014 - 12:00
We consider sequential decision making in a setting where regret is measured with respect to a set of stateful reference policies, and feedback is limited to observing the rewards of the actions performed (the so called "bandit" setting). If either the reference policies are stateless rather than stateful, or the feedback includes the rewards of all actions (the so called "expert" setting), previous work shows that the optimal regret grows like O(\sqrt(T)) in terms of the number of decision rounds T.
The difficulty in our setting is that the decision maker unavoidably loses track of the internal states of the reference policies, and thus cannot reliably attribute rewards observed in a certain round to any of the reference policies. In fact, in this setting it is impossible for the algorithm to estimate which policy gives the highest (or even approximately highest) total reward. Nevertheless, we design an algorithm that achieves expected regret that is sublinear in T, of the form O(T/polylog T). Our algorithm is based on a certain local repetition lemma that may be of independent interest. We also show that no algorithm can guarantee expected regret better than O(T/polylog T).
Joint work with Tomer Koren and Moshe Tennenholtz
27/11/2014 - 12:00
Handling increasingly large datasets is one of the major problems faced by machine learning today. One approach is to distribute the learning task, and split the data among several machines which can run in parallel. Ideally, a distributed learning algorithm on k machines should provably (1) Run k times faster than an algorithm designed for a single machine; (2) Reach the same statistical learning performance with the same amount of training data; And (3) Use minimal communication between the machines, since it is usually much slow than internal processing. In other words, such an algorithm should combine computational efficiency, statistical efficiency, and communication efficiency. In this talk, I'll survey the challenges of designing such algorithms for convex learning problems, and describe some recent advances as well as fundamental limitations.
Includes joint work with Andrew Cotter, Ofer Dekel, Ran Gilad-Bachrach, Nathan Srebro, Karthik Sridharan, Lin Xiao and Tong Zhang.
Prof. Noam Shental from the open university
20/11/2014 - 12:00
Noam Shental, The department of Computer Science, The Open University of Israel
High resolution microbial profiling
The emergence of massively parallel sequencing technology has revolutionized microbial profiling, allowing the unprecedented comparison of microbial diversity across time and space in a wide range of host-associated and environmental ecosystems. Although the high throughput nature of such methods enables the detection of low frequency bacteria, these advances come at the cost of sequencing read length, limiting the phylogenetic resolution possible by current methods.
We present a generic approach for integrating short reads from large genomic regions, thus enabling phylogenetic resolution far exceeding current methods. The approach is based on a mapping to a statistical model that is later solved as a constrained optimization problem.
We demonstrate the utility of this method by analyzing human saliva and Drosophila samples, using Illumina single-end sequencing of a 750bp amplicon of the 16S rRNA gene. Phylogenetic resolution is significantly extended while reducing the number of falsely detected bacteria, as compared to standard single-region Roche 454 Pyrosequencing.
Our approach can be seamlessly applied to simultaneous sequencing of multiple genes providing a higher resolution view of the composition and activity of complex microbial communities.
Joint work with Amnon Amir, Amit Zeisel, Michael Elgart, Shay Stern, Ohad Shamir and Yoav Soen, from Weizmann Institute of Science, Or Zuk from the Hebrew University and Peter J. Turnbaugh, Harvard University.
References
1. Amnon Amir, Amit Zeisel, Or Zuk, Michael Elgart, Shay Stern, Ohad Shamir, Peter J. Turnbaugh, Yoav Soen and Noam Shental. High resolution microbial community reconstruction by integrating short reads from multiple 16S rRNA regions. Nucleic Acids Research, Nov, 2013
2. Or Zuk, Amnon Amir, Amit Zeisel, Ohad Shamir and Noam Shental, Accurate Profiling of Microbial Communities from Massively Parallel Sequencing Using Convex Optimization. SPIRE, 2013
3. Yael Fridmann-Sirkis, Shay Stern, Michael Elgart, Matana Galili, Amit Zeisel, Noam Shental and Yoav Soen, Delayed development induced by toxicity to the host can be inherited by a bacterial-dependent, transgenerational effect. Frontiers in Genetics, 5:27, Jan, 2014
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13/11/2014 - 12:00
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06/11/2014 - 12:00 - 13:00
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This week colloquium will be given by two researchers from Waze
30/10/2014 - 12:00 - 13:30
Waze
.They are going to tells us how to fix maps and how to estimate time of arrival
Peter Stone from the University of Texas
10/07/2014 - 11:30
Over the past half-century, we have transitioned from a world with just a handful of mainframe computers owned by large corporations, to a world in which private individuals have multiple computers in their
homes, in their cars, in their pockets, and even on their bodies This transition was enabled by computer science research in multiple areas such as systems, networking, programming languages, human computer interaction, and artificial intelligence. We are now in the midst of a similar transition in the area of
robotics. Today, most robots are still found in controlled, industrial settings. However, robots are starting to emerge in the consumer market, and we are rapidly transitioning towards a time when private individuals will have useful robots in their homes, cars, and workplaces. For robots to operate robustly in such dynamic, uncertain environments, we are still in need of multidisciplinary research advances in many areas such as computer vision, tactile sensing, compliant motion, manipulation, locomotion, high-level decision-making, and many others. This talk will focus on two essential capabilities for robust autonomous intelligent robots, namely online learning from experience, and the ability to interact with other robots and with people. Examples of theoretically grounded research in these areas will be highlighted, as well as concrete applications in domains including robot soccer and autonomous driving.
מיקום - בניין 507 כתה 1
Arie Zaban from the Department of Chemistry at Bar Ilan University
19/06/2014 - 12:00
All oxide PV are expected to be very inexpensive, stable and environmentally safe. However, the
electronic properties of most known MOs, i.e. short lifetime of excited electronic states and low
mobility of charge carriers, prevented their use as active solar cell materials. To bring all-oxide
PV to a breakthrough, new materials have to be developed. The prospect of finding these unique
materials lies in combinatorial material science which can produce novel MOs consisting of
two, three, four or more elements. While most binary MOs are known, the number of unknown
compositions is drastically increasing with the number of components. The new materials are
tested in combinatorial PV device libraries to investigate them under PV operating conditions.
Data analysis, storage and handling are organized using a dedicated database. Tools for data
mining are developed to reveal complex correlations between the material properties and the
device performance and to improve our understanding of all-oxide PV device operation. The
methodology, new photoactive MOs and opportunities will be reported.
Arie Zaban, BIU, Chemistry Department
12/06/2014 - 12:00 - 13:00
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All oxide PV are expected to be very inexpensive, stable and environmentally safe. However, the electronic properties of most known MOs, i.e. short lifetime of excited electronic states and low mobility of charge carriers, prevented their use as active solar cell materials. To bring all-oxide PV to a breakthrough, new materials have to be developed. The prospect of finding these unique materials lies in combinatorial material science which can produce novel MOs consisting of two, three, four or more elements. While most binary MOs are known, the number of unknown compositions is drastically increasing with the number of components. The new materials are tested in combinatorial PV device libraries to investigate them under PV operating conditions. Data analysis, storage and handling are organized using a dedicated database. Tools for data mining are developed to reveal complex correlations between the material properties and the device performance and to improve our understanding of all-oxide PV device operation. The methodology, new photoactive MOs and opportunities will be reported.
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12/06/2014 - 12:00
No talk this week
Prof. Doron Peled, BIU
29/05/2014 - 12:00 - 13:00
We show how the use of genetic programming, in combination of model checking and testing, provides a powerful way to synthesize programs. Whereas classical algorithmic synthesis provides alarming high complexity and undecidability results, the genetic approach provides a surprisingly successful heauristics. We describe several versions of a method for synthesizing sequential and concurrent systems. To cope with
the constraints of model checking and of theorem proving, we combine such exhaustive verification methods with testing. We show several
examples where we used our approach to synthesize, improve and correct code.
David M. Mount, University of Maryland
26/05/2014 - 11:00
Nearest neighbor searching is among the most fundamental retrieval problems in geometry. Given a set of points in d-dimensional space, the objective is to preprocess these points so that the closest point to a given query point q can be computed efficiently. Provably efficient solutions are known in only one- and two-dimensional space, and research over two decades has focused on solving the problem approximately.
In this talk I will survey developments in approximate nearest neighbor searching in Euclidean space of constant dimension. Recent work on establishing the best computational bounds has been based on the solution to a completely different problem, namely approximating a convex polytope with a minimum number of facets. We will see how a number of classical concepts from convex geometry can be applied to obtain the best known bounds for approximate nearest neighbor searching.
faculty room (next to the dean office)
David M. Mount University of Maryland
22/05/2014 - 12:00
Suppose you are given a map that subdivides the earth into regions (countries, states, voting districts, and so on). Given your GPS coordinates, you would like to know, as fast as possible, which region you are in. Your goal is to build an efficient data structure for answering such queries.
Suppose there are n regions. Generalizing binary search, an ideal solution uses space O(n) and answers queries in O(log n) time. But what if you know that some regions (such as highly populated cities) are much more likely to be sought than others (such as desolate wastelands). Rather than worst-case time, you want to minimize AVERAGE query time. Here, an ideal solution runs in time proportional to the ENTROPY of the access distribution.
In this talk, we will derive data structures for answering this problem both in 1D and 2D space. We will explain what entropy is and how it is related to this problem. Finally, we will present a simple, randomized algorithm that achieves a running time that is within a constant factor of the entropy bound.
Ronen Basri from the Faculty of Mathematics and Computer Science in Weizmann Institute
15/05/2014 - 12:00
Finding corresponding points between images is challenging, particularly when objects change their pose non-rigidly, in wide-baseline conditions, or when instances of a perceptual category are compared. In this talk I will present an algorithm for finding a geometrically consistent set of point matches between two images. Given a set of candidate matches that may include many outliers, our method seeks the largest subset of these correspondences that can be aligned using a non-rigid deformation that exerts a bounded distortion. I will discuss theoretical guarantees and show experimentally that this algorithm produces excellent results on a number of test sets, in comparison to several state-of-the-art approaches. In the second part of the talk I will introduce a convex framework for problems that involve singular values. Specifically, the framework enables optimization of functionals and constraints expressed in terms of the extremal singular values of matrices. I will show applications of this framework in geometry processing problems such as non-rigid shape registration and computing extremal quasi-conformal maps.
This is joint work with Yaron Lipman, Stav Yagev, Roi Poranne, David Jacobs, Shahar Kovalsky and Noam Aigerman.
David Sarne, BIU
01/05/2014 - 12:00
In many multi-agent systems we find information brokers or information technologies aiming to provide the agents with more information or reduce the cost of acquiring new information. In this talk I will show that better information can hurt: the presence of an information provider, even if the use of her services is optional, can degrade both individual agents' utilities and overall social welfare. The talk will focus on two specific domains: auctions (where the provided information relates to the common value of the auctioned item) and cooperative information gathering (where costly information is shared between the agents). For the first, I'll show that with the information provider in the market, in conflict with classic auction theory, the auctioneer may prefer to limit the number of bidders that participate in the auction and similarly bidders may prefer to have greater competition. Also, bidders' unawareness of the auctioneer's option to purchase the information does not necessarily play into the hands of the auctioneer and, similarly, bidders may sometimes benefit from not knowing that the auctioneer has the option to purchase such information. For cooperative information gathering I'll present three methods for improving overall and individual performance, all based on limiting and constraining information sharing. Along the talk we will also discuss ways that charities could benefit from group buying; and why it makes sense to pay someone to over-price information she wants to sell you
Seth Pettie from the Department of EECS, University of Michigan Ann Arbor
24/04/2014 - 12:00
The 3SUM problem is to decide, given a set of N real numbers, whether
any three of them sum to zero. A simple algorithm solves 3SUM in
O(N^2) time and it has long been conjectured that this is the best
possible.
The significance of the 3SUM problem does not lie with its practical
applications (roughly zero) but with the long list of problems in
computational geometry that are reducible from 3SUM. Some examples
include deciding whether a point set contains 3 colinear points,
calculating the area of the union of a set of triangles, and
determining whether one convex polygon can be placed within another
convex polygon. If 3SUM requires N^2 time then all 3SUM-hard problems
require N^2 time. More recently Patrascu gave many conditional lower
bounds on triangle enumeration and dynamic data structures that rely
on the hardness of 3SUM over the integers.
In this talk I'll present a non-uniform (decision-tree) algorithm for
3SUM that performs only N^{3/2} comparisons and a uniform 3SUM
algorithm running in O(N^2/polylog(N)) time. This result refutes the
3SUM conjecture and casts serious doubts on the optimality of many
O(N^2) geometric algorithms.
This is joint work with Allan Gronlund. A manuscript is available at
arXiv:1404.0799.
Nir Ailon from the Computer Science faculty of the Technion
03/04/2014 - 12:00
Obtaining a non-trivial (super-linear) lower bound for computation of
the Fourier transform in the linear circuit model has been a long
standing open problem for over 40 years.
An early result by Morgenstern from 1973, provides an $\Omega(n \log n)$
lower bound for the UNNORMALIZED Fourier transform when the constants
used in the computation are bounded. The proof uses a potential function
related to a determinant. The determinant of the unnormalized Fourier
transform is $n^{n/2}$, and thus by showing that it can grow by at most
a constant factor after each step yields the result. This classic
result does not explain why the NORMALIZED Fourier transform (of unit
determinant) should require $\Omega(n\log n)$ steps.
More recently, Ailon (2013) showed that if only unitary 2-by-2 gates are
used, one obtains an $\Omega(n\log n)$ lower bound, using a notion of
matrix entropy. In this work we take another step and show an
$\Omega(n\log n)$ lower bound for computing the Fourier transform, under
ANY scaling, in a model allowing the gates to perform any nonsingular
(not necessarily unitary) transformation involving 2 variables. Our
restriction is that the composition of all gates up to any given step is
uniformly well conditioned. This is a practical requirement because an
ill conditioned transformation introduces numerical instability.
The main technical contribution is extension of the matrix entropy used
in Ailon (2013) to "negative probabilities".
Yacov Hel-Or, IDC
20/03/2014 - 12:00
A fast pattern matching scheme termed Matching by Tone Mapping (MTM) is introduced which allows matching
under non-linear tone mappings. We show that, when tone mapping is approximated by a piecewise constant/linear function, a fast computational scheme is possible requiring computational time similar to the fast implementation of Normalized Cross Correlation (NCC). In fact, the MTM measure can be viewed as a generalization of the NCC for non-linear mappings and actually reduces to NCC when mappings are restricted to be linear. We empirically show that the MTM is highly discriminative and robust to noise with comparable performance capability to that of the well performing Mutual Information, but on par with NCC in terms of computation time. If time permits, I’ll show that MTM can be formulated as a graph Laplacian where pattern’s pixels are represented by a weighted graph and fast computation is possible due to low-rank decomposition. Generalizing the Laplacian distance results in distance measures which are tolerant to various visual distortions.
Eran Halperin, TAU
06/03/2014 - 12:00
I will present a few approaches for the detection of the ancestry of an individual based on the individual's genome. The general approach for ancestry inference is either based on generative probabilistic models that are dichotomous (e.g., mixture models, clustering), or are based on linear methods such as PCA. In this talk I will review these methods and describe two new methods that are based on a hybrid of these two approaches.
Dr. David Traum from ICT&USC
23/02/2014 - 11:45 - 13:00
By analyzing the user's utterance while it is still in progress, dialogue
systems and virtual humans can provide more natural, human-like
conversational behavior. This includes (non-verbal) listening behavior,
backchannels, quick responses and interruptions and collaborative
completions.
I will present recent work at the Institute for Creative Technologies
aimed at incremental understanding and dialogue management
while someone is speaking. A particular focus will be on a model of
grounding (reaching assumed mutual understanding of what is said) that is
updated very rapidly and facilitates feedback decisions while listening.
These models have been implemented within a multi-party virtual human
negotiation applications. I will also present work on providing additional
listener feedback to indicate attitudes toward the content as well as level
of understanding.
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02/01/2014 - 12:00
There will not be a talk today.
Michal Armoni from the Science Teaching Department ofWeizmann Institute of Science
26/12/2013 - 12:00
The research presented in this talk is within the discipline of computer science (CS) education. This is a sub-field of mathematics and science education, which in turn is a sub-field of the discipline of education. Thus, research in CS education is based on methodologies of social science, as well as on understanding the special nature of CS – as a discipline that has something in common with mathematics, science and engineering, but also has unique characteristics of its own. The research of CS education focuses on two educational processes: the teaching process (what should we teach our students in order to better prepare them to be CS graduates, and how should we teach it?), and the learning process (how do our students interpret and perceive what we teach them?).
This talk will focus on non-determinism, a fundamental idea in CS that is relevant to many CS areas, and is strongly connected to abstraction, another CS fundamental idea. I will describe a series of studies. The findings of some of these studies indicate difficulties students have in perceiving and using non-determinism, while others indicate that certain teaching strategies can affect students' perception of non-determinism. I will discuss issues concerning the manner in which non determinism can be taught and the implications of this on students' perception.
Iddo Tzameret from Institute for Theoretical Computer Science, Tsinghua University
19/12/2013 - 12:00
Proof complexity is the field that provides the foundation for proof-search algorithms (such as industrial DPLL-based SAT-solvers), as well as an established direction towards the central questions of computational complexity; namely, the fundamental questions about the limits of efficient computation. In its heart is the notion of a propositional proof system: each proof system defines a specific way to write proofs (or certify) tautologies, and the complexity of a given proof is the number of symbols it takes to write it down. Establishing strong size lower bounds on strong enough propositional proof systems, that is, showing that a certain
family of tautologies do not have small proofs in the system, is an extremely hard problem in itself which has striking implications to computational complexity.
In recent years, a new purely algebraic approach to proof complexity has emerged via the notion of arithmetic proofs, introduced by Hrubes and myself (CCC'09). This approach sets out to provide new structures and tools to old problems in the field; as well as providing an elegant solution to some long-standing open problems and giving rise to new seemingly fundamental open problems. In this talk I will give a short background into propositional proof complexity and discuss the algebraic approach, some of its achievements, prospects, and basic open problems.
Based on joint works with Pavel Hrubes (CCC'09, STOC'12), Fu Li ('13) and Stephen A. Cook ('13).
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Bio: Dr. Iddo Tzameret is an Assistant Professor at the Institute for Interdisciplinary Information Sciences, Tsinghua University. He received his PhD degree from Tel Aviv University in 2008 under the supervision of Ran Raz (Weizmann Institute) and Nachum Dershowitz (Tel Aviv). He was awarded a postdoctoral fellowship by the Eduard Cech Center for Algebra and Geometry. He has been conducting extensive research on the foundations of computer science, computational complexity, satisfiability (in practice and theory), and applications of logic in computer
science. He has been most successful in applying methods from computational complexity, algebraic complexity and logic in the area of efficient reasoning and proof complexity. His research is funded by the National Natural Science Foundation of China.
Keren Censor-Hillel from the CS faculty of the Technion
05/12/2013 - 12:00
Vertex and edge connectivity are fundamental graph-theoretic concepts,
as they give measures for flows and robustness. While there is
extensive knowledge in the literature about edge connectivity, using
the essential tool of edge-disjoint spanning trees, much less is known
about vertex connectivity, mainly due to the exponential number of
vertex cuts, as opposed to the polynomial number of edge cuts.
In this talk I will present new results that propose CDS (connected
dominating set) partitions and packings as tools that help us argue
about the vertex connectivity of graphs. I will show algorithms for
obtaining an optimal CDS packing of size Omega(k/logn) and a CDS
partition of size Omega(k/log^5n) in k-vertex connected graphs. Our
results also apply to analyzing vertex connectivity under random
vertex sampling, showing an almost optimal bound of
\tilde{Omega}(kp^2) for the vertex connectivity of a graph whose
vertexes are sampled with probability p from a k-vertex connected
graph. Finally, I will discuss applications to the throughput of
Based on joint work with Mohsen Ghaffari and Fabian Kuhn.
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28/11/2013 - 12:00
No Colloquium Today
Mooly Sagiv from CS TAU
21/11/2013 - 12:00
We describe an approach for synthesizing data representations for
concurrent programs. Our compiler takes as input a program
written using concurrent relations and synthesizes a representation of
the relations as sets of cooperating data structures as well as the
placement and acquisition of locks to synchronize concurrent access
to those data structures. The resulting code is correct by construction:
individual relational operations are implemented correctly and the
aggregate set of operations is serializable and deadlock free. The
relational specification also permits a high-level optimizer to choose
the best performing of many possible legal data representations
and locking strategies, which we demonstrate with an experiment
autotuning a graph benchmark.
This is joint work with Alex Aiken and Peter Hawkins(Stanford),
Katleen Fisher(DARPA), and Martin Rinard(MIT)
This work is part of Petrer Hawkins thesis
http://theory.stanford.edu/~hawkinsp/
Please also look into the December 12 CACM article
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14/11/2013 - 12:00
Aviv Zohar from CS HUJI
07/11/2013 - 12:00
Joint work with Yonatan Sompolinsky
Bitcoin is a potentially disruptive new cryptocurrency based on a
decentralized open-source cryptographic protocol. The Bitcoin network
consists of nodes who contribute their computational power to approve
transactions. Transactions are approved in batches that are called
blocks once every 10 minutes (in expectation). Transactions need
multiple such approvals to occur before they can be considered
irreversible with sufficiently high probability. This implies a long
waiting time for each individual transaction (a typical transaction
may wait for an hour or so).
Additionally, blocks are currently restricted in size to 1MB and thus
limit the average number of transactions that can be processed per
second.
We seek to improve both the waiting time for transaction authorization
and the number of transactions processed per second by lowering the
block authorization time to far below 10 minutes, and by increasing
the maximal allowed block size. We analyze the effects such changes
would have on the security of the protocol.
Using the typical block propagation time in the Bitcoin network
(recently measured by Decker & Wattenhoffer) our findings indicate
that:
1) At today’s transaction rates, the waiting time for approval can be
significantly decreased with negligible compromise with regards to the
security of the protocol.
2) Bitcoin’s ability to scale up to greater transaction volumes is
inherently limited by the propagation delay in the network. Our
analysis allows us to derive estimates regarding this limit.
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31/10/2013 - 12:00
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20/06/2013 - 12:00
There will be no colloquium this week.
BISFAI takes place on Wed-Fri.
http://websrv.cs.biu.ac.il/bisfai13//
See you next year.
Idan Szpektor from Yahoo Research
13/06/2013 - 12:00
Yahoo! Answers is one of the largest community-based question answering sites today, with over a billion answers for hundreds of millions of questions.
In this talk I will present two works that leverage this amount of data. The first work, presented in WWW 2013, is a personalized recommender system that recommends open questions for users to answer, based on the answering activity of each user. This task contains unique attributes: most of the users and questions are new with hardly any information on them. In addition, proposing relevant questions is not the only requirement by the users, where diversity and freshness play a significant role in the quality of recommendations. This system was recently deployed in Yahoo! Answers.
The second work, presented in WWW 2012, is an automatic question answering algorithm that matches past answers to new open questions. Prior question answering systems focused on factoid questions for which a single correct answer exists. However, our algorithm needs to address many types of open-ended questions, including asking for suggestions, opinions and social feedback.
The first work is a joint work with Dan Pelleg and Yoelle Maarek.
The second work is a joint work with Anna Shtok, Gideon Dror and Yoelle Maarek.
Ori Rottenstreich, Department of Electrical Engineering Technion
06/06/2013 - 12:00
Bloom filters and counting Bloom filters (CBFs) are widely used in networking device algorithms. They implement fast set representations to support membership queries with limited error. Unlike Bloom filters, CBFs also support element deletions.
In the first part of the talk, we introduce a new general method based on variable increments to improve the efficiency of CBFs and their variants. We demonstrate that this method can always achieve a lower false positive rate and a lower overflow probability bound than CBF in practical systems.
In the second part of the talk, we uncover the Bloom paradox in Bloom filters: sometimes, it is better to disregard the query results of Bloom filters, and in fact not to even query them, thus making them useless. We analyze conditions under which the Bloom paradox occurs, and demonstrate that it depends on the a priori probability that a given element belongs to the represented set.
Short Bio:
Ori Rottenstreich is a Ph.D. student at the Electrical Engineering department of the Technion, Haifa, Israel. His current research interests include exploring novel coding techniques for networking applications. Ori is a recipient of the Google Europe Fellowship in Computer Networking, the Andrew Viterbi graduate fellowship, the Jacobs-Qualcomm fellowship, the Intel graduate fellowship and the Gutwirth Memorial fellowship.
Ori Rottenstreich from Department of Electrical Engineering Technion
06/06/2013 - 12:00
Bloom filters and counting Bloom filters (CBFs) are widely used in networking device algorithms. They implement fast set representations to support membership queries with limited error. Unlike Bloom filters, CBFs also support element deletions.
In the first part of the talk, we introduce a new general method based on variable increments to improve the efficiency of CBFs and their variants. We demonstrate that this method can always achieve a lower false positive rate and a lower overflow probability bound than CBF in practical systems.
In the second part of the talk, we uncover the Bloom paradox in Bloom filters: sometimes, it is better to disregard the query results of Bloom filters, and in fact not to even query them, thus making them useless. We analyze conditions under which the Bloom paradox occurs, and demonstrate that it depends on the a priori probability that a given element belongs to the represented set.
Guy Even School of Electrical Engineering Tel-Aviv University
04/06/2013 - 12:00
In this talk we present relations between linear programming, dynamic programming, and the min-sum algorithm (also known as the sum-product algorithm).
Linear programming is an algorithmic technique for optimization problems over the real numbers. Dynamic programming is an algorithmic technique for optimization problems over trees and acyclic graphs. The min-sum algorithm is a variant of Belief Propagation and is used for
performing inference under graphical models.
The study of relations between these algorithmic techniques is motivated by decoding of error correcting codes. Many variants of the min-sum algorithm have been proposed for decoding starting with Gallagher in 1963. Viterbi's algorithm from 1967 for decoding convolutional codes is a dynamic programming algorithm. Feldman, Karger, and Wainwright proposed in 2005 to use linear programming for decoding binary linear codes.
The talk will focus on two results:
1. Inverse polynomial bounds on the decoding error for a wide class of error correcting codes with logarithmic girth (including LDPC codes, and irregular repeat-accumulate codes with both even and odd repetition factors). The bounds apply both to linear-programming decoding as well as a variant of the min-sum algorithm.
2. A proof that the min-sum algorithm for integer packing and covering problems is not better than linear programming.
Both results use dynamic programming to interpret the min-sum algorithm. In addition, graph covers are used as tool that enables a comparison between optimal solutions of linear programming and optimal solutions of dynamic programming.
Joint work with Nissim Halabi.
Building 507, room 061
ד"ר שרון אהרוני – גולדנברג, מרצה, בית הספר למשפטים, המכללה האקדמית נתניה
30/05/2013 - 11:00
אחת מתופעות הלוואי השליליות של מהפכת המחשבים היא תופעת המחשוב הפולשני, המתבטאת בין היתר, בחדירה למחשב. השלכות התופעה אינן עוצרות בפגיעה בזכויות אדם כגון, הזכות לאוטונומיה, לפרטיות, לקנין ולחופש הביטוי; אלא יש להן היבטים חברתיים נרחבים יותר, כגון תקינות הפעולה של מערכות מופעלות מחשב או תשתיות מסחר, שעליהן נסמכת הפעילות החברתית התקינה.
אולם, מהבחינה המשפטית יש קושי רב בהתמודדות עם תופעת המחשוב הפולשני שכן לרוב, קורבנות העברה כלל אינם מודעים לעצם החדירה למחשבם, מעטים מהם מתלוננים למשטרה, שסובלת מכח אדם מועט וגם לחוק עצמו קשה להגדיר את פרישת עברת החדירה למחשב, בשל היבטיה הערטילאיים. כך למשל, לאחרונה בית משפט השלום זיכה נאשם שחדר לחשבונות בנק של אחרים מעברת החדירה למחשב (האוסרת חדירה שלא כדין לחומר מחשב הנמצא במחשב), שכן העברה ותחולתה אינם ברורים דיים (מדינת ישראל נ' ניר עזרא)!
בשל קשיים כבדי משקל אלה, הפתרון הראוי לתופעת המחשוב הפולשני צריך להתבטא, לא רק באיסור פלילי ונזיקי של עצם החדירה הלא מורשית למחשב, אלא גם בראייה עונשית מניעתית. כלומר, על הדין לאסור את עצם עריכת התוכנה החודרנית ואת הפצתה לציבור הרחב. אכן, בתיקון מספר 1 לחוק המחשבים המחוקק ניסה להתמודד בצורה ראויה עם תופעת המחשוב הפולשני על ידי אימוץ עברות הכנה והפצה; סעיף 6 של החוק המתוקן אוסר עריכת תוכנה המשבשת מחשב, מוחקת מידע ממוחשב, מאפשר זיוף של מידע ממוחשב, חדירה למחשב והאזנת סתר והכל "במטרה לבצען שלא כדין".
בנוגע לאיסור תכנות תוכנה המאפשרת חדירה למחשב, נשאלת השאלה האם היא כוללת גם איסור על פיתוח תוכנה חודרנית במטרה לסגלה לצרכי ריגול במדינות אויב; האם היא אוסרת תכנות תוכנה חודרנית לצרכי לימוד והדגמה; האם למעשה האיסור צובע באדום גם פיתוח קוקיס וסרגלים? מושג המפתח למענה על שאלה זו הוא המונח "שלא כדין". אלא שקשה ביותר לעמוד על מהותו של מונח זה ולאורך השנים בתי המשפט התלבטו רבות בניסיון לפרשו; יש שפירשוהו כ"ללא צידוק בדין החיצוני לחוק שבו מדובר", כלומר כסוג של "קישורית" המאפשרת לייבא הגנה שקבועה בחוק כלשהו לחוק שבו טבוע המונח, למשל לחוק המחשבים. ויש שפירשו את המונח "שלא כדין" כמתייחס לסוגיית ההסכמה. כלומר, חדירה שלא כדין לחומר מחשב היא חדירה שנעשית ללא הסכמה של בעל המידע הממוחשב (פס"ד מדינת ישראל נ' בדיר).
לדעתי, כדי למנוע הפללת יתר על פי סעיף 6 לחוק המחשבים, צריך יהיה לפרש את המונח "שלא כדין" בצורה רחבה מאוד, כזו שתכלול גם הגנה על פיתוח תוכנות למטרות מחקר. כלומר, כאשר מרצה למחשבים מדגים לסטודנטים שלו איך לערוך ולתכנת תוכנה חודרנית, לא יהא זה מעשה "שלא כדין" ובוודאי שלא עברה פלילית. אשר לקוקיס ולסרגלים פולשניים. אף שלרוב החדרתן למחשב היעד נעשית ללא הסכמה וללא ידיעה של בעל המחשב, ואף שבחלק ניכר מהמקרים התוכנה שמאפשרת את החדרתם מטרידה את בעל המחשב, דומתני שמן הראוי שהמחוקק ייתן עליהם את הדעת באופן פרטני. קרי, שאיסור תוכנות פולשניות נפוצות כל כך ייעשה באמצעות חקיקה ספציפית, אשר תתייחס אליהן ותדרוש הסכמה מדעת של בעל המחשב ואפשרויות זמינות להסרתן מהמחשב הנחדר, ולא באמצעו הפעלה של סעיף 6 לגביהם.
Roded Sharan from Tel Aviv University
23/05/2013 - 12:00
Protein networks have become the workhorse of biological research in recent years, providing mechanistic explanations for
basic cellular processes in health and disease. However, these explanations remain topological in nature as the underlying logic
of these networks is to the most part unknown. In this talk I will
describe the work in my group toward the automated learning of
the Boolean rules that govern network behavior under varying conditions.
I will highlight the algorithmic problems involved and demonstrate how they can be tackled using integer linear programming techniques.
The talk is self contained.
Danny Harnik from IBM Research Israel
09/05/2013 - 12:00
Real-time compression for primary storage is quickly becoming
compression on the data path consumes scarce CPU and memory resources
on the storage system.
In this talk I'll present different approaches to efficient estimation
of the potential compression ratio of data and how these methods can
be applied in advanced storage systems. Our work targets two
granularities: the macro scale estimation which is backed up by
analytical proofs of accuracy, and the micro scale which is heuristic.
Based on joint work with Oded Margalit, Ronen Kat, Dmitry Sotnikov and
Avishay Traeger, from IBM Research - Haifa. This work has recently
appeared in FAST 2013.
Amichai Shulman and Tal Beery, IMPERVA
02/05/2013 - 12:00
On 2012, security researchers shook the world of security with their CRIME attack against the SSL encryption protocol. CRIME (Compression Ratio Info-leak Made Easy) attack used an inherent information leakage vulnerability resulting from the HTTP compression usage to defeat SSL’s encryption.
However, the CRIME attack had two major practical drawbacks. The first is the attack threat model: CRIME attacker is required to control the plaintext AND to be able to intercept the encrypted message. This attack model limits the attack to mostly MITM (Man In The Middle) situation.
The second issue is the CRIME attack was solely aimed at HTTP requests. However, most of the current web does not compress HTTP requests. The few protocols that did support HTTP requests compression (SSL compression and SPDY) had dropped their support following the attack details disclosure, by thus rendering the CRIME attack irrelevant.
In our work we address these two limitations by introducing the TIME (Timing Info-leak Made Easy) attack for HTTP responses.
By using timing information differential analysis to infer on the compressed payload’s size, the CRIME attack’s attack model can be simplified and its requirements can be loosened. In TIME’s attack model the attacker only needs to control the plaintext, theoretically allowing any malicious site to launch a TIME attack against its innocent visitors, to break SSL encryption and/or Same Origin Policy (SOP).
Changing the target of the attack from HTTP requests to HTTP responses significantly increases the attack surface, as most of the current web utilizes HTTP response compression to save bandwidth and latency.
In particular, we:
Introduce the TIME attack
Show an actual POC of timing differential analysis to infer on the compressed payload’s size and subsequently the cipher-text’s underlying plaintext
Show the relevancy of compression ratio information leakage for HTTP responses
Suggest mitigation steps against the TIME attack
Michael Schapira. Huji
25/04/2013 - 12:00
The Border Gateway Protocol (BGP) establishes routes between the smaller networks that make up today's Internet (Google, AT&T, etc.) and can be viewed as the glue that holds the Internet together. As evidenced by recent major Internet outages, BGP is unbelievably vulnerable to configuration errors and attacks. To remedy this, several security-enhancements of BGP have been proposed. Yet, despite over a decade of work, deployment is still not yet on the horizon. While at first it seemed that the main obstacle en routing to deployment is technical infeasibility, it is now clear that the main challenges lie elsewhere: (1) lingering disagreements about which of the major proposals to adopt; and (2) lack of economic incentives for adoption. I will survey recent results along these lines including ways to quantify Internet routing security and economic mechanisms for driving transition to BGP security
Joint work with Sharon Goldberg, Pete Hummon and Jennifer Rexford (2010), with Phillipa Gill and Sharon Goldberg (2011), and with Robert Lychev and Sharon Goldberg (2013)
Michael Schapira from the Hebrew Uninversity
25/04/2013 - 12:00
The Border Gateway Protocol (BGP) establishes routes between the smaller networks that make up today's Internet (Google, AT&T, etc.) and can be viewed as the glue that holds the Internet together. As evidenced by recent major Internet outages, BGP is unbelievably vulnerable to configuration errors and attacks. To remedy this, several security-enhancements of BGP have been proposed. Yet, despite over a decade of work, deployment is still not yet on the horizon. While at first it seemed that the main obstacle en routing to deployment is technical infeasibility, it is now clear that the main challenges lie elsewhere: (1) lingering disagreements about which of the major proposals to adopt; and (2) lack of economic incentives for adoption. I will survey recent results along these lines including ways to quantify Internet routing security and economic mechanisms for driving transition to BGP security.
Joint work with Sharon Goldberg, Pete Hummon and Jennifer Rexford (2010), with Phillipa Gill and Sharon Goldberg (2011), and with Robert Lychev and Sharon Goldberg (2013)
Eyal Ackerman, Oranim
18/04/2013 - 12:00
The Crossing Number of a graph $G$ is the minimum number of
edge crossings in a drawing of $G$ in the plane. According to the
Crossing Lemma, there is a constant $c$ such that the crossing number
of every (not too sparse) graph $G=(V,E)$ is at least $c|E|^3/|V|^2$..
This bound is tight, apart from the multiplicative constant $c$..
The first result that I will present states that if an $n$-vertex
graph can be drawn such that each of its edges is involved in at most
four crossings, then it has at most $6n-12$ edges. This settles a
conjecture of Pach, Radoicic, Tardos, and G. Toth, and implies a
better lower bound for the constant $c$ of the Crossing Lemma.
The second result that I will present considers the degenerate
crossing number of a graph $G$, namely, the minimum number of crossing
\emph{points} in a drawing of $G$ in the plane. It has been an open
problem whether this crossing number behaves like the usual crossing
number. In a recent joint work with Rom Pinchasi we proved that it
does.
Eyal Ackerman from Oranim college
18/04/2013 - 12:00
The Crossing Number of a graph $G$ is the minimum number of
edge crossings in a drawing of $G$ in the plane. According to the
Crossing Lemma, there is a constant $c$ such that the crossing number
of every (not too sparse) graph $G=(V,E)$ is at least $c|E|^3/|V|^2$.
This bound is tight, apart from the multiplicative constant $c$.
The first result that I will present states that if an $n$-vertex
graph can be drawn such that each of its edges is involved in at most
four crossings, then it has at most $6n-12$ edges. This settles a
conjecture of Pach, Radoicic, Tardos, and G. Toth, and implies a
better lower bound for the constant $c$ of the Crossing Lemma.
The second result that I will present considers the degenerate
crossing number of a graph $G$, namely, the minimum number of crossing
\emph{points} in a drawing of $G$ in the plane. It has been an open
problem whether this crossing number behaves like the usual crossing
number. In a recent joint work with Rom Pinchasi we proved that it
does.
Judea Pearl, UCLA
14/04/2013 - 12:00
Recent developments in graphical models and the logic of counterfactuals have had a marked effect on the way scientists treat problems involving cause-effect relationships. Paradoxes and controversies have been resolved, slippery concepts have been demystified, and practical problems requiring causal information, which long were regarded as either metaphysical or unmanageable can now be solved using elementary mathematics.
I will review concepts, principles, and mathematical tools that were found useful in this transformation, and will demonstrate their applications in several data-intensive sciences.
These include questions of policy evaluation, model testing, mediation mechanisms, missing data, generalizations from experiments, and integrating data from diverse sources,.
Reference: J. Pearl, Causality (Cambridge University Press, 2000,2009) Background material:
http://ftp.cs.ucla.edu/pub/stat_ser/r379.pdf
http://ftp.cs.ucla.edu/pub/stat_ser/r355-reprint.pdf
http://ftp.cs.ucla.edu/pub/stat_ser/r391.pdf
http://ftp.cs.ucla.edu/pub/stat_ser/r400.pdf
Working papers:
http://bayes.cs.ucla.edu/csl_papers.html
Building 410 Beck Auditorium
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Avishai Wool, School of Electrical Engineering, Tel Aviv University
11/04/2013 - 12:00
Modbus/TCP is used in SCADA networks to communicate between the Human Machine Interface (HMI) and the Programmable Logic Controllers (PLCs). Therefore, deploying Intrusion Detection Systems (IDS) on Modbus networks is an important security measure. In this paper we introduce a model-based IDS specifically built for Modbus/TCP. Our approach is based on a key observation: Modbus traffic to and from a specific PLC is highly periodic. As a result, we can model each HMI-PLC channel by its own unique deterministic finite automaton (DFA). Our IDS looks deep into the Modbus packets and produces a very detailed model of the traffic. Thus, our method is very sensitive, and is able to flag anomalies such as a message appearing out of its position in the normal sequence, or a message referring to a single unexpected bit. We designed an algorithm to automatically construct the channel’s DFA based on about 100 captured messages. A significant contribution is that we tested our approach on a production Modbus system. Despite its high sensitivity, the system enjoyed a super-low false-positive rate: on 5 out of the 7 PLCs we observed a perfect match of the model to the traffic, without a single false alarm for 111 hours. Further, our system successfully flagged realanomalies that were caused by technicians troubleshooting the HMI system---and the system also helped uncover one incorrectly configured PLC
Amir Herzberg from our department
04/04/2013 - 12:00
Everyone is concerned about Internet security, yet most traffic is not cryptographically protected. The usual justification is that most attackers are only {\em off-path} and cannot intercept traffic; hence, challenge-response mechanisms suffice to ensure authenticity. Usually, the challenges re-use existing `unpredictable' protocol header fields; this allows use of existing, widely-deployed protocols such as TCP and DNS.
We argue that this practice may only give an {\em illusion of security}. We present recent off-path TCP injection and DNS poisoning attacks, allowing circumvention of existing challenge-response defenses.
Both TCP and DNS attacks are non-trivial, yet very efficient and practical. The attacks allow circumvention of widely deployed security mechanisms, such as a Same Origin Policy, and allow a wide range of exploits, e.g., long-term caching of malicious objects and scripts.
We hope that this article will motivate adoption of cryptographic mechanisms such as SSL/TLS, IPsec and DNSSEC, and of correct, secure challenge-response mechanisms.
(Joint work with Yossi Gilad and Haya Shulman)
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28/03/2013 - 12:00
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Sivan Toledo from Tel Aviv University
14/03/2013 - 12:00
Matrix computations over the real and complex numbers are immensely important algorithmic building blocks (for physical simulations, optimization, machine learning, and many other applications). These computations include the solution of systems of linear equations, minimization of quadratic forms, and computation of spectral objects, such as eigen/singular values and vectors. All of these problems can be solved in polynomial time and algorithms for solving them have been investigated extensively for over 60 years (with some algorithms going back to Gauss). But deep and important open questions remain. The talk will explain what is still not known, why these open questions are important, and what was discovered about them recently. The selection of open problems will be fairly subjective and highly influenced by my own research; it will not be the most objective and comprehensive survey of open problems in numerical linear algebra.
Michael Segal from Ben-Gurion University
07/03/2013 - 12:00
A wireless ad-hoc network consists of several transceivers (nodes) located in the plane, communicating by radio. Unlike wired networks, in which the link topology is fixed at the time the network is deployed, wireless ad-hoc networks have no fixed underlying topology. The temporary physical topology of the network is determined by the distribution of the wireless nodes, as well as the transmission range of each node. The ranges determine a directed communication graph, in which the nodes correspond to the transceivers and the edges correspond to the communication links. The key difference between wireless ad-hoc networks and "conventional" communication structures, from the designer's point of view, is in the power assignment model, where each node decides on a transmission power level. In this talk we will concentrate on efficient power assignment strategies that produce communication graphs with various desired properties such as: k-connectivity, small energy consumption and interferences, bounded diameter, limited latency and short paths.
Vadim Indelm robotics and vision lab(the BORG lab) at Georgia Tech
06/03/2013 - 16:00
This talk will focus on efficient methods for single- and multi-robot autonomous localization and structure from motion (SfM) related problems such as mobile vision, augmented reality and 3D reconstruction. High-rate performance and high accuracy are a challenge, in particular when operating in large scale environments, over long time periods and in presence of loop closure observations. This challenge is further enhanced in multi-robot configurations, where communication and computation budgets are limited and consistent information fusion should be enforced. In this talk, I will describe approaches that address these challenges.
First, I will present a computationally efficient method for incremental bundle adjustment. The method, incremental light bundle adjustment (iLBA), incorporates two key components to substantially reduce computational complexity: the observed 3D points are algebraically eliminated, leading to a cost function that is formulated in terms of multiple view geometry constraints. The second component is incremental smoothing, which uses graphical models to adaptively identify the variables that should be re-eliminated at each step. The described method will be demonstrated both in SfM and robot navigation scenarios. Since high-rate performance for general loop closure observations is not guaranteed, I will overview an approach that addresses this issue by parallelized computations, partitioning the underlying graphical structure of the problem at hand. Next, I will focus on multi-agent configurations and discuss approaches that exploit commonly observed 3D points to both perform cooperative localization, improving pose estimation of the individuals in the group, and to extend sensing horizon. Two methods will be described: sharing the estimated distributions of observed 3D points, and sharing the actual (image) observations of these points, thereby employing distributed multiple view geometry and extending iLBA. A special attention will be given to consistent information fusion, i.e. avoid double counting measurements
### הקולוקוויום יתקיים בבניין 202 חדר 106
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28/02/2013 - 12:00
Next week Michael Segal from Communication Systems Engineering Department
Anna Zamansky from TU Wien will give a talk in our colloquium.
24/01/2013 - 12:00 - 13:00
In recent decades a vast variety of non-classical logics have been introduced, driven by various CS applications. Temporal logics, separation logics, fuzzy logics and paraconsistent logics are just a few prominent examples, used in verification of software and hardware, medical expert systems, data and knowledge bases, etc. A useful logic should ideally have two components: a simple and intuitive semantics, which can provide real insights into the logic, and a corresponding analytic proof system which is the key to effective proof search strategies for automated deduction methods. Obtaining these components for a given logic is a challenging process, which is usually tailored to the particular logic at hand. However, due to the increasing number of new application-driven logics, there is a need for a systematic approach to obtaining these components, which could be used for developing tools for automatic support for the design and investigation of logical systems.
In this talk we show that this goal can be achieved at least for some useful families of non-classical logics. We provide a uniform and modular method for a systematic generation of effective semantics and analytic proof systems for a very large family of paraconsistent logics used for reasoning with inconsistent information, thus making a substantial step towards the development of efficient paraconsistent theorem provers. The method, implemented by the Prolog system PARAlyzer, has been extended to infinitely many other logics formulated in terms of axiomatic systems of a certain natural form.
Mark Silberstein University of Texas at Austin.
17/01/2013 - 12:00 - 13:00
The processor landscape has fractured into latency-optimized CPUs,
throughput-oriented GPUs, and soon, custom accelerators. Future applications
will need to cohesively use a variety of hardware to achieve their performance
and power goals. However building efficient systems that use
accelerators today is incredibly difficult.
In this talk I will argue that the root cause of this complexity lies in
the lack of adequate operating system support for accelerators. While
operating systems provide optimized resource management
and Input/Output (I/O) services to CPU applications, they make no such
services available to accelerator programs.
I propose GPUfs - an operating system layer which enables access to files directly from programs
running on throughput-oriented accelerators, such as GPUs.
GPUfs extends the constrained GPU-as-coprocessor programming model,
turning GPUs into first-class computing devices with full file I/O support.
It provides a POSIX-like API for GPU programs, exploits parallelism for efficiency,
and optimizes for access locality by extending a CPU buffer cache into physical memories
of all GPUs in a single machine.
Using real benchmarks I show that GPUfs simplifies the development
of efficient applications by eliminating the GPU management complexity,
and broadens the range of applications that can be accelerated by GPUs.
For example, a simple self-contained GPU program which searches for a set of strings in the entire
tree of Linux kernel source files completes in about third of the time of an
8-CPU-core run.
Joint work with Idit Keidar (Technion), Bryan Ford (Yale) and Emmett Witchel
(UT Austin)
Joseph Keshet from TTI-Chicago
10/01/2013 - 12:00 - 13:00
The goal of discriminative learning is to train a system to optimize a certain desired measure of performance. In simple classification we seek a function that assigns a binary label to a single object, and tries to minimize the error rate (correct or incorrect) on unseen data. In structured prediction we are interested in the prediction a structured label, where the input is a complex object. Typically, each structured prediction task has its own measure of performance, or task loss, such as word error rate in speech recognition, the BLEU score in machine translation or the intersection-over-union score in object segmentation. Not only that those task losses are much more involved than the binary error rate, the structured prediction itself spans an exponentially large label space. In the talk, I will present two algorithms each designed to the minimize a given task loss, and applied to speech recognition.
In the first part of the talk, I will present an algorithm which aims at achieving a high area under the receiver operating characteristic (ROC) curve. This measure of performance is often used to evaluate the detection of events over time. We will give some theoretical bounds that relate the performance of the algorithm with the expected area under the ROC curve, and will demonstrate its efficiency to that task of keyword spotting, i.e., that detection of all occurrences of any given word in a speech signal.
In the second part of the talk, I will describe a new algorithm which aims to minimize a regularized task loss. The algorithm is derived by directly minimizing a generalization bound for structured prediction, which gives an upper-bound on the expected task loss in terms of the empirical task loss. The resulting algorithm is iterative and easy to implement, and as far as we know, the only algorithm that can handle a non-separable task loss. We will present experimental results on the task of phoneme recognition, and will show that the algorithm achieves the lowest phoneme error rate (normalized edit distance) compared to other discriminative and generative models with the same expressive power.
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# Kinematics question on a rock
Intr3pid
## Homework Statement
A rock is dropped from a sea cliff, and the sound of it striking the ocean is heard 3.2s later. If the speed of sound is 340 m/s, how high is the cliff.
d= v1t + 1/2at^2
## The Attempt at a Solution
I tried using d = vt since we're given the speed of sound and the time taken for the sound to reach our ears. Since sound is unaffected by gravity, i thought i could straightaway use that formula, but I guess I was wrong. This is how far I got up to. Can someone give me a hand?
Thanks
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# mwohlauer.d-n-s.name / www.mobile-infanterie.de
### Site Tools
en:games:quake_3_arena
# Quake 3 Arena
## Info
Multiplayer Information
• Internet play: yes
• LAN play: yes
• Lobby search: yes
• Direct IP: yes
• Play via GameRanger: no
• Coop: yes
• Singleplayer campaign: yes
• Hotseat: no
The third reincarnation of Quake is one of the most revered first person shooters in gaming history. It was released on December 5th, 1999 by Activision. Its idTech 3 engine is also the basis for a lot of other games, for instance Star Trek - Voyager Eliteforce. In 2005 idTech released the source files of this engine under the GPL, giving rise to even more new games, developed and maintained by the open source community.
## Installation
The game still installs well even under Windows 10 (2019-03). The only drawback is the not startable setup.exe file, when running a 64 bit version of Windows. In this case simply use the file Quake3\Setup.exe from the CD. It works just fine.
In case you still have problems, you may always fall back to simply copying the required files. The only really required files are quake3.exe and baseq3\pak0.pk3. They can be found in the folder Quake3.
## Graphics Hack
The game engine does offer a rather limited set of graphics resolutions. Especially the wide screen resolutions are only a few. This however, only goes for the presented resolutions in the game menu. Actually the engine does provide all available resolutions of your graphics card. The article Q3A Graphics Hack describes, how you can use them. These steps also work for most of the other Quake 3 Based Games.
## Settings That Affect Gameplay Mechanics
Q3 is known to have »side effects« caused by certain settings. Here are some candidates that might improve your experience, or worsen it.
### com_maxfps
This value limits your maximum frame rate, at which you computer updates the graphical representation of the game state (read: what you can see on your monitor). It is an integer value, that theoretically can take any number but in practice can only take values of (1000 / x), where x is an integer number greater than 1. So values such as 500 (x = 2) or 125 (x = 8) are OK. Values that do not hit these spots will be rounded up (e. g. for com_maxfps = 120 set the taken value will still be 125). Note: This is the maximum frame rate your client will allow itself to produce, even when your hardware might allow for more. On the other hand, if your hardware is not potent enough for this value, it might also produce lower actual frame rates.
Certain values of this cvar my result in higher jumps or lower jumps. The optimal value is 125. This video explains it in more detail: Quake 3's frame rate dependent physics.
### rate
This value defines the bytes per second your client will allow to receive at max. It is send to the server and regarded as your upper limit. If the server itself has set a lower limit, it will us its own limit. When nothing is going on on the server (e. g. nobody moves or shoots) this rate must not necessarily be used up entirely. A very common value for this is 25,000. The higher the better for you, unless you reach your downstream capacity. If you set this too high for your downstream, you might experience higher pings, loss and in general a bad experience up to the point where you drop off from the server.
### snaps
The server does not send you a continuous stream of changes as it processes them. It sends you snap shots of the current situation. This variable defines how often the server should send you an update on the current game state (kind of a frame rate but for the game state). The higher you set this, the more accurate will the representation of the current game flow be on your end. But at the same time it will consume more bandwidth. So if you reach the upper limit of your connection, you may again experience lagg, loss and drop outs. A very common value would be 40. However, just like with rate, the server caps this value, depending on its sv_fps value. You will never receive more snaps than the server allows.
### cl_maxpackets
This defines how many packages per second you send to the server. Similar to the snaps, this is an update from you (the client) to the server. The higher you set this value, the more smooth your game experience will be. However, if you use a too high value for your upstream bandwidth, you will again experience loss which might result in situations such as shooting (and from your end actually hitting the target) but not hitting the target anyways as the shot never reaches the server and therefore does not count at all. And of course, lagg, ping spikes and stuff like that will happen more frequently. This is a sensitive value. The max value is 125. For really low bandwidth connections such as dial up, 20-30 might be sensible.
Also note: Setting cl_maxpackets to a lower value than your com_maxfps value will reduce your effective cl_maxpackets. See section Value Compatibility on the matter.
### cl_packetdup
This cvar controls how often you re-send the same packet (duplication). This is meant as a means against a lossy connections. If one of the duplicated packages is lost, it is covered by another one of the bunch. The downside of this is that of course it increases your bandwidth by the given factor. 0 means it is off. 1 means, a package is send 1 more time than necessary, 2 means two duplicates, and so on. So bandwidth strain is also increased to 2, 3 or even more times. Similar to a too high cl_maxpackets values, this may eventually cause higher pings and laggy results when you hit your bandwidth limit. So it is a tradeoff of bandwidth strain and loss occuring. For a healthy connection a value of 1, maybe 2 should suffice, if needed at all. On a connection with potent upstream you might be able to totally cut occuring loss to zero this way. But it might be more advisable to not use your laggy wireless connection or fix that faulty LAN cable.
Any network game has the problem, that the actual game flow is chopped into snapshots or frames and that the rate of this chopping may vary for all involved computers. This quantization also means, that there is kind of a dead time in-between two snaps. But the game continues for everyone locally anyways. Hence, this dead time may be interpolated on the client (so that during those 3 frames on your 125 fps PC you get not stuck playing on a 40 fps server until the next frame updates from the server). This also means, the client has to do predictions on what will be going on in the meantime. (This goes especially for the server, too, as it is kind of the master of what the game states of the clients should be using as a reference.) Only when the next frame update from the server arrives, your game state will be »set straigt«. (You know this from laggy games, where someone moves through the map and all of a sudden does a big jump or stuff like that. That's when the interpolation is dropped and the updated frame takes effect.)
cl_timenudge defines in frames relative to the next expected frame from the server, when the client begins to extrapolate. You can set negative values, which will cause the prediction to happen even before the update from the server can kick in. In the example above, you having 125 fps, the server using 40 it means you will have to wait 3-4 client frames until the server will send you another update. So your window of premature interpolation would be -3 to -4 tops. -5 and lower will not make sense, as this will have the same effect as -4. -2 would be somewhere in the middle, meaning half the way your client will start interpolating. The values allowed are calculated as 1000/sv_fps. So sv_fps = 40 would mean -1 to -3. If a server would have sv_fps = 125, the value might be between -1 and -8.
But you can also use positive values, meaning the interpolation starts after the expected server update frame would happen. So in essence it would only kick in, if the server's update frame comes in delayed (or not at all) somehow. When having loss or a high ping, that might make sense, too. It would ensure that under normal circumstances, the client waits at least until the server had a chance to update the client. This will give you the most accurate game state (no interpolation without loss or update delays) as the interpolation only kicks in, when something seems to be going a little bit sideways.
Important: This will by no means fix any ping problems or loss. It just tries to compensate for it.
### r_swapinterval
If you set this to 1, the fps value is synced with the actual hardware refresh rate of your monitor. For many players this is 60, which will act exactly as if you set the com_maxfps value to that value yourself (with all side effects that may have). If you turn it off, you may be able to set higher values than can actually be effective on your hardware. In the worst case scenario your screen will turn black until you quit Quake 3.
### cg_predict
This affects how your client predicts what will go on (see cl_timenudge). This may be 0, 1 or 2. 0 shuts it off, 1 is standard. 2 is an optimized variation, slightly more prone to faulty predictions but a lot faster. This should not be necessary on modern systems but on older machines it might give you an fps boost. Do not set cg_predictItems = 1 with cg_predict = 2.
### cg_predictItems
If set to 1, items being taken by a player are also predicted. Should not be necessary on a good connection.
### Value Compatibility
If you use cl_maxpackets, make sure to set compatible values. Internally, the cl_maxpackets value will be reduced to the next lowest value, according to your max_fps value: cl_maxpackets = com_maxfps / x with x being a positive integer number.
So for your generic 125 FPS, the allowed cl_maxpackets values are: 125 (1), 63 (2), 42 (3), 32 (4), 25 (125),… So if setting cl_maxpackets = 100, you are below 125, it takes the next lower value, which is 63. That's quite a big step down.
For a computer that cannot handle 125 but 100 cl_maxpackets steps would look like this: 100 (1), 50 (2), 34 (3), 25 (4),…. So when in doubt, set cl_maxpackets on the same or higher value as com_maxfps. Q3 will lower cl_maxpackets automatically anyways.
(Source)
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## SWATting
Lies, damned lies, and statistics.
Anaxagoras
Posts: 22906
Joined: Wed Mar 19, 2008 5:45 am
Location: Yokohama/Tokyo, Japan
### Re: SWATting
Los Angeles man arrested in ‘swatting’ call that preceded fatal police shooting in Kansas
I think the other guy who gave him the address should also be arrested.
A fool thinks himself to be wise, but a wise man knows himself to be a fool.
William Shakespeare
Grammatron
Posts: 33929
Joined: Tue Jun 08, 2004 1:21 am
Location: Los Angeles, CA
### Re: SWATting
Mentat wrote:
Grammatron wrote:
Mentat wrote:
the prank may have originated as part of a dispute between Call of Duty gamers
Why am I not surprised? The whole gaming community has gotten so toxic that it really was just a matter of time.
It's rather like painting a whole group of people by the action of few, and you don't seem to keen on that usually.
Fair enough, except for the "few" part. But it's not a demographic one is born or indoctrinated into. It is defined by the people it attracts, and the type of people they attract tend to behave a certain way. Or misbehave. Same with clubs or gangs.
You don't hear about country club members swatting each other. Or just about any other group.
I don't hear about country club members beating their wives in elevators but I still don't think all football players are cunts.
Rob Lister
Posts: 20193
Joined: Sun Jul 18, 2004 7:15 pm
Title: Incipient toppler
Location: Swimming in Lake Ed
### Re: SWATting
Anaxagoras wrote:Los Angeles man arrested in ‘swatting’ call that preceded fatal police shooting in Kansas
I think the other guy who gave him the address should also be arrested.
He even looks the part.
What will they charge him with?
Anaxagoras
Posts: 22906
Joined: Wed Mar 19, 2008 5:45 am
Location: Yokohama/Tokyo, Japan
### Re: SWATting
Rob Lister wrote: What will they charge him with?
Throw the book at him I guess. That's for prosecutors to figure out I suppose, but there must be something. Maybe they need a new law.
https://en.wikipedia.org/wiki/Swatting
Laws
United States – It can be prosecuted through federal criminal statutes.
"Conspiracy to retaliate against a witness, victim or informant".[11][12]
"Conspiracy to commit access device fraud and unauthorized access of a protected computer".[11][13]
An accomplice may be found guilty of "conspiring to obstruct justice".[14][15]
In the State of California pranksters bear the "full cost" of the response which can range up to $10,000.[16] Not sure which one of those is applicable. I did find one example: On January 31, 2016, at around 10pm, U.S. Representative Katherine Clark was swatted by an anonymous caller who claimed there was an active shooter in the home. Melrose Police responded to the home, and left after determining the call was a hoax. Rep. Clark had sponsored a bill called the 'Interstate Swatting Hoax Act of 2015', aimed at increasing the penalties for swatting, as well as making swatting a federal crime. After the incident, she "said she had been very sympathetic to people who have been the victims of swatting before Sunday night but now fully understands what it’s like." She further stated that the swatting "will really cause me to double down." [47] Unfortunately, the Interstate Swatting Hoax Act of 2015 doesn't seem to have become a law yet, so they can't charge him with that. I'll let the prosecutors figure it out. A fool thinks himself to be wise, but a wise man knows himself to be a fool. William Shakespeare Giz Posts: 1555 Joined: Mon Jul 12, 2004 5:07 pm Location: UK ### Re: SWATting Anaxagoras wrote:Yeah, when you tell the police that somebody in there is armed and dangerous, and in fact has already killed someone, they're going to be on a hair trigger. The person who called in the false report should be charged with murder. The police probably have some culpability too, but they were deceived. I feel bad for the officer who shot the guy. Yeah, I agree with this. Make some high profile examples. Pour discourager les autres. Mentat Posts: 10271 Joined: Tue Nov 13, 2007 11:00 pm Location: Hangar 18 ### Re: SWATting Grammatron wrote:I don't hear about country club members beating their wives in elevators but I still don't think all football players are cunts. Football doesn't make a highly profitable industry out of domestic abuse. It's "pea-can", man. Lapis Sells . . . But Who's Buying? Pyrrho Posts: 26348 Joined: Sat Jun 05, 2004 2:17 am Title: Man in Black Location: Division 6 ### Re: SWATting https://www.washingtonpost.com/news/pos ... ill-a-man/ In tweets and interviews, a man known online as “Swautistic” said he had placed the 911 call — which in his view was a routine hoax gone badly wrong. “Bomb threats are more fun and cooler than swats in my opinion and I should have just stuck to that,” Swautistic told reporter Brian Krebs on Friday. “But I began making$ doing some swat requests.”
Several hours later, Los Angeles police arrested a 25-year-old named Tyler Barriss in connection with Finch’s death. According to KABC, he had been arrested two years earlier for making hoax bomb threats to their TV station.
Well, fuck him. Period.
The flash of light you saw in the sky was not a UFO. Swamp gas from a weather balloon was trapped in a thermal pocket and reflected the light from Venus.
Mentat
Posts: 10271
Joined: Tue Nov 13, 2007 11:00 pm
Location: Hangar 18
### Re: SWATting
Sadly, it won't stop as long as it is profitable. And it's profitable as long as there's demand.
It's "pea-can", man.
Lapis Sells . . . But Who's Buying?
Anaxagoras
Posts: 22906
Joined: Wed Mar 19, 2008 5:45 am
Location: Yokohama/Tokyo, Japan
### Re: SWATting
Mentat wrote:Sadly, it won't stop as long as it is profitable. And it's profitable as long as there's demand.
People do this for money now?
Police could crack down on this better if there was a more specific law against it.
And if people are doing it for money, well just have undercover agents posing as butthurt gamers "hire" them and catch them in act.
A fool thinks himself to be wise, but a wise man knows himself to be a fool.
William Shakespeare
Rob Lister
Posts: 20193
Joined: Sun Jul 18, 2004 7:15 pm
Title: Incipient toppler
Location: Swimming in Lake Ed
### Re: SWATting
http://www.cnn.com/2017/12/31/us/swatti ... index.html
Article gives examples of how he might be charged. Reckless homicide looks promising.
WildCat
Posts: 14465
Joined: Tue Jun 15, 2004 2:53 am
Location: The 33rd Ward, Chicago
### Re: SWATting
I don't think the cop who shot him should get off the hook, he shot him for no good reason.
Do you have questions about God?
you sniveling little right-wing nutter - jj
Nyarlathotep
Posts: 48102
Joined: Fri Jun 04, 2004 2:50 pm
### Re: SWATting
Now that SWATting has a body count, it will be more popular than ever.
We live in interesting times
Bango Skank Awaits The Crimson King!
sparks
Posts: 14470
Joined: Fri Oct 26, 2007 4:13 pm
Location: Friar McWallclocks Bar -- Where time stands still while you lean over!
### Re: SWATting
No shit. Happy New Year Nyar.
You can lead them to knowledge, but you can't make them think.
Captain
Posts: 277
Joined: Thu Aug 23, 2012 11:47 pm
Title: Captain
Location: On the verandah in the cool breeze.
### Re: SWATting
Gonna need a third set of chains...
You run one time, you got yourself a set of chains. You run twice you got yourself two sets. You ain't gonna need no third set, 'cause you gonna get your mind right.
ed
Posts: 33919
Joined: Tue Jun 08, 2004 11:52 pm
Title: Rhino of the Florida swamp
### Re: SWATting
Bag o' Dicks Productions presents:
SWATing
A new raucous comedy that follows a wacky nerd phony phone caller and the inept cops that respond to his absurd calls
Starring:
Jimmy "JJ" Walker as Captain McGregor
Don Knotts as "The Landlord"
Wenn ich Kultur höre, entsichere ich meinen Browning!
Rob Lister
Posts: 20193
Joined: Sun Jul 18, 2004 7:15 pm
Title: Incipient toppler
Location: Swimming in Lake Ed
### Re: SWATting
Charged with involuntary manslaughter. 11 years potentially. I hope he serves every day of it.
http://beta.latimes.com/local/lanow/la- ... story.html
Pyrrho
Posts: 26348
Joined: Sat Jun 05, 2004 2:17 am
Title: Man in Black
Location: Division 6
### Re: SWATting
The guy's lawyer let him give an interview in which he admitted to things that will help to ensure that result.
The flash of light you saw in the sky was not a UFO. Swamp gas from a weather balloon was trapped in a thermal pocket and reflected the light from Venus.
WildCat
Posts: 14465
Joined: Tue Jun 15, 2004 2:53 am
Location: The 33rd Ward, Chicago
### Re: SWATting
Not a good idea to get a lawyer whose primary interest is promoting himself instead of the acting in the best interests of the client.
Do you have questions about God?
you sniveling little right-wing nutter - jj
gnome
Posts: 22448
Joined: Tue Jun 29, 2004 12:40 am
Location: New Port Richey, FL
### Re: SWATting
His public defender?
"If fighting is sure to result in victory, then you must fight! Sun Tzu said that, and I'd say he knows a little bit more about fighting than you do, pal, because he invented it, and then he perfected it so that no living man could best him in the ring of honor. Then, he used his fight money to buy two of every animal on earth, and then he herded them onto a boat, and then he beat the crap out of every single one. And from that day forward any time a bunch of animals are together in one place it's called a zoo! (Beat) Unless it's a farm!"
--Soldier, TF2
WildCat
Posts: 14465
Joined: Tue Jun 15, 2004 2:53 am
Location: The 33rd Ward, Chicago
### Re: SWATting
gnome wrote:His public defender?
Assigned by the court, or swept in to represent for free while primarily looking for publicity for himself since it's a high-profile case?
Do you have questions about God?
you sniveling little right-wing nutter - jj
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### Data Structures
Learn most important basic data structures used in programming computers. Such as Arrays, Linked List, Queues, Stacks, Trees…
Learn most important basic algorithms used on these data structures. Such as Sorting, Searching, and Hashing algorithms …
Learn basic tools to measure/compare the efficiency of these algorithms
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## Topic 1C
$E_{n}=\frac{h^{2}n^{2}}{8mL^{2}}$
Guzman_1J
Posts: 72
Joined: Wed Sep 18, 2019 12:16 am
### Topic 1C
I did not see Topic 1C homework problems in the syllabus. Does this mean we do not have to know the stuff on that part for any tests?
905416023
Posts: 54
Joined: Thu Jul 25, 2019 12:17 am
### Re: Topic 1C
I believe we do need to know 1C. On the syllabus, it says to read Appendix 1B, 1C, 1D, 1E. So we should probably know it if we were expected to read it.
Guzman_1J
Posts: 72
Joined: Wed Sep 18, 2019 12:16 am
### Re: Topic 1C
For the Quantum World? Where it says to "Omit 1C and Table 1D.1"? I'm just wondering because En =((h^2)(n^2))/(8 m L^2) is an equation that is in the Constant and Equations sheet on the website, but Appendix 1C is about scientific notation, I mean Topic 1C on page 23 because we were also not assigned any hw problems from there and have not used that equation as far as I remember.
Emma Popescu 1L
Posts: 105
Joined: Wed Sep 11, 2019 12:16 am
### Re: Topic 1C
I don't think we need to know the material from Topic 1C. He did not talk about most of the information from that section.
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## Saturday, November 08, 2014 ... /////
### A balanced world map: more than cute arts?
Frank Jacobs blogging for Big Think posted (and Sean Carroll reposted) a fascinating distorted map of the world drawn by Gaicomo Faiella:
What's so fascinating about it, Carroll asks? If you want to think yourself, please don't open the rest of this blog post, and/or don't read anything beneath the ads.
In fact, I can tell you that the map contains infinitely many coincidences or fine-tunings relatively to a generic set of continents you could draw according to the same template. Don't you still know what is special? Try to think harder and really don't read anything below the other ad. ;-)
What are those coincidences? The first one that many people notice is that the ocean has been suppressed and the continents were expanded – in such a way that their ratio is 50:50. You might think it's just approximately true. But it's actually exactly half.
And this exact coincidence is just one among infinitely many similar coincidences. The truth is that the map is actually completely symmetric – or antisymmetric, if you wish. Rotate the distribution of the continents and oceans, the image, by 180 degrees, and you will obtain this image. It isn't just a "rotated Earth": it is the same Earth with continents and oceans interchanged!
How does the symmetry work? Look at the picture once again:
Obviously, the boundaries have some kind of a $\ZZ_2$ symmetry. But how does the symmetry act on the surface of the Earth? Look at Caracas, Venezuela. You probably don't know exactly where it is. OK, so find the Panama Canal – the thinnest point separating the Americas – and the nearby point where South America and Africa nearly touch. Find the center of this line interval. That's Caracas.
Caracas is a center of the world! Hugo Chávez would have been thrilled to read such a comment on a conservative blog.
If you rotate the map around Caracas by 180 degrees, you get the same (or "opposite") map. In particular, now you see that the South America is the rotated version of the North Atlantic Ocean! And Africa is the rotated image of the North Pacific Ocean! If you still haven't seen the pattern with your eyes, look at South America; at North Atlantic; at South America; at North Atlantic; at South America... do you already see it? ;-) Similarly, do it with Africa and the North Pacific.
And you may continue like that. The Mediterranean Sea has the same shape as the 180-degree-rotated Australia. And the North and Baltic seas between Europe and Scandinavia are the image of New Zealand. And so on. Feel free to find the image of the place where you are right now. But be careful: Chances are that the image will be in the ocean. :-) Central Europe ends up being some boring place of the sea South of Australia.
If you get familiar with these maps, you will notice that the 180-degree rotation has another center beyond Caracas, of course. It's Vishakhapatnam, India. OK, that word sounds complicated so what I mean is the center of the Eastern Coast of India. If you rotate the map around this place by 180 degrees, you get the "anti-same" map, too.
I have some rough idea how I could generate this "best $\ZZ_2$-symmetric distortion of the Earth" using a computer, in Mathematica. (The artist has probably created it manually.) But it could be somewhat difficult for me who has no experience with similar manipulations. I think that Mathematica can do many of the steps concisely, by a single command, but I don't know these commands.
Fine. So because this "slightly distorted Earth" has a $\ZZ_2$ symmetry and it rotates the Earth by 180 degrees around an axis, it is useful to redefine the polar coordinates so that this axis plays a special role. In particular, let's use the term "New North Pole" for Caracas and "New South Pole" for the Visha-city. (I apologize to my dear Indian readers for shortening the name for the sake of the convenience of us, the barbarian non-Indians. My special apologies go to Balasubramaniam Sethuramanathan Sreejahannathan and his four brothers.)
Let's introduce the new latitude and the new longitude. Caracas has $\theta=0$ and the V-city has $\theta=\pi$. Note that on the (new) poles, the longitude is ill-defined. However, the $\ZZ_2$ symmetry is generated simply by$(\theta,\varphi) \mapsto (\theta, \varphi+\pi)$ So if the point $(\theta,\varphi)$ sits in the water, the point $(\theta, \varphi+\pi)$ is a point on the land, and vice versa! The existence of Feiella's image – and its relative proximity to the actual map of the world – suggests that this rule is "approximately" true for our mostly blue, not green planet (it is exactly true for the distorted half-blue, half-green planet).
Now, I think that a good question (OK, at least my question!) is whether there is any reason to think that this statement should be true?
My model will be weirder than an average science-fiction movie but of course I can offer you a mechanism that would produce a planet with this property. ;-) To prepare you, let's try something simpler. From the Earth, the Moon and the Sun look almost equally large. That's also a shocking coincidence, isn't it? (But in this case, it's just one fine-tuning, not infinitely many.) How could you explain this one?
Well, it's easy. Millions of years ago, the Moon was significantly larger. Or smaller. But there was a civilization on Earth based on a two-party system. One party wanted to preserve the full solar eclipses because these eclipses were a pillar of their religion. The other party wanted to sell the material from the Moon because the extraterrestrials paid a significant amount of cash for Earth to afford some cute things. What was the outcome? Of course, the second party sold as much as it could and the Moon was reduced to the minimum size allowing full solar eclipses.
Alternatively, the Moon materialized in the cone connecting a point P on Earth with the Sun out of the special miraculous solar photons that arrive to P. I don't know how it could happen but the Moon is the largest ball at the same distance that fits into this shining cone. :-)
Now you've been calibrated to the "generalized physics" we allow here. Fine. What about the symmetric distorted Earth? You may start with an exactly spherical (or ellipsoidal) Earth. But it is bombarded by meteors whose trajectories are always perpendicular to the axis of the Earth's rotation and they actually intersect this axis, too. When they hit the surface, they cause a depression at the point $(\theta,\varphi)$ where they hit the surface – this point becomes more ocean-like – but they add matter on the opposite side, $(\theta,\varphi+\pi)$. So the opposite point's altitude is increased – it becomes more continent-like.
Of course, I can't tell you why meteoroids' paths should be perpendicular to (and intersecting with) the axis of the Earth's spin. And I can't explain many other things that my "explanation" apparently depends upon. But it's fun to think about these possible reasons. Perhaps, it could be more than a coincidence that this $\ZZ_2$ symmetry is "almost" possible for Earth. Perhaps you might invent a somewhat more plausible mechanism that generates such a pattern on Earth?
(It would be slightly less awkward to invent mechanisms if the symmetry were a $\ZZ_2$ that reflects all three Cartesian coordinates' signs. But note that the orbifold $S^2/\ZZ_2$ would be an ${\mathbb R \mathbb P}^2$ in that case: on the surface, the continents and their wet images would look like mirror images of each other, not simple rotations.)
Even without a more plausible mechanism to generate the pattern, you may want to ask the question: Does the actual shape of the Earth's oceans and continents nontrivially support the claim that the $\ZZ_2$ symmetry holds more accurately than it does for a random distribution of the continents? I roughly know how I would construct a quantity that measures the degree at which this symmetry is preserved. But I don't want to tell you. Maybe you can offer your own, simpler rule? And what is the answer? Is there something special about the distribution we have on the Earth? I have no idea!
Sometimes, arts is just arts but sometimes arts adds lots of interesting questions if you have some fantasy that is waiting to be liberated. I will add one more comment.
Note that in all the comments about mechanisms, I was neglecting continental drift. I was assuming the current distribution and locations of the continents to be scientifically relevant but the continents have actually been moving and reshaping due to plate tectonics. An actual viable explanation would have to address the distribution of the continents as they existed billions of years ago, and the current distribution would simply be evolved out of the old one. It's strange because it's the current distribution, and not necessarily the old one, that supports the symmetry.
But couldn't it be true that the symmetry is "approximately true" at every point of the continental drift? If true, the continental drift would sort of have to respect the symmetry axis connecting Caracas and the V-city. Isn't this axis going through the Earth – approximately (and, in the idealization, exactly) belonging to the equatorial plane – the axis of some internal circulation in the Earth that dominate the continental drift?
The distribution of continents and oceans as we know them from the world map may be dismissed as a "random picture". And when most people hear "random", they no longer ask any questions. They think that if chance (a random generator) decides about something, it's no longer our business. However, if you're scientific, you know that there isn't anything such as a universal "random" thing. When something is random, you want to know what are the odds, what are the probability distributions (and what are the correlations in these distributions etc.), and those may reveal or hide lots of additional patterns and explanations that wait to be discovered.
Note that quantum mechanics also says that the results of all observations are "random" but this single adjective isn't everything that quantum mechanics tells us. Instead, it tells us the probability of any possible outcome for any possible experimental setup. And those calculable numbers ultimately end up being "similarly constrained" as the outcomes are in a deterministic theory. So when something looks "seemingly random", it just doesn't mean that there is nothing else to be discovered and it isn't true that there can be no way to "localize the right answer" by pure thought almost exactly!
#### snail feedback (3) :
That is a truly beautiful rendering! The areas of the land masses are well ratioed, too. Another planetary coincidence: Between the poles, little dry land is antipodal to other dry land. It is almost always land, through the center of the Earth, water or water, through the center of the Earth, land. Here is your antipodes utility. Click drag to move and mouse wheel to magnify,
http://www.findlatitudeandlongitude.com/antipode-map/
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### Home > GC > Chapter 8 > Lesson 8.1.5 > Problem8-53
8-53.
1. For each equation below, solve for w, if possible. Show all work. Homework Help ✎
1. 5w2 = 17
2. 5w2 − 3w − 17 = 0
3. 2w2 = −3
$\frac{5w^{2}}{5}= \frac{17}{5}$
$w^{2}= \frac{17}{5}$
$\sqrt{w^{2}}= \sqrt{\frac{17}{5}}$
w ≈ ± 1.84
Remember that since each equation has w² you should be looking for two possible answers.
It may help to use the Quadratic Formula.
w ≈ 2.17 and w ≈ 1.57
No real solution because you can't take the square root of a negative number.
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# VCI2016 - The 14th Vienna Conference on Instrumentation
Feb 15 – 19, 2016
Vienna University of Technology
Europe/Vienna timezone
## A 65 nm CMOS analog processor with zero dead time for future pixel detectors
Not scheduled
15m
Vienna University of Technology
#### Vienna University of Technology
Gusshausstraße 27-29, 1040 Wien
Board: 81
Poster Electronics
### Speaker
Luigi Gaioni (INFN - National Institute for Nuclear Physics)
### Description
Next generation pixel chips at the High-Luminosity (HL) LHC will be exposed to extremely high levels of radiation and particle rates. In the so-called Phase II upgrade, ATLAS and CMS will need a completely new tracker detector, complying with the very demanding operating conditions and the delivered luminosity (up to 5 × 10$^{34}$cm$^{−2}$s$^{−1}$ in the next decade). This work is concerned with the design of a synchronous analog processor with zero dead time developed in a 65 nm CMOS technology, conceived for pixel detectors at the HL-LHC experiment upgrades. It includes a low noise, fast charge sensitive amplifier featuring a detector leakage compensation circuit, and a compact single ended comparator that guarantees very good performance in terms of channel-to-channel dispersion of threshold without needing any pixel-level trimming. A 3-bit Flash ADC is exploited for digital conversion immediately after the preamplifier. The conference paper will provide a thorough discussion on the design of the different blocks making up the analog front-end channel.
### Primary author
Luigi Gaioni (INFN - National Institute for Nuclear Physics)
### Co-authors
David Christian (Fermilab) Davide Braga (Fermi National Accelerator Laboratory) Farah Fahim (Fermilab) Grzegorz Deptuch (Fermi National Accelerator Laboratory) Lodovico Ratti (University of Pavia) Thomas Zimmerman (Fermi National Accelerator Lab. (US)) Valerio Re (INFN)
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### AuthorTopic: In Pacific Skies - Woodlark and Trobiand Islands (Read 42109 times)
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#### agracier
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##### Re: In Pacific Skies - Woodlark and Trobiand Islands
« Reply #120 on: May 06, 2013, 03:00:43 PM »
Hi Raptor, could you possibly be missing these objects, because I just put this map into a new HSFX Install and while loading I got a loading error so I did a Missing Object's Search with Universal Static Ini Checker, and this is what it says:
buildings.House$storage_003 buildings.House$Rus_House_006
buildings.House$FurnitureTreeolive buildings.House$Rus_House_007
buildings.House$Rus_Barn_001 buildings.House$Rus_House_005
buildings.House$Rus_House_002 From what I can see in my static.ini, the above objects are all part of the standard objects in DBW ... so in theory they should be in Boomer's Object pack ... or so I would surmise ... they are not any recent new objects from the past year ... The treeolive is quite an old object even, it was part of a set of several tree objects made years ago ... I think that Kapteeni and I even used it on the old Andalucia map ... Logged #### BravoFxTrt • "BIGFOOT" • Modder • member • Offline • Posts: 13433 • Flying Ass Clown #13 ##### Re: In Pacific Skies - Woodlark and Trobiand Islands « Reply #121 on: May 06, 2013, 03:24:15 PM » OK, in my new HSFX Install I was using the Static.ini from HSFX, which intern cause Loading error, What I did was replaced the Staic.ini with one from DBW and Woodlark started right up. Hope this may be of some help for you Raptor. Logged MSI R9 280X 6GB x2/Crossfire /ASUS M5A99FX PRO R2.0 Mobo/AMD FX 4170 Bulldozer 4core CPU/RAM Kingston HyperX FURY 16GB/ Corsair RM 1000WATT PSU/Windows 10 Enterprise 64bit #### sphantom • retired Fighter Pilot • member • Offline • Posts: 766 • If you wake up and see grass roots growing ##### Re: In Pacific Skies - Woodlark and Trobiand Islands « Reply #122 on: May 06, 2013, 04:21:09 PM » I have the maps and they work great thru qmb and fmb and my install is dbw1.71 full monty and thanks to bravo have hsfx with all in one install. Logged #### BravoFxTrt • "BIGFOOT" • Modder • member • Offline • Posts: 13433 • Flying Ass Clown #13 ##### Re: In Pacific Skies - Woodlark and Trobiand Islands « Reply #123 on: May 06, 2013, 04:23:16 PM » Awesome Sphantom, glad you like it. Logged MSI R9 280X 6GB x2/Crossfire /ASUS M5A99FX PRO R2.0 Mobo/AMD FX 4170 Bulldozer 4core CPU/RAM Kingston HyperX FURY 16GB/ Corsair RM 1000WATT PSU/Windows 10 Enterprise 64bit #### X-Raptor • member • Offline • Posts: 238 ##### Re: In Pacific Skies - Woodlark and Trobiand Islands « Reply #124 on: May 06, 2013, 06:25:25 PM » OK, in my new HSFX Install I was using the Static.ini from HSFX, which intern cause Loading error, What I did was replaced the Staic.ini with one from DBW and Woodlark started right up. Hope this may be of some help for you Raptor. hO jes!. This may be the problem: static.ini need be of DBW .. but using this I think I could affect my HFSX "genuine" version may be?..now I go to check. Thank bravo, and agracier however for all help. Logged #### agracier • Modder • member • Offline • Posts: 3048 ##### Re: In Pacific Skies - Woodlark and Trobiand Islands « Reply #125 on: May 07, 2013, 06:16:12 AM » I extracted the actors.static for this map to see if the objects missing above could be hand edited out ... but verily, there are just too many of them on the map to do by hand ... unless you wish to do a session of an an hour or 3 of hand editing. I can send the extracted buildings file as a text file if you wish, but it's really abit of a job to do by hand. Installing the missing objects would be far easier ... once you find them that is ... Logged #### X-Raptor • member • Offline • Posts: 238 ##### Re: In Pacific Skies - Woodlark and Trobiand Islands « Reply #126 on: May 07, 2013, 06:22:02 AM » agracier I can have pacience to control your edit actors.static step by step.. the only doubt I have is IF missing obyect may be the cause because I repeat I have not found any indication of that into my log of FMB consolle .. Logged #### BravoFxTrt • "BIGFOOT" • Modder • member • Offline • Posts: 13433 • Flying Ass Clown #13 ##### Re: In Pacific Skies - Woodlark and Trobiand Islands « Reply #127 on: May 07, 2013, 06:45:37 AM » Your Log might not indicate an error so it is best you download and do a missing object search your self with the Universal Static .ini checker, Raptor. Logged MSI R9 280X 6GB x2/Crossfire /ASUS M5A99FX PRO R2.0 Mobo/AMD FX 4170 Bulldozer 4core CPU/RAM Kingston HyperX FURY 16GB/ Corsair RM 1000WATT PSU/Windows 10 Enterprise 64bit #### agracier • Modder • member • Offline • Posts: 3048 ##### Re: In Pacific Skies - Woodlark and Trobiand Islands « Reply #128 on: May 07, 2013, 09:10:40 AM » Here's the buildings file extracted from the actors.static for Woodlark. You can unzip and then manually edit the lines ... when you find an object that is missing from your install, then the entire line on which it appears in the outbuildings.txt file needs to be removed ... Then save and I'll see above recombining into a working actors.static. But I would follow the advice of first checking exactly which objects you are missing by using the Universal Static checker ... http://www.sendspace.com/file/aab4og Logged #### X-Raptor • member • Offline • Posts: 238 ##### Re: In Pacific Skies - Woodlark and Trobiand Islands « Reply #129 on: May 07, 2013, 03:56:00 PM » Ok roger that..go for universal static checker. Logged #### BravoFxTrt • "BIGFOOT" • Modder • member • Offline • Posts: 13433 • Flying Ass Clown #13 ##### Re: In Pacific Skies - Woodlark and Trobiand Islands « Reply #130 on: May 07, 2013, 03:59:59 PM » You wont regret it Raptor. Logged MSI R9 280X 6GB x2/Crossfire /ASUS M5A99FX PRO R2.0 Mobo/AMD FX 4170 Bulldozer 4core CPU/RAM Kingston HyperX FURY 16GB/ Corsair RM 1000WATT PSU/Windows 10 Enterprise 64bit #### X-Raptor • member • Offline • Posts: 238 ##### Re: In Pacific Skies - Woodlark and Trobiand Islands « Reply #131 on: May 07, 2013, 04:05:41 PM » ok m8's I've found this with universal static .ini checker: MISSING OBJECT >>> buildings.House$FurniturePalmAg3
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Hi, I'm Jesse. I like programming, math & theorem proving, the web, philosophy and history, and, strangely, email. (NB "like" doesn't necessarily imply "good at".) Work mainly in Racket & JavaScript. American.
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Please note: Early Bird tickets for #BOBkonf2023 are available for one more week -- after that, it's regular price for all of them.
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You could argue that using one-off purchases rather than subscriptions is leaving money on the table. But if I’ve got people using Cheyenne for just a couple of weeks and then canceling, multiple times, this pollutes churn statistics. And I think it’s betyer to be up-front with users about how the service will likely be used, instead of trying to get them to sign up for Yet Another Subscription.
For other use cases, a subscription does make sense because you’d be actively monitoring your site with Cheyenne. Maybe not every day, but certainly a couple times a month, for the indefinite future.
Indeed, “subscription” might not even be the right word for what I have in mind. Basically, it’s a one-off purchase. You get a seat, but just for, say, a month. No renewal, though you can of course come back again and use Cheyenne for another month later.
The idea is that, for some use cases, it makes sense to use Cheyenne for, say, a couple of weeks to get your site in shape; after that, you don’t need it. This is a scenario where auto-cancellation might make sense. You may want to just use the service once or twice a year, and that’s it. No need to get locked into a monthly subscription you don’t use.
For Cheyenne, I’m thinking of building in some logic for auto-canceling subscriptions. If you don’t use the service for, say, 90 days, you’re wasting money. Your account doesn’t get deleted, but your subscription gets cancelled. A bit of email automation can help survey the customer, find out why they’re not using the service.
Ah! The joys of writing your own webhook handles for billing.
Enjoying a bourbon recommended to me by. ChatGPT.
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I was an #emacs user before moving to #neovim. This made me chuckle.
project-mage.org/emacs-is-not-
Really enjoying lazygit. Magit in Emacs—my main tool for using git— seems to be suffering from a bug in a recent release, so I fired up lazygit just to see what it is. Pleasantly surprised!
github.com/jesseduffield/lazyg
You can create a webhook for new purchases in ConvertKit.
In my view, validation is one of those topics on the web that bores most people, but actually gets me fired up. I can think of all sorts of ways validation can be improved.
If there are some pains you've encountered with HTML validation, reply here or DM me. I can think of all sorts of possible features, but I would rather launch Cheyenne in a minimal form before adding features few people care about.
If you do HTML, let me know what you think about markup validation. The first version of Cheyenne will be pretty bare bones, but takes on a couple core pains I've had with validation:
(1) how to validate a whole website (not just a single page), and
(2) ignore certain validator messages.
I'm about to launch a new HTML markup validator I call Cheyenne. It's made for web folks of all kinds. I've been working on this for a while and am excited to get it out there.
cheyenne.dev/
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Some Racket news: Release process for Racket 8.8 is on its way and will be available early February from download.racket-lang.org/
Racket gets a release every quarter in Feb, May, Aug & Nov.
If you are a package maintainer get ready to update, if you are a user, remember you can use raco pkg migrate 8.7 to update your packages.
* Racket is a modern lisp and a descendent of scheme. The racket distribution includes an incremental native-code compiler*, the Racket language, Typed Racket, Datalog & several other languages, an IDE**, documentation and a variety of other tools. The distribution is suitable for new learners, application development, or language design and implementation.
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Asking ChatGPT to recommend bourbons similar to ones I like.
Happy to see that @Horse_ebooks is here on Mastodon! Looking forward to some great stuff there.
@mattmight Doing anything in Racket these days? Racketfest, the little Racket conference I organize, is coming up in March — we’d love to have you! (It’ll be in Berlin, far from AL, but I thought I’d ask.)
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A Mastodon instance for programming language theorists and mathematicians. Or just anyone who wants to hang out.
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Gabriel Infante-Lopez Homepage
# PAC-Learning Unambiguous k,l-NTS<= Languages
### Authors:
Luque, F.; Infante-Lopez, G.
### Source:
ICGI 2010, Springer, Valencia (2010)
### Abstract:
In this paper we present two hierarchies of context-free languages: the $k,l$-NTS languages and the $k,l$-NTS$^\leq$ languages. $k,l$-NTS languages generalize the concept of Non-Terminally Separated (NTS) languages by adding a fixed size context to the constituents, as $k,l$-substitutable languages generalize substitutable languages. $k,l$-NTS$^\leq$ languages are $k,l$-NTS languages that also consider the edges of sentences as possible contexts. We then prove that Unambiguous $k,l$-NTS$^\leq$ ($k,l$-UNTS$^\leq$) languages be converted to plain old UNTS languages over a richer alphabet. Using this and the result of polynomial PAC-learnability with positive data of UNTS grammars proved by Clark, we prove that $k,l$-UNTS$^\leq$ languages are also PAC-learnable under the same conditions.
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# How big can a chi-square table be?
For example, i know a traditional table would be something like male / female and yes / no.
I have conducted a survey on opinions of animal research. One of my questions is "do you have ethical misgivings" which is a yes or no response. I would like to compare responses of six different job types, to see if there is a statistically significant difference among those job types as to whether they have ethical misgivings. Is this 6 x 2 permittable in a Chi squared test?
To take it further, say I again want to compare the same job categories with a nominal outcome with 4 levels (Likert scale), would a 6 x 4 table be permitted? I have a small sample size (n=81), and I have tried Ordinal Regression though fail the assumptions. This is a simple project and I would just like some basic comparisons. I am using SPSS.
Thanks so much
• Actually, your example is a fairly trivial one as the sample size, number of factors and levels within factors is small. So, nothing to worry about there, unless there are structural zeros or small cell sizes in the tables. Bishop in her 1975 book, Discrete Multivariate Analysis, proposed an adjustment to chi-square when the sample sizes were quite large. More recently, David Dunson has written papers on tensor regression with massive numbers of categorical features (arxiv.org/abs/1509.06490). Jul 27 '17 at 13:37
• +1. One common rule of thumb is to have at least 5 expected entries in each cell. With $n=81$ and $6\times 4=24$ cells, we'd expect $81/24=3.4$ entries, so I'd be careful with the $\chi^2$ approximation here. However, the example is still small enough that one can approximate the $p$ value by simulations, e.g., using R's chisq.test function with simulate.p.value=TRUE. Jul 27 '17 at 12:42
• @Stephan quotes the rule correctly. However, many have found that this rule is unnecessarily strict. The $\chi^2$ distribution often works well provided not too many of the cells have expectations less than $5$ (and watch out for cells with expectations of zero!). "Not too many" may be less than $25\%$ or so. I suspect that the larger the table gets, the more you can tolerate low-expectation cells for this analysis. At some point the whole exercise becomes pointless: with a huge table, the model grows too complicated and should be replaced (typically) by a mixed model.
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# Fundamentals of Music Acoustics
Any signal that may be represented as an amplitude varying over time has a corresponding frequency spectrum. This applies to concepts (and natural phenomena) that most human beings encounter daily, without giving them a second thought. Such as visible light (and colour perception), radio/TV channels, wireless communications… Even the regular rotation of the Earth. Even the sound of music…
When these physical phenomena are represented in the shape of a frequency spectrum, certain physical descriptions of their internal processes become much simpler. A lot of the time, the frequency spectrum clearly shows harmonics, visible as distinct spikes or lines at particular frequencies, that provide insight into the mechanisms that generate the entire signal.
Music acoustics, the science behind music, is in fact the branch of acoustics concerned with researching and describing the physics of music. For example… How can individual sounds mesh together and create music? How do musical instruments work? What is in a human voice, in terms of the physics of speech and singing? How can a computer analyse a melody? What is the clinical use of music in music therapy?
# What is that Sound?
Acoustics is the general area of science that deals with the study of mechanical waves, whether in gases, liquids, and solids, including oscillation, vibration, sound, ultrasound and infrasound.
The perception of sound is a subjective experience.
Air pressure variations against the ear drum, and their subsequent physical and neurological signal processing in the human brain, are left open to interpretation at each individual level.
Most sounds that people do recognise as ‘musical’ are dominated by periodic or regular vibrations rather than non-periodic ones. In other words, musical sounds typically have a definite pitch.
Sounds with definite pitch have harmonic frequency spectra or close to harmonic spectra. The transmission of these pressure variations through air is via a sound wave.
Using the simplest example, the sound of a sine wave, which is considered to be the most basic model of a sound waveform, causes the air pressure to increase and decrease regularly, and is heard as a very pure tone.
# Perfect Pitch
Pure tones can be produced by tuning forks or whistling. The rate at which the air pressure oscillates is the frequency f of the tone, measured in oscillations (or cycles) per second. The typical unit often used is called a hertz (Hz). Frequency is the primary determinant of the perceived pitch.
Pitch is a perceptual property that allows the ordering of sounds on a frequency-related scale. Pitches are qualitatively compared as being “higher” or “lower” in the sense associated with musical melodies. Normally, this requires sound whose frequency is clear and stable enough to distinguish it from noise.
Pitch is a major auditory attribute of musical tones, along with the duration, the loudness, and the timbre. Pitch can be quantified as a frequency, but it is not a purely objective physical property. Indeed, it is known as a subjective ‘psycho-acoustical’ attribute of sound. The study of pitch and pitch perception has been a central problem in the domain of psycho-acoustics, instrumental in forming and testing theories of sound representation, processing, and perception in the auditory system.
### The perception of sound is a subjective experience.
It is worthy of noting that the frequency of musical instruments can change with altitude due to changes in air pressure.
# Tarzan’s Sound Spectrum
The majority of sounds are a complex mix of vibrations.
Ever heard of Tarzan? His jungle call had no clear pitch.
A sound spectrum displays the different frequencies present in a sound. Depending on where you are reading this, you will hear a variety of sounds in your surrounding environment. Can you hear the wind outside, the rain or the rumble of traffic? Or perhaps there is music in the background, in which case there is a mixture of high notes and low notes, and some sounds (drum beats and cymbal crashes) with no clear pitch.
The MGM studio that made the first Tarzan movies with Johnny Weissmuller, famously claimed to have enhanced the yell in post-production. Reportedly, they added and mixed a second track of Weismuller’s voice, somewhat amplified, a hyena howl, played backwards, a note sung by a female soprano, with the speed varied to produce a fluttery sound, the growl of a dog, the raspy note of a violin’s G-string being bowed, and finally… the bleat of a camel !!
Anyhow…
Sound spectra are produced using a microphone to measure the sound pressure over a certain time interval, an analogue-digital converter to convert it to a series of numbers (corresponding to the microphone voltage) as a function of time, and a computer to perform a digital Fourier transform calculation upon these numbers.
Broadly speaking, a sound spectrum is a graphic representation of a sound sample, in terms of the amount of vibration produced at each individual frequency. Usually, the data is presented as a graph of either power (or pressure) as a function of frequency.
The power is usually measured in decibels, while the frequency is measured in vibrations per second (or hertz, Hz), or in thousands of vibrations per second (or kilohertz, kHz).
Check out Tarzan’s sonogram:
If the loudness changes, the size (or amplitude) of the spectral components gets larger.
If the pitch changes, the frequency of all of the components increases.
If the sound changes, but loudness and pitch remains the same, the timbre – that is the sum of all the qualities that are different in two different sounds which have the same pitch and loudness – changes.
Singing “ah” and “eeh” at the same pitch and loudness, will result in a significant difference between the two spectra.
## Quantifying Sound: How Many Decibels?
The decibel (dB) is used to measure sound level, but it is also widely used in electronics, signals and communication. The decibel is a logarithmic unit used to describe a ratio. This ratio may involve power, sound pressure, voltage, intensity or several other variables.
The phon and the sone are additional units related to loudness.
For example, let us assume that we have two loudspeakers, the first playing a sound with power P1, and another playing a louder version of the identical sound with power P2. Everything else being kept the same.
The difference in decibels between the two speakers is defined as
$10 log (\frac {P_2} {P_1}) dB$,
with a log to the base 10.
If the second loudspeaker produces twice as much power than the first speaker, the difference in dB is then
$10 log (\frac {P_2} {P_1}) = 10 log (2) = 3 dB$.
If the second speaker had 10 times the power of the first one, the difference in dB would be
$10 log (\frac {P_2} {P_1}) = 10 log (10) = 10 dB$.
If the second one had a million times the power of the first, the difference in dB would be
$10 log (\frac {P_2} {P_1}) = 10 log (10^6) = 60 dB$.
# Pipes and Harmonics: Open and Closed Waves
The flute is a nearly cylindrical instrument, open to the outside air at both ends. The player leaves the embouchure hole open to the air, and blows across it.
The clarinet, on the other hand, is a roughly cylindrical instrument, open to the outside air at the bell, but closed by the mouthpiece, reed and the player’s mouth and lips at the other end.
The two instruments have roughly the same length. (The bore of the clarinet is a little narrower than that of the flute, but this difference is irrelevant here.)
The left diagram shows the type of sound waves occurring in the case of a flute. On top, the pressure curve (red) is only half a cycle of a sine wave. Hence the longest sine wave fitting into the open pipe is twice as long as the pipe. A flute is about 0.6 m long, so it can produce a wavelength that is about twice as long, which is about $2 L = 1.2 m$. The longest wave is also its lowest note, so let’s try calculating the frequency – that is, the speed divided by the wavelength.
## The Speed of Sound
$c_s = 340 m s^-1$.
The sound frequency is given by
$f_s = \frac {c_s}{\lambda}$,
where λ is the wavelength.
This gives a frequency $f = \frac {c_s}{2 L} = 280 s^-1 = 280 Hz$.
Given the approximations, this is close to the frequency of middle C, the lowest note on a flute. We can also fit in waves that are equal to the length of the flute (half the fundamental wavelength, therefore twice the frequency of the fundamental), 2/3 the length of the flute (one third the fundamental wavelength, so 3 times the frequency of the fundamental), 1/2 of the length of the flute (one quarter the wavelength, so 4 times the frequency of the fundamental).
This set of frequencies is the complete harmonic series.
In the top right diagram, the blue curve is only a quarter of a cycle of a sine wave. Hence the longest sine wave that fits into the closed pipe is four times as long as the pipe. Therefore, a clarinet can produce a wavelength that is about four times as long as a clarinet, which is about $4 L = 2.4 m$.
This gives a frequency of $f = \frac {c_s}{\lambda} = \frac {c_s}{4 L} = 140 s^{-1} = 140 Hz$one octave lower than the flute. Now, the lowest note on a clarinet is either the D or the C# below middle C. Again, given the approximations, this works out. We can fit in a wave if the length of the pipe is 3/4 of the wavelength, i.e. if wavelength is 1/3 that of the fundamental and the frequency is 3 times that of the fundamental. But we cannot fit in a wave with a 1/2 or 1/4 of the fundamental wavelength (twice or four times the frequency). So the second register of the clarinet is a musical 1/12 above the first.
## Try your Fingers at the Virtual Flute
According to the physics website of UNSW (University of New South Wales, Australia), there are approximately 40,000 different fingerings for the Boehm flute. Results are based on theoretical calculations, using a theory that agrees well with experiment for the hundred or so acoustic impedance spectra measured at UNSW.
All flutes can vary. Slide and cork positions vary. Key adjustments and embouchures vary. Temperature varies. Already, we saw that the frequency of musical instruments varies with altitude due to simple changes in air pressure.
Theory only makes approximations.
More details are available from the UNSW extensive site on Flute Music Acoustics and Clarinet Music Acoustics… and much more.
Play The Virtual Flute
And there’s even more fun stuff here. Prepare for the amazing…
# Clarinet Robot Wins International Competition!
This clarinet-playing robot was built by a NICTA − UNSW team for the Artemis Orchestra Competition in 2008. The contest rules require that embedded device robots, having a mass of less than 20 kg, play unmodified musical instruments. The clarinet robot was the international competition winner.
However, the aim is not to replace human musicians. So, why a clarinet robot? The design and development of such a robot provides:
• an interesting challenge to understand and implement a few of the complicated things that humans do while performing music,
• an interesting way for the music acoustics lab of the NICTAUNSW team to see how well we understand clarinet playing.
The clarinet robot has been the subject of experimentation to analyse how the pitch, loudness, timbre and transients produced by the clarinet depend on the fingering, mouth pressure, lip force and damping, bit position, reed hardness and mouth geometry.
Look at results and sound files.
# As for the Didgeridoo…
The didgeridoo (or didjeridu), or yidaki (or yiraki) in the language of the Yolngu, one of the peoples of Northern Australia, where the instrument originated, is a deceptively simple sound instrument. A wooden tube, of about 1.2 m to 1.5 m in length, that was hollowed out by termites in the thin trunk of a eucalyptus tree, and with a ring of beeswax around the mouthpiece for sealing and player’s comfort. Taken altogether, the didgeridoo seems a very unusual tribal music instrument. Although it plays only one note, the didgeridoo is capable of a spectacular range of different sounds, and the rhythmic variation of these sounds is its chief musical interest.
The instrument is closed at one end by the player’s lips and face. The difference between closed and open pipes has been explained above by comparing both using wave diagrams. However, we considered models of ideal cylindrical pipes. Most didgeridoos are flared, and they may have complicated surface geometries.
What makes the sound of the didgeridoo so unique and varied is the powerful interaction between
### In the case of didgeridoo-playing, it is essential.
The didgeridoo player’s lips produce a sound wave that travels into the instrument, but it also travels back in the other direction, into the vocal tract. In normal speech, the vocal tract is a resonator designed to assist the radiation of some frequency bands, but not others. Its resonances are what permits the production of different speech sounds. (See voice music acoustics for an introduction.)
For a didjeridu player, the vocal tract is working backwards: it still has resonances, but the vibration is coming from the lips, rather than from the vocal folds. Whether in speech, singing, or didgeridoo-playing, the frequencies at which the vocal tract resonates are determined by the shape of the tract, especially by the position and shape of the tongue.
And that gives a range of sounds like that…
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# B Thin film interference - why thin, exactly?
Tags:
1. Mar 10, 2017
### FranzDiCoccio
Hi,
in every explanation of thin film interference I came across, little or nothing is said as to why the layer of transparent material creating the effect should be thin.
What would go wrong if that is not the case?
I'm asking because it seems to me that, in principle, the mathematic explanation of the phenomenon would work for (admittedly very ideal) "thick films" too.
What is the point here? Perhaps that a macroscopic layer, however smooth, has "imperfections" on a scale much bigger than the wavelength of the light? I guess that these would spoil any interference.
I think that with careful deposition techniques one could create a very even "macroscopic" layer. That should exhibit the same interference patterns of a thin film, right?
I also thought of light absorption, but I think that this is not a good explanation... After all some macroscopically thick materials are pretty transparent.
Thanks a lot for any insight
Franz
2. Mar 10, 2017
### Grinkle
I don't think there is any deeper understanding to be gained by thinking of films that are more than 1/2 wavelength of light thick may be the reason.
I am not sure, but I think you are right that the interference pattern can occur for thicker films as well at integral multiples of wavelenghts, but its not any different physics.
3. Mar 10, 2017
Besides the need for surface flatness and parallelism, the angle of incidence requirement to get the same interference (e.g. in a bandpass filter) becomes much more stringent if the film becomes much thicker. The film will greatly change its passband of constructive interference with a small change in incident angle if the film is anything but very thin. $\\$ Editing.. Also, spectrally, the interference maxima become much closer together as the material becomes thicker. I believe $\Delta \lambda=\lambda^2/(2nd)$ between the interference peaks. Running a spectral scan and observing these interference peaks becomes very difficult for a sample as thick as $d=1 \, mm$ that has parallel faces. It requires very high a very high resolution spectral measurement to observe the peaks (and valleys) of the transmission spectrum in such a case. (graphing transmission vs. wavelength). Generally thin film filters are made of several layers of the same interference so that the peaks become sharper. They generally will have harmonics(e.g. 2x the frequency which is one half the wavelength) of "pass" wavelengths or spectral regions, but unwanted harmonics can often be blocked with materials that do not pass these unwanted harmonics.
Last edited: Mar 10, 2017
4. Mar 10, 2017
### FranzDiCoccio
Hi Grinkle and Charles,
@ Charles.
Correct me if I'm wrong. You're pointing out that an ideal macroscopic layer would work only for monochromatic light, in the sense that the maxima for the different components of, say, white light would be so close to each other that basically one would see white light anyways.
I'm trying to keep things simple. I'd like to hash out a reasonable and intuitive explanation about what goes wrong with "thick films".
For the sake of simplicity I'd like to limit this to almost zero angle of incidence.
So could I say "in principle one could grow an ideal, macroscopically thick slab and everything would be the same with a monochromatic light, provided that the thickness of the sample is the suitable (possibly very large) integer or half-integer multiple of the wavelength. However, when using white light, the interference maxima of different wavelengths would be so packed together that it would be extremely hard to resolve them".
Does this work?
5. Mar 10, 2017
That is correct (comments about white light), but even with a very monochromatic source, (such as a laser), for a macroscopically thick slab with precisely parallel faces, it could also require very good collimation, i.e. a beam that had one single angle of incidence to observe the interference. There are Fabry-Perot interferometers in use in Optics labs that have a significant distance (perhaps 1/2") between the half-silvered mirrors. It is possible to establish interference with such an apparatus (or a similar one which is a Michelson interferometer) with a monochromatic source. In one experiment we did in college with a (editing this) Michelson interferometer, we used the two lines (5889 and 5996 Angstroms) in the sodium doublet (source was a sodium arc lamp). The source was incident on a diffuser plate so that interference rings (instead of a single bright or reduced intensity across the plane) were observed. By varying the distance between the plates, the ring interference patterns from the 5889 and 5996 lines were made to cycle through each other. From this info, we were able to get a value for the difference between the two wavelengths. This experiment took much effort to align the mirrors=get them parallel so that the interference pattern was centered. In general, experimentation on thin films is far easier. Interference can be observed on macroscopic thicknesses, but the experimental requirements, both spectral (laser lines and even spectral lines from electronic transitions in atoms can work for such interference) and alignment requirements, are much more stringent.
Last edited: Mar 10, 2017
6. Mar 10, 2017
### FranzDiCoccio
Cool example... Yes, I remember doing the same experiment, but that was such a long time ago I did not make the connection. Very helpful, thanks again!
7. Mar 10, 2017
I edited the above=I believe the apparatus we used was a Michelson, but the principles are similar.
8. Mar 10, 2017
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# NAG CL Interfaceg01euc (prob_vavilov)
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CL Name Style:
## 1Purpose
g01euc returns the value of the Vavilov distribution function ${\Phi }_{V}\left(\lambda \text{;}\kappa ,{\beta }^{2}\right)$.
It is intended to be used after a call to g01zuc.
## 2Specification
#include
double g01euc (double x, const double comm_arr[])
The function may be called by the names: g01euc, nag_stat_prob_vavilov or nag_prob_vavilov.
## 3Description
g01euc evaluates an approximation to the Vavilov distribution function ${\Phi }_{V}\left(\lambda \text{;}\kappa ,{\beta }^{2}\right)$ given by
$ΦV(λ;κ,β2)=∫-∞λϕV(λ;κ,β2)dλ,$
where $\varphi \left(\lambda \right)$ is described in g01muc. The method used is based on Fourier expansions. Further details can be found in Schorr (1974).
## 4References
Schorr B (1974) Programs for the Landau and the Vavilov distributions and the corresponding random numbers Comp. Phys. Comm. 7 215–224
## 5Arguments
1: $\mathbf{x}$double Input
On entry: the argument $\lambda$ of the function.
2: $\mathbf{comm_arr}\left[322\right]$const double Communication Array
On entry: this must be the same argument comm_arr as returned by a previous call to g01zuc.
None.
## 7Accuracy
At least five significant digits are usually correct.
## 8Parallelism and Performance
g01euc is not threaded in any implementation.
g01euc can be called repeatedly with different values of $\lambda$ provided that the values of $\kappa$ and ${\beta }^{2}$ remain unchanged between calls. Otherwise, g01zuc must be called again. This is illustrated in Section 10.
## 10Example
This example evaluates ${\Phi }_{V}\left(\lambda \text{;}\kappa ,{\beta }^{2}\right)$ at $\lambda =0.1$, $\kappa =2.5$ and ${\beta }^{2}=0.7$, and prints the results.
### 10.1Program Text
Program Text (g01euce.c)
### 10.2Program Data
Program Data (g01euce.d)
### 10.3Program Results
Program Results (g01euce.r)
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# Tomasz Downarowicz (Wroclaw) Multiorder in countable amenable groups; a promising new tool in entropy theory.
Abstract: Let G be a countable group. A mutliorder is a collection of
bijections from G to Z (the integers) on which G acts by a special
"double shift". If G is amenable, we also require some uniform Folner
property of the order intervals. The main thing is that mutiorder exists
on every countable amenable group, which can be proved using tilings.
For now, multiorder provides an alternative formula for entropy of a
process and we are sure in the nearest future it will allow at produce
an effective formula for the Pinsker sigma-algebra. And we hope for
many more applications.
This is work in progress with Piotr Oprocha (Kraków) and Guohua
Zhang (Shanghai).
## Date:
Tue, 21/04/2020 - 14:00 to 15:00
## Location:
E-seminar Zoom meeting ID 914-412-92758
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