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# 8.2: Factoring Polynomials
You should be familiar with the quadratic formula, which allows us to factor any polynomial of degree two, into linear factors. Specifically, it tells us that the roots of $$ax^2 + bx + c$$ are
$\dfrac{-b ± \sqrt{b^2 -4ac}}{2a}$
Notice that this does not tell us immediately how to factor $$ax^2 + bx + c$$, because it’s missing a constant factor of a. So if we want to factor $$ax^2 + bx + c$$, we actually get
$ax^2 + bx + c = a \left(x - \left( \dfrac{-b + \sqrt{b^2 -4ac}}{2a} \right) \right) \left(x - \left( \dfrac{-b - \sqrt{b^2 -4ac}}{2a} \right) \right)$
Recall that in order to use the Generalised Binomial Theorem, we need the constant term to be $$1$$. If you are very comfortable with algebraic manipulations, you can use the quadratic formula to factor as above, and then divide each factor by the appropriate value so as to make the constant term $$1$$. This may create a messy constant outside the whole thing, and a messy coefficient of $$x$$ in each term, but if you are careful, you can get the correct answer this way.
If you are more confident in memorising another formula (closely related to the quadratic formula) for factoring $$ax^2 + bx + c$$, you can also factor a quadratic polynomial directly into the form we want, using the following formula:
$ax^2 + bx + c = c \left(1 - \dfrac{-b + \sqrt{b^2 -4ac}}{2c} x \right) \left(1 - \dfrac{-b - \sqrt{b^2 -4ac}}{2c} x \right)$
Sometimes a denominator will already be factored in the formula for a generating function, but when it isn’t, either of the above methods can be used to factor it.
Example $$\PageIndex{1}$$
Factor $$3x^2 − 2x + 1$$ into linear factors.
Solution
We will use the formula given above. We have $$a = 3$$, $$b = −2$$, and $$c = 1$$. Then
$$$$\begin{split} 3x^2 − 2x + 1&= \left(1 - \dfrac{2 + \sqrt{4 -12}}{2} x \right) \left(1 - \dfrac{2 - \sqrt{4 -12}}{2} x \right) \\ &= (1 − (1 + i\sqrt{2})x)(1 − (1 − i\sqrt{2})x). \end{split}$$$$
It is always a good idea to check your result, by multiplying the factors back out.
When coefficients in the factorisation get ugly (even complex, as in the example above), you might find the algebra involved in working out the coefficients hard to deal with. Let’s work through an example of this, using the factorisation we’ve just completed.
Example $$\PageIndex{2}$$
Find the coefficient of $$x^r$$ in $$f(x)$$, where
$$f(x) = \dfrac{1}{3x^2 − 2x + 1}$$
Solution
We have determined in the previous example, that
$$3x^2 − 2x + 1 = (1 − (1 + i \sqrt{2})x)(1 − (1 − i \sqrt{2})x),$$
so we need to solve for $$A$$ and $$B$$, where
$$$$\begin{split} f(x)&= \dfrac{1}{3x^2 − 2x + 1} \\ &= \dfrac{A}{1 − (1 + i\sqrt{2})x} + \dfrac{B}{1 − (1 - i\sqrt{2})x} \\ &= \dfrac{A(1 − (1 − i\sqrt{2})x) + B(1 − (1 + i\sqrt{2})x)}{3x^2 − 2x + 1}, \end{split}$$$$
Thus,
$$A(1 − (1 − i\sqrt{2})x) + B(1 − (1 + i\sqrt{2})x) = 1 + 0x$$,
so the constant term gives $$A+B = 1$$, while the coefficient of $$x$$ gives $$A(1−i \sqrt{2})+B(1+i \sqrt{2}) = 0$$. Substituting $$B = 1 − A$$ into the latter equation, gives
$$A − i \sqrt{2}A + 1 + i \sqrt{2} − A − i \sqrt{2}A = 0$$,
so $$1 + i \sqrt{2} = i2 \sqrt{2}A$$. Hence
$$A = \dfrac{1 + i \sqrt{2}}{i2 \sqrt{2}} = \dfrac{1}{2 \sqrt{2}i} + \dfrac{1}{2}$$
We make the denominator of the first fraction rational, by multiplying numerator and denominator by $$\sqrt{2}i$$, giving
$$A = -\dfrac{\sqrt{2}i}{4} + \dfrac{1}{2}$$
Now since $$B = 1 − A$$, we have
$$B = \dfrac{1}{2} + \dfrac{\sqrt{2}i}{4}$$
To make things a bit simpler, we’ll rewrite $$A$$ as $$\dfrac{(2 − \sqrt{2}i)}{4}$$, and $$B = \dfrac{(2 + \sqrt{2}i)}{4}$$.
Thus we have
$$f(x) = \dfrac{\dfrac{(2 − \sqrt{2}i)}{4}}{1 − (1 + i \sqrt{2})x} + \dfrac{\dfrac{(2 + \sqrt{2}i)}{4}}{1 − (1 - i \sqrt{2})x}$$
Using the Generalised Binomial Theorem and $$y = (1 + i \sqrt{2})x$$, we see that the first fraction expands as
$$\left[\left(\dfrac{(2 − \sqrt{2}i)}{4}\right)\right](1 + y + y^2 + y^3 + . . .)$$,
and the coefficient of $$x^r$$ in this, will be $$\left[\left(\dfrac{(2 − \sqrt{2}i)}{4}\right)\right](1 +i \sqrt{2})^r$$. Similarly, with $$y = (1−i \sqrt{2})x$$, the second fraction expands as
$$\left[\left(\dfrac{(2 + \sqrt{2}i)}{4}\right)\right](1 + y + y^2 + y^3 + . . .)$$,
and the coefficient of $$x^r$$ in this, will be $$\left[\left(\dfrac{(2 + \sqrt{2}i)}{4}\right)\right](1 − i \sqrt{2})^r$$.
So the coefficient of $$x^r$$ in $$f(x)$$ is
$$\left[\left(\dfrac{(2 − \sqrt{2}i)}{4}\right)\right](1 +i \sqrt{2})^r + \left[\left(\dfrac{(2 + \sqrt{2}i)}{4}\right)\right](1 − i \sqrt{2})^r$$
You can see from this example that the algebra can get ugly, but the process of finding the coefficient of $$x^r$$ is nonetheless straightforward.
Exercise $$\PageIndex{1}$$
For each of the generating functions given, factor the denominator and use the method of partial fractions to determine the coefficient of $$x^r$$.
1. $$\dfrac{x}{x^2+5x−1}$$
2. $$\dfrac{2+x}{2x^2+x−1}$$
3. $$\dfrac{x}{x^2−3x+1}$$
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# the number 69 , 300 , 000 in scientific notation.
### Precalculus: Mathematics for Calcu...
6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071
### Precalculus: Mathematics for Calcu...
6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071
#### Solutions
Chapter 1.2, Problem 77E
a)
To determine
## To write: the number 69,300,000 in scientific notation.
Expert Solution
69,300,000=6.93×107
### Explanation of Solution
Given information:
Given number
69,300,000
Concept used:
Scientific notation:
A notation in which a given quantity can be expressed as a number with significant digits required for a specified degree of accuracy and multiplied by 10 to the appropriate power.
Calculation:
Consider the given number.
69,300,000
So, the number can be expressed in scientific notation as
69,300,000=6.93×10000000=6.93×107
Hence, 69,300,000=6.93×107 .
b)
To determine
### To write: the number 7,200,000,000,000 in scientific notation.
Expert Solution
7,200,000,000,000=7.2×1012
### Explanation of Solution
Given information:
Given number
7,200,000,000,000
Concept used:
Scientific notation:
A notation in which a given quantity can be expressed as a number with significant digits required for a specified degree of accuracy and multiplied by 10 to the appropriate power.
Calculation:
Consider the given number.
7,200,000,000,000
So, the number can be expressed in scientific notation as
7,200,000,000,000=7.2×1000000000000=7.2×1012
Hence, 7,200,000,000,000=7.2×1012 .
c)
To determine
### To write: the number 0.000028536 in scientific notation.
Expert Solution
0.000028536=2.8356×105
### Explanation of Solution
Given information:
Given number
0.000028536
Concept used:
Scientific notation:
A notation in which a given quantity can be expressed as a number with significant digits required for a specified degree of accuracy and multiplied by 10 to the appropriate power.
Calculation:
Consider the given number.
0.000028536
So, the number can be expressed in scientific notation as
0.000028536=2.8536÷100000=2.8536÷105=2.8356×105
Hence, 0.000028536=2.8356×105 .
d)
To determine
### To write: the number 0.0001213 in scientific notation.
Expert Solution
0.0001213=1.213×104
### Explanation of Solution
Given information:
Given number
0.0001213
Concept used:
Scientific notation:
A notation in which a given quantity can be expressed as a number with significant digits required for a specified degree of accuracy and multiplied by 10 to the appropriate power.
Calculation:
Consider the given number.
0.0001213
So, the number can be expressed in scientific notation as
0.0001213=1.213÷10000=1.213÷104=1.213×104
Hence, 0.0001213=1.213×104 .
### Have a homework question?
Subscribe to bartleby learn! Ask subject matter experts 30 homework questions each month. Plus, you’ll have access to millions of step-by-step textbook answers!
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# Optimal Confidence for Monte Carlo Integration of Smooth Functions
Presented by:
Robert J. Kunsch Universität Osnabrück
Date:
Thursday 21st February 2019 - 13:40 to 14:15
Venue:
INI Seminar Room 1
Abstract:
We study the complexity $n(\varepsilon,\delta)$ of approximating the integral of smooth functions at absolute precision $\varepsilon > 0$ with confidence level $1 - \delta \in (0,1)$ using function evaluations as information within randomized algorithms. Methods that achieve optimal rates in terms of the root mean square error (RMSE) are not always optimal in terms of error at confidence, usually we need some non-linearity in order to suppress outliers. Besides, there are numerical problems which can be solved in terms of error at confidence but no algorithm can guarantee a finite RMSE, see [1]. Hence, the new error criterion seems to be more general than the classical RMSE. The sharp order for multivariate functions from classical isotropic Sobolev spaces $W_p^r([0,1]^d)$ can be achieved via control variates, as long as the space is embedded in the space of continuous functions $C([0,1]^d)$. It turns out that the integrability index $p$ has an effect on the influence of the uncertainty $\delta$ to the complexity, with the limiting case $p = 1$ where deterministic methods cannot be improved by randomization. In general, higher smoothness reduces the effort we need to take in order to increase the confidence level. Determining the complexity $n(\varepsilon,\delta)$ is much more challenging for mixed smoothness spaces $\mathbf{W}_p^r([0,1]^d)$. While optimal rates are known for the classical RMSE (as long as $\mathbf{W}_p^r([0,1]^d)$ is embedded in $L_2([0,1]^d)$), see [2], basic modifications of the corresponding algorithms fail to match the theoretical lower bounds for approximating the integral with prescribed confidence.
Joint work with Daniel Rudolf
[1] R.J. Kunsch, E. Novak, D. Rudolf. Solvable integration problems and optimal sample size selection. To appear in Journal of Complexity.
[2] M. Ullrich. A Monte Carlo method for integration of multivariate smooth functions. SIAM Journal on Numerical Analysis, 55(3):1188-1200, 2017.
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# Hydro-electric generation from river beds
## Recommended Posts
i ment building and matenecice cost but i see the piont given the world we live in if some bright spark has a good idea then everyone has to have or do what they do
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i ment building and matenecice cost but i see the piont given the world we live in if some bright spark has a good idea then everyone has to have or do what they do
It has nothing to do with "having to do the same as everyone else".
Well... at least not directly. Everybody follows an idea that is the most profitable.
At least two people here suggested that hydro power is cheaper than the other options. I claimed that hydro power is even cheaper than fossil fuels. That price includes the building and maintenance.
Source (look under "costs")
So, to continue the discussion, perhaps you can respond to our points. Talk about the economics of the river-bed-turbines. Talk about efficiency. We need to compare it. This is a science forum, we need data!
Edited by CaptainPanic
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hiw big do you think these turbine are they are only small so that they dont desturb anything for ages
also i was talking about the turbines in the water being cheaper then dams not fossil feuls
only an idiot would think they are cheap with the comsupmtion rate they will most definatly run or sooner rather then later
this teard was onrianly called globel warming in the genetics area so i was thinking about the enviroment when i posted this
whats better to save the ecomimy that can sort it self out or the enviroment that has to have lots of help being sorted out because unless a imagenery fairy pixie thing will save who will. we the adluts have to be resonceible for making sure children inherit a clean safe beautiful world not some stink hole full of pollution
plus i am responding your your points i just might have said the wrong words to exspress my self
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hiw big do you think these turbine are they are only small so that they dont desturb anything for ages
also i was talking about the turbines in the water being cheaper then dams not fossil feuls
only an idiot would think they are cheap with the comsupmtion rate they will most definatly run or sooner rather then later
this teard was onrianly called globel warming in the genetics area so i was thinking about the enviroment when i posted this
whats better to save the ecomimy that can sort it self out or the enviroment that has to have lots of help being sorted out because unless a imagenery fairy pixie thing will save who will. we the adluts have to be resonceible for making sure children inherit a clean safe beautiful world not some stink hole full of pollution
plus i am responding your your points i just might have said the wrong words to exspress my self
First of all, a minor complaint. I am having trouble reading and understanding your post because of all the misspellings. Please try to do better (I know we aren't all perfect) so I can understand.
Now, about what I think you said. Certainly the river ecology is an important consideration. A turbine does screw the the river ecology; its effect is not zero. To generate energy from the kinetic energy of the flow must slow the waterflow down. When all is said and done, there is always a tradeoff. Which is more harmful to the river ecology; one middle sized dam at only one point in the river or turbines throught the entire river plus a coal fired plant for when the river is dry/flood stage?
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but i have a point about the econimy or environment plus the trubines a small about the size of a car tire so they would not cause upsets to the local ecosystem
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but i have a point about the econimy or environment plus the trubines a small about the size of a car tire so they would not cause upsets to the local ecosystem
Well... you are right about one thing: a turbine the size of a car tire has a small impact on the ecology. Let's assume that the car-tire-sized turbine generates 1 kW (kilowatt). That's probably overestimating it, but who cares.
A 1 kilowatt turbine has a much smaller ecological impact than a 1 Gigawatt dam. That's true.
The point is that we still need electricity... and using less electricity seems to be no option (I personally disagree with that, but the majority of the people on earth just use more and more).
So: You are right that one small turbine has a small impact on the ecology...
But we need 1 million (*puts pinky finger at corner of mouth*) of such turbines to replace one big dam! One million of them will have a massive impact.
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I'm guessing it would be more than a million. Not sure what the design you have in mind is, but 1 kilowatt seems like a lot for something the size of a car tire, harnessing just the horizontal flow of a river. I could be wrong about that, though.
I just looked up the soon to be completed Three Gorges Dam in China. It will be the largest dam in the world, and is very controversial because of the flooding it caused, destroying habitats (though creating others), forcing the mass relocation of over a million people, and drowning archeological sites. However, it will have a mind-boggling capacity of 22.5 gigawatts (about 11 times the Hoover Dam), supplying tens of millions of people with electricity and reducing the amount of coal they need to burn by about 40 million tonnes per year, including the more efficient ship transport it allows via ship elevator and smoothing out droughts and floods. It will also pay for itself in ten years.
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I'm guessing it would be more than a million. Not sure what the design you have in mind is, but 1 kilowatt seems like a lot for something the size of a car tire, harnessing just the horizontal flow of a river. I could be wrong about that, though.
$Power = flow * head * g$
The flow is equal to: $flow= area(frontal) * velocity$
So, indeed, there are plenty of cases where a frontal area the size of a car tire will generate less than 1 kW. But there might also be cases where it will generate even more.
But the actual point I wanted to make is that we're comparing the ecological impact of one single 1 kW turbine with a 1 gigawatt dam... which isn't really a fair comparison.
We must compare one million, or more, or less (but certainly a lot!) with one dam. Then we have a decent comparison.
One more issue:
I just realized one more problem challenge with the turbines: since they use no difference in height at all, the energy comes from a difference in velocity of the water. Therefore, water behind the turbine flows slower than in front of it. Therefore, the river must get wider.
I'm not saying that this is equal to a lake behind a dam... but lots and lots of turbines will severely disrupt the flow of the water.
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But the actual point I wanted to make is that we're comparing the ecological impact of one single 1 kW turbine with a 1 gigawatt dam... which isn't really a fair comparison.
We must compare one million, or more, or less (but certainly a lot!) with one dam. Then we have a decent comparison.
I agree.
One more issue:
I just realized one more problem challenge with the turbines: since they use no difference in height at all, the energy comes from a difference in velocity of the water. Therefore, water behind the turbine flows slower than in front of it. Therefore, the river must get wider.
I'm not saying that this is equal to a lake behind a dam... but lots and lots of turbines will severely disrupt the flow of the water.
That is a good point. It can't not disrupt the flow, in fact, since that is where its getting its energy from. One million 1kw turbines have to extract the same kinetic energy as a 1 gigawatt dam. I don't know for sure, but it seems at least plausible that the overall effect would actually be a lot greater if you spread them out over a long stretch of river. (Or it might be the other way around.) [/speculation]
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thanks for the tips
the dam would have a larger impact because of the siz of the damp and the flooding of the area
also would small pencil size turbine in pipes and sewers be a good idea
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thanks for the tips
the dam would have a larger impact because of the siz of the damp and the flooding of the area
also would small pencil size turbine in pipes and sewers be a good idea
I first want to see some sign that you actually read what we posted. then I'll reply. Respond to one of the previous comments here please (something more than just "thanks"). I don't get the feeling you read them, or understood them.
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i allready know that there would not be much power be i had put 10000 leters of water into drains in a year with 4 other people i live with that has to turn the trubins a little bit and even slitly cut down on the gas prices
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i allready know that there would not be much power be i had put 10000 leters of water into drains in a year with 4 other people i live with that has to turn the trubins a little bit and even slitly cut down on the gas prices
You can't be serious here.
I think using the natural fall of the sewer lines to generate a miniscule amount of energy is a "good" idea. [/sarcasm] Until these turbines plug up the works (how good are you at plumbing? With this you will probably get a lot of practice). Or until these turbines cause a leak (keep in mind raw sewage is a health hazard).
There are better ways to generate energy; why do something hard for so little return. For much less work (and less raw sewage you have to clean up) you could simply install a small windmill or solar panel on your roof. I'll wager either of these will generate many, many times the energy that a sewage turbine would. The maintenance costs for these would be very considerably less (plumbers are expensive you know) as well.
But maybe I am mistaken on the energy potential from this idea. You should prove me wrong. The equations for calculating the energy that can be generated are in this thread (post # 33). Do the math and show us please.
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i allready know that there would not be much power be i had put 10000 leters of water into drains in a year with 4 other people i live with that has to turn the trubins a little bit and even slitly cut down on the gas prices
Let's assume the average water consumption of a household is about 100 m3/yr, so 100000 liters, 10x more than you said, or 100000 kg/yr.
Let's assume that it falls 10 meters down.
Let's assume we have a 100% efficient turbine.
$Energy = m*g*h$
$Energy = 100000*9.81*10=9810000 J$ in 1 year.
$Power = energy/time$
$Power = 9810000/(3600*24*365)=0.31 W$
0.31 W isn't even enough for your phone charger.
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was a good idea but too costly
still my piont still stands about the turines in the beds they ae eco friendly effincant and safe for the enviroment please dont openly disrespect me or my ideas as i am only trying to helpthe cause to susseed and make the world a better place
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was a good idea but too costly
still my piont still stands about the turines in the beds they ae eco friendly effincant and safe for the enviroment please dont openly disrespect me or my ideas as i am only trying to helpthe cause to susseed and make the world a better place
Openly disrespect? Where did that come from? I hope you understand that all the arguments given here are objective. This means that you can read them, check them, counter them (with equally objective arguments).
The point is:
If you want to make the world a better place ("help the cause"), then you must do the most environmentally friendly and ecologic thing.
Did it occur to you that 1 million turbines in the river bed might actually be a really bad idea? You have to place 1 million machines, with moving parts in a river. They will break and litter the river with broken parts.
In addition, I have reason to assume that the construction will require more material than a single dam (because those turbines will need a serious foundation, if they are not to be washed away):
this means that your 1 million turbines can use a maximum of 6 ton of concrete per turbine. That's 3 m3 / turbine. I believe that that would actually be realistic... because you'll need to drill poles into the soil. Therefore 1 million turbines might use the same amount of concrete as 1 dam.
I am helping "the cause" here... and you're "openly ignoring" all arguments given here.
Edited by CaptainPanic
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sorry about the comment but i was not thinkking about 1million trubines just about ten in diffrent river and diffrent pionts puls there would not be moving parts(other then the turbines) in the atuall river
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# Optimize matrix chain multiplication
This challenge is to compute the most efficient multiplication order for a product of several matrices.
The size of the matrices is specified on a single line of standard input. You should print to standard output a list of integers indicating the order in which to do the multiplications to minimize the total multiplication cost.
## example 1
### input
5x6 6x12 12x100 100x7
## output
3 2 1
The input line will be a space-separated list of matrix sizes, each of which is the number of rows, followed by an x, followed by the number of columns. For the example, there are 4 matrices to multiply together (so 3 total multiplications), and since matrix multiplication is associative they can be done in any order.
The output should be the order in which to perform the multiplications to minimize total cost. This should be a space-separated list of integers representing the index of the multiplication to perform next. For N matrices, this list should contain the numbers 1 through N-1, inclusive. For example 1, the output 3 2 1 means you should do the 12x100 * 100x7 multiplication first, then the 6x12 * 12x7 multiplication (the second matrix times the result of the previous step), then finally the resulting 5x6 * 6x7 multiplication.
The matrix multiplications will always be compatible, i.e. the number of columns of a matrix will match the number of rows of the subsequent matrix. Assume the cost of multiplying two matrices AxB * BxC is A*B*C.
Your code must handle lists of up to 100 matrices, each of dimension up to 999, and do so in a reasonable time.
## example 2
### input
5x10 10x5 5x15 15x5
1 3 2
or
3 1 2
## example 3
### input
22x11 11x78 78x123 123x666 666x35 35x97 97x111 111x20 20x50
### output
2 3 4 5 6 7 8 1
Note: for verification, the best total cost for the three examples is 9114, 750, and 1466344.
Shortest code wins!
• Are you sure about the last example? The total cost given by my code is 1466344. – Howard Sep 17 '11 at 7:31
• @Howard: Yep, you're right, a bug in my code. Fixed. – Keith Randall Sep 17 '11 at 16:23
q=(gets.split<<$_[/\d+$/]).map &:to_i
$><<r[1][q.size-1][1]*' ' First version: straight recursive implementation in Ruby. It does a complete search and thus might be slow on large inputs. k=->m{m[2]?(1..m.size-2).map{|l|s=k[m[0,l]+m[l+1..-1]];[m[l-1]*m[l]*m[l+1]+s[0],[l]+s[1].map{|u|u<l ?u:u+1}]}.min: [0,[]]}$><<k[(gets.split<<$_[/\d+$/]).map &:to_i][1]*' '
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HAL : in2p3-00640993, version 1
arXiv : 1110.0208
Physical Review D 85 (2012) 022001
All-sky Search for Periodic Gravitational Waves in the Full S5 LIGO Data
VIRGO Collaboration(s)
(2012)
We report on an all-sky search for periodic gravitational waves in the frequency band 50-800 Hz and with the frequency time derivative in the range of 0 through -6e-9 Hz/s. Such a signal could be produced by a nearby spinning and slightly non-axisymmetric isolated neutron star in our galaxy. After recent improvements in the search program that yielded a 10x increase in computational efficiency, we have searched in two years of data collected during LIGO's fifth science run and have obtained the most sensitive all-sky upper limits on gravitational wave strain to date. Near 150 Hz our upper limit on worst-case linearly polarized strain amplitude $h_0$ is 1e-24, while at the high end of our frequency range we achieve a worst-case upper limit of 3.8e-24 for all polarizations and sky locations. These results constitute a factor of two improvement upon previously published data. A new detection pipeline utilizing a Loosely Coherent algorithm was able to follow up weaker outliers, increasing the volume of space where signals can be detected by a factor of 10, but has not revealed any gravitational wave signals. The pipeline has been tested for robustness with respect to deviations from the model of an isolated neutron star, such as caused by a low-mass or long-period binary companion.
équipe(s) de recherche : APC - Cosmologie et GravitationOptique et PhotoniqueAPC - ADAMIS
Thème(s) : Physique/Relativité Générale et Cosmologie Quantique
Lien vers le texte intégral : http://fr.arXiv.org/abs/1110.0208
in2p3-00640993, version 1 http://hal.in2p3.fr/in2p3-00640993 oai:hal.in2p3.fr:in2p3-00640993 Contributeur : Nicole Berger <> Soumis le : Lundi 14 Novembre 2011, 15:51:22 Dernière modification le : Jeudi 28 Novembre 2013, 16:45:03
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# Random walks and diffusion limits
Imagine a long and narrow cylinder of radius r and a point particle that moves in the region bounded by the cylinder. The motion is specified as follows: starting at a point on the inner wall of the cylinder, choose at random a direction and let the particle move with constant speed until it hits another point of the cylinder. Once there, choose a new direction at random and repeat the process. The problem is to determine the probability that the particle will be given distance away from the initial point at a given time in the future.
I realized that t is hard to find such a probability explicitly, but if the cylinder is very narrow and the particle moves very fast (with speed proportional to the reciprocal of the radius) you can use the central limit theorem to obtain an explicit (Gaussian) approximation. What is the variance of the resulting normal law? How does the variance change if the cross section of the tube is, say a square, instead of a circle?
Can Anyone help me here?
-
I believe that polar coordinates and some calculus are used for this problem. – NasuSama Jan 3 at 2:49
Not sure the CLT applies at all...
In the analogous dynamics in dimension $2$, the particle moves in the plane $(x,y)$ bouncing back and forth between the walls $y=0$ and $y=1$, choosing an angle $t$ in $(0,\pi)$ uniformly at random and getting a displacement $x=\cot t$. This implies that $P[|x|\geqslant u]\sim 2/(\pi u)$ when $u\to\infty$, hence $|x|$ is not integrable.
In dimension $3$, assume without loss of generality that the cylinder has equation $y^2+z^2=z$ in the coordinate system $(x,y,z)$ (thus, $y=z=0$ is a line on the surface of the cylinder and the diameter is $1$). Starting from $(0,0,0)$, the particle moves along the line $x=r\cos t\cos s$, $y=r\cos t\sin s$, $z=r\sin t$, where $t$ and $s$ are independent and uniform on $(0,\pi)$. The length of the displacement until the particle hits the cylinder again is $r=\sin t/(\sin^2t+\cos^2t\sin^2s)$, for a distance along the $x$-axis of $|x|=r\cos t\cos s=\sin t\cos t\cos s/(\sin^2t+\cos^2t\sin^2s)$.
Now, $|x|$ is large when $(t,s)\to(0,0)$, and then $|x|\sim t/(t^2+s^2)$. Using polar coordinates for $(t,s)$, that is, introducing $(\varrho,\alpha)$ such that $t=\varrho\cos\alpha$, $s=\varrho\sin\alpha$, one gets $|x|\sim \cos\alpha/\varrho$. Thus, $P[|x|\geqslant u\mid\alpha]\sim C\int\limits_0^{\cos\alpha/u}\varrho\mathrm d\varrho=C\cos^2\alpha/u^2$ and $P[x\geqslant u]\sim C/u^2$ when $u\to\infty$, where the various occurrences of $C$ are absolute constants whose value can vary from line to line. In particular, $|x|$ is not square integrable.
It seems that for a cylinder in $\mathbb R^{d+1}$ whose section is a ball in $\mathbb R^d$, the displacement $|x|$ in one step is such that $E[|x|^\nu]$ is finite if and only if $\nu\lt d$. In particular one would expect CLT for cylinders in dimension at least $d+1=4$.
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Choosing the two angles $t$ and $s$ to be uniform on $(0,\pi)$ doesn't give a uniformly random direction, does it? – mjqxxxx Jan 1 at 3:22
No it doesn't, since the surface element is proportional to $\cos t\mathrm ds\mathrm dt$. – Did Jan 1 at 15:34
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Multifluid magnetohydrodynamic turbulent decay
# Multifluid magnetohydrodynamic turbulent decay
T.P. Downes11affiliation: School of Mathematical Sciences, Dublin City University, Glasnevin, Dublin 9, Ireland 22affiliation: School of Cosmic Physics, Dublin Institute for Advanced Studies, 31 Fitzwilliam Place, Dublin 2, Ireland 33affiliation: National Centre for Plasma Science and Technology, Dublin City University, Glasnevin, Dublin 9, Ireland and S. O’Sullivan44affiliation: School of Mathematical Sciences, Dublin Institute of Technology, Kevin Street, Dublin 8, Ireland
###### Abstract
It is generally believed that turbulence has a significant impact on the dynamics and evolution of molecular clouds and the star formation which occurs within them. Non-ideal magnetohydrodynamic effects are known to influence the nature of this turbulence. We present the results of a suite of resolution simulations of the decay of initially super-Alfvénic and supersonic fully multifluid MHD turbulence.
We find that ambipolar diffusion increases the rate of decay of the turbulence while the Hall effect has virtually no impact. The decay of the kinetic energy can be fitted as a power-law in time and the exponent is found to be for fully multifluid MHD turbulence. The power spectra of density, velocity and magnetic field are all steepened significantly by the inclusion of non-ideal terms. The dominant reason for this steepening is ambipolar diffusion with the Hall effect again playing a minimal role except at short length scales where it creates extra structure in the magnetic field.
Interestingly we find that, at least at these resolutions, the majority of the physics of multifluid turbulence can be captured by simply introducing fixed (in time and space) resistive terms into the induction equation without the need for a full multifluid MHD treatment.
The velocity dispersion is also examined and, in common with previously published results, it is found not to be power-law in nature.
MHD â ISM: kinematics and dynamics â ISM: magnetic fields â methods: numerical â turbulence
## 1 Introduction
Turbulence is recognized as a possible source of support against gravitational collapse for molecular clouds. The precise role and source of the observed motions interpreted as evidence of turbulence in these clouds has been studied extensively by many researchers (see the reviews of Mac Low & Klessen 2004; Elmegreen & Scalo 2004). Clearly, if turbulence can support molecular clouds then it can influence star formation in terms of rate, efficiency and initial mass function (Elmegreen, 1993; Klein et al., 2003).
Many studies of turbulence in molecular clouds have focused on ideal magnetohydrodynamics (MHD) as an approximation of the physics governing this system (Mac Low et al., 1998; Mac Low, 1999; Ostriker et al., 2001; Vestuto et al., 2003; Gustaffson et al., 2006; Glover & Mac Low, 2007; Lemaster & Stone, 2008, 2009). The assumption of ideal MHD, while desirable for technical reasons, is perhaps risky in the context of turbulence. The reason for this is that while ideal MHD is valid in molecular clouds on fairly large length scales, on shorter length scales non-ideal effects are thought to become significant (Wardle, 2004; Oishi & Mac Low, 2006). Given that turbulence in 3 dimensions involves the transfer of energy from large scales to ever smaller scales, the assumption of ideal MHD will be invalid below some critical spatial scale and the correct nature of the energy cascade may not be observed at this range.
The most important of the non-ideal effects for molecular cloud dynamics is ambipolar diffusion. Some authors (Oishi & Mac Low, 2006; Li et al., 2008; Kudoh & Basu, 2008) have studied driven MHD turbulence in the presence of ambipolar diffusion. All these authors find that ambipolar diffusion produces significant differences in the properties of the turbulence.
While most likely of lesser significance, it has been suggested that although the Hall resistivity is generally at least an order of magnitude lower than the ambipolar resistivity in molecular clouds (Wardle, 2004), its effect should not be ignored. Although relatively weak, it is capable of inducing topological changes in the magnetic field which are quite distinct to any influence caused by ambipolar diffusion. In support of this assertion, we note that researchers working on reconnection and the solar wind have studied the Hall effect in the context of turbulence and found that, although the overall decay rate appears not to be affected, the usual coincidence of the magnetic and velocity fields seen in MHD does not occur at small scales (Matthaeus et al., 2003; Mininni et al., 2006; Servidio et al., 2007). Almost no work has been done on comparing the influences of this effect coupled with that of ambipolar diffusion on turbulence with the exception of Downes & O’Sullivan (2009, hereafter Paper I).
In Paper I a series of simulations of decaying supersonic non-ideal MHD turbulence incorporating both ambipolar diffusion and the Hall effect were performed. These simulations, however, were constrained in that the resistivities associated with each of ambipolar diffusion, the Hall effect and the Pederson resistivity were kept fixed in both space and time. The authors found that, at length scales of 0.2 pc, ambipolar diffusion has a significant impact on the decay of the turbulence. The Hall effect was less significant in this respect but does have an influence on the magnetic field at short length scales. Here we present simulations in which the resistivities are self-consistently calculated from the evolution of both the magnetic field and the densities of all of the component species of the fluid. Using these dynamically evolving resistivities we study the decay of fully multifluid MHD turbulence. This is the first such study presented in the literature, with the exception of the low resolution simulations presented by Downes & O’Sullivan (2008).
The aim of this work is to examine in detail the differences between the decay of ideal MHD turbulence and that of multifluid MHD turbulence with a full tensor resistivity incorporating the effects of ambipolar diffusion, the Hall effect and Ohmic resistivity. We will use the results of Paper I in our discussion of these differences as it represents an intermediate stage between the calculations presented here and those of ideal MHD. This work is new in two respects: notwithstanding Paper I, no previous work has focused on decaying (i.e. un-driven) multifluid MHD turbulence and, in addition, no previous work has addressed the issue of turbulence in the presence of both ambipolar diffusion and the Hall effect simultaneously.
In section 2 we outline the numerical techniques used in this work, as well as the initial conditions and general set-up for the simulations while in section 3 we describe the methods used to analyze the simulation data. In section 4 we present and discuss the results of our simulations of turbulent decay. Finally, section 5 contains a summary of our results.
## 2 Numerical method
As in Paper I, we use the code HYDRA (O’Sullivan & Downes, 2006, 2007) to integrate the equations of weakly ionized multifluid MHD (see section 2.1). We assume that the molecular cloud material we are simulating can be treated as isothermal and that initially the density and magnetic field are uniform. We use the capabilities of HYDRA to extend the physics incorporated in the simulations here beyond those presented in Paper I so that the turbulence here is fully multifluid MHD.
### 2.1 Equations and algorithm
We briefly outline the equations and assumptions in our model here but refer the reader to O’Sullivan & Downes (2006, 2007) for a comprehensive description of the underlying assumptions for the weakly ionized model of multifluid MHD.
We assume that the cloud material can be treated as weakly ionized. This is clearly valid for molecular clouds and allows us to ignore the inertia of the charged species (Ciolek & Roberge, 2002; Falle, 2003). For a system composed of fluids, one of which is neutral, the equations to be solved are then
∂ρi∂t+\boldmath∇⋅(ρi\boldmathqi)=0, (1≤i≤N), (1) ∂ρ1\boldmathq1∂t+∇⋅(ρ\boldmathq1\boldmathq1+a2ρ% \boldmathI)=\boldmathJ×\boldmathB, (2) ∂\boldmathB∂t+∇⋅(% \boldmathq1\boldmathB−\boldmathB\boldmathq1)=−∇×\boldmathE′, (3) αiρi(\boldmathE+\boldmathqi×\boldmathB)+ρiρ1Ki1(\boldmathq% 1−\boldmathqi)=0 (4) ∇⋅\boldmathB=0, (5) ∇×\boldmathB=\boldmathJ, (6) N∑i=2αiρi=0, (7)
where , , , and are the neutral mass density, neutral velocity, sound speed, magnetic field and current density respectively and unless otherwise noted. , and ( are the collision coefficients between species and the neutrals, the charged fluid charge-to-mass ratios and mass densities respectively. Equations (1) to (7) express conservation of mass for each fluid, conservation of neutral momentum, the induction equation, force-balance for the charged species, the inadmissibility of magnetic monopoles, Faraday’s law and charge neutrality respectively.
The electric field in the frame of the fluid, , is calculated from the generalized Ohm’s law for weakly ionized fluids (e.g. Falle 2003; O’Sullivan & Downes 2006) and is given by
\boldmathE′=\boldmathEO+\boldmathEH+\boldmathEA, (8)
where
\boldmathEO = (\boldmathJ⋅\boldmathaO)% \boldmathaO, (9) \boldmathEH = \boldmathJ×\boldmathaH, (10) \boldmathEA = −(\boldmathJ×\boldmathaA)×\boldmathaA, (11)
using the definitions , , , where , , . , and are the Ohmic, Hall and ambipolar resistivities respectively and are given by
rO = 1σO, (12) rH = σHσ2H+σ2A, (13) rA = σAσ2H+σ2A, (14)
with the conductivities given by
σO = 1BN∑i=2αiρiβi, (15) σH = 1BN∑i=2αiρi1+β2i, (16) σA = 1BN∑i=2αiρiβi1+β2i, (17)
where is the Hall parameter for species and is given by
βi=αiBK1iρ1. (18)
As noted by Falle (2003) and O’Sullivan & Downes (2006), the main difficulty with standard numerical techniques for integrating equation (3) lies with the Hall term. As this term becomes dominant the stable time-step goes to zero. However, O’Sullivan & Downes (2006; 2007) presented a novel, explicit numerical method for integrating this term such that the limit on the stable time-step is not overly restrictive. We use this “Hall Diffusion Scheme” in this work. Of course, all explicitly differenced diffusion terms give rise to a stable time-step which is proportional to , where is the resolution of the simulation. To ameliorate this we use standard subcycling of the Hall terms and super time-stepping to accelerate the ambipolar diffusion terms (see Alexiades et al., 1996; O’Sullivan & Downes, 2006, 2007).
Equations (1) – (3) are solved using a standard shock-capturing, second order, finite volume, conservative scheme. The numerical techniques employed in this work are slightly different to those used in Paper I in one respect: we have altered the calculation of the advective fluxes in equation (3) to use the method suggested by Falle (2003). In Paper I these fluxes were derived from interface values of the neutral gas velocity and the magnetic field. We find that at high resolutions with variable resistivities the latter method is prone to introducing grid scale features in the solution while the former is not. This undesirable effect was not an issue for the investigations carried out in Paper I since resistivities were fixed. The downside of the described variation between the numerical approaches is that it must be considered as a possible source of discrepancy in comparisons between the results of Paper I and this work. However, in order to provide evidence of the small influence, we have also run a fixed resistivity simulation for this work (see section 2.2).
Equation (5) is enforced using the method of Dedner et al. (2002). The effects of the diffusive terms in equation (3) are then incorporated in an operator split fashion.
### 2.2 Initial conditions
We examine the decay of MHD turbulence in conditions suitable for dense regions of molecular clouds. The conditions we use are similar to those used in Paper I. We briefly review them here for completeness.
The computational domain is set up as a cube of side pc. Periodic boundary conditions are enforced on all faces of the simulation domain. The sound speed is set to 0.55 km s, the initial density is chosen to be uniform with a value of cm and the magnetic field is also initially uniform in the direction with a magnitude of 1 mG. For these conditions, suitable conductivities are s, s and s (Wardle & Ng, 1999). We choose a 3-fluid set-up for our multifluid simulation: 1 neutral species and 2 charged species. The densities, charge-to-mass ratios and collisional coefficients of the charged species are chosen in order to achieve these conductivities. We choose these particular physical conditions with a view to maximizing the influence of the Hall effect in our simulations (Wardle & Ng, 1999). In this way we hope to determine whether the Hall effect is ever likely to be important in molecular cloud turbulence.
The initial velocity field is defined to be the sum of waves with 64 wave-vectors each with random amplitude and phase - i.e.
qα=64∑j=0Aα,jcos(\boldmathkj⋅% \boldmathx+ϕα,j) (19)
where defines the component (, or ) of the appropriate quantity, and are the random amplitudes and phases and is the position vector. We restrict the velocity field to be solenoidal (i.e. non-compressional). By construction the mean velocity over the domain is zero.
Table 1 presents a complete list of the various simulations carried out in this work. The nomenclature we employ in referencing the simulations is xx-c where xx denotes the type of physics (e.g. a standard molecular cloud run is “mc”, ideal MHD is “mhd” etc) and c is the resolution used. The initial root-mean-square (rms) of the field is chosen to be 5 with a corresponding Alfvénic Mach number of approximately 1.9. In addition to the 4 multifluid MHD simulations run at different resolutions, we also run 4 further simulations. The first is an ideal MHD simulation (mhd-512) which we use for comparison purposes and the other two (ambi-512 and hall-512) only incorporate one of ambipolar diffusion or the Hall effect, respectively. The final case (fr-512) is a fixed resistivity simulation used to make contact with the simulations of Paper I. We use these latter 4 simulations to investigate separately the influence of each non-ideal effect.
## 3 Analysis
In this section we discuss the method of analysis of the output of the simulations described in section 2.2. The aim of this paper is to investigate the decay rate of supersonic turbulence in molecular clouds. Hence, the main analysis carried out is of the kinetic, magnetic and total energy as functions of time. These quantities are defined respectively as
ek = ∫domainρ|\boldmathq|2dV (20a) eb = ∫domain|\boldmathB|22dV−<\boldmathB>22V (20b) etot = eb+ek (20c)
where is the volume of the computational domain.
We also calculate the mass-weighted average Mach number, defined by
M=1a{σ2x+σ2y+σ2z}1/2 (21)
where is the sound speed and the velocity dispersions, , are defined by
σα={⟨ρq2α⟩⟨ρ⟩}1/2 (22)
where is either , or and the angle brackets denote averaging over the computational domain (see Lemaster & Stone 2009).
In section 4.3 we present the power spectra for the velocity, density and magnetic field for each of the simulations. These spectra are calculated by taking the power spectra in the , and directions separately and summing the power over the interval (where we take ). This gives us some insight into the scale of structures being formed by the turbulence for the various initial conditions and range of physics examined.
Finally, in section 4.4 we calculate the velocity dispersion as a function of length scale, . For these purposes we define the velocity dispersion to be
σ(l)={<σ2x(l)>domain+<σ2y(l)>domain +<σ2z(l)>domain}12 (23)
where
σα(l)={⟨q2α⟩l−⟨qα⟩2l}12 (24)
where indicates an average taken over a cube of side in the simulation domain and indicates averaging of the quantity over all such non-overlapping cubes within the domain.
## 4 Results
Each of the simulations detailed in table 1 was run for one sound crossing time, yrs, of the simulation domain. All analysis was carried out for (i.e. one flow crossing time) at which point we expect significant turbulent mixing to have taken place and the system’s memory of the initial state to be largely forgotten.
As an illustration of the differences between an ideal MHD turbulence simulation and a multifluid MHD simulation, figure 1 contains plots of the density distribution at in a slice through the computational domain for simulations mhd-512 and mc-512. It is clear that there is much less fine structure in mc-512. Also shown is the same slice for simulation fr-512 (i.e. fixed resistivities). While the similarities in terms of the levels of structure are relatively small between mc-512 and fr-512, there are clear differences between the distributions indicating that calculating the resistivities self consistently has some impact on the dynamics of the system.
In figure 2 we present plots of the ambipolar and Hall resistivities for simulation mc-512 for the same times and slices as figure 1. It is clear that the resistivities vary considerably throughout the computational domain with the features strongly correlated with the features in the density distribution. We also show to give an indication of the relative importance of each of the resistivities. This, as we shall see, is an important parameter. Finally, figure 3 is the same as 2 except that the data is taken at time - i.e. after one flow crossing time. Here we can see that the variation of the resistivities in space is dramatic with, for example, the ambipolar resistivity varying by almost 4 orders of magnitude with varying by around 2 orders of magnitude.
### 4.1 Resolution study
Four simulations identical in every way except for the resolution were run. Specifically, the resolutions used were , , , and . We now focus our attention on how the energy decay behaves with resolution. Figure 4 contains plots of the kinetic energy as a function of time for each of the simulations in the resolution study. It is clear that the lower the resolution, the faster the decay - this is what one would expect since lower resolution results in a higher numerical viscosity and hence one expects faster dissipation of energy.
Simulations mc-256 and mc-512 are, however, quite similar in terms of the energy decay with a maximum relative difference of around 10% between the kinetic energies in the simulations at any one time - an almost identical result to that obtained from the resolution study in Paper I. This is notable since in Paper I the resistivities were kept constant in space and time whereas here the resistivities locally increase significantly during the course of the simulations. A reasonable inference is that the influence of local variations in resistivities averages out in some sense on the global scale.
As we shall see later, however, the effect of spatially varying resistivities is noticeable in properties such as the power spectrum of the density and magnetic field.
The various energy decay rates can be modeled approximately as power-laws in time, i.e. . Fitting the kinetic energy, , magnetic energy, , and total energy, , as functions of time in this way we obtain the values given in table 2. This data confirms quantitatively what can be observed in Figure 5 and extends it to the decay of the energy in magnetic perturbations: increasing resolution decreases the rate of energy decay, but the difference between the and simulations is relatively minor. We note in passing that the decay in the energy in magnetic perturbations is considerably more sensitive to resolution than kinetic energy: varies between 1.43 and 1.28 while only varies between 1.38 and 1.34. We know that the resistivities are a critical factor in determining the decay of the magnetic energy since they facilitate loss of magnetic energy through reconnection. We attribute the extra sensitivity to resolution of to the necessity to properly resolve the diffusive effects in the induction equation, including the variation of the resistivities themselves. The variation of the resistivities throughout the computational domain is significant (see figure 2) and it is interesting to note that in simulations with fixed resistivities is not so sensitive to resolution (see Paper I).
### 4.2 Energy decay
We now discuss the behavior of the kinetic and magnetic energy in our multifluid simulations and compare with those in Paper I.
#### 4.2.1 Kinetic energy decay
Figure 5 contains plots of the decay of kinetic energy with time for the resolution simulations outlined in table 1. Note that the kinetic energy decay in all of the simulations is very similar until around . This is because at such early times compressions are only just starting to form and so the non-ideal terms in the induction equation have had almost no effect on the dynamics. The subsequent energy decay of simulations mhd-512 and hall-512 are almost identical to each other. The energy decay of simulations mc-512 and ambi-512 are virtually identical over the full plotted range while the data plotted for fr-512 coincide with the former simulations for times in the range . We have fitted the kinetic energy decay as a power-law in the range (i.e. after approximately one initial flow crossing time) and the exponents are given in the first column of table 2.
It is clear that the presence of ambipolar diffusion has a significant impact on the behavior of the kinetic energy in the turbulent system. This is a result of the exchange of energy between kinetic and magnetic energies as will be discussed in section 4.2.2.
From figure 5 and table 2 it is evident that the Hall effect has almost no impact on the kinetic energy decay in turbulence in molecular clouds. In order to emphasize any possible impact of the Hall effect we have plotted the time evolution of the ratio of the kinetic energy in each of our simulations to that in mc-5-512 in figure 6 on a linear scale. It is clear even in this figure that the Hall effect has little impact on the evolution of the turbulence. This result is also reproduced if we examine the evolution of the magnetic energy (see section 4.2.2). This supports our conclusion from Paper I in which the simulations were run using fixed resistivities (see also fr-512 in this work).
Also shown in figure 5 is the energy decay for simulation fr-512. Given the wide variation of the resistivities in both space and time (see figures 2 and 3) it is somewhat surprising that the energy decay is so similar to that of mc-512. In fact, the volume average of the ambipolar resistivity at in mc-512 is approximately 60% higher than that in the fr-512 simulation. It would appear that, while ambipolar diffusion enhances energy loss, the expected spatial and temporal variation of it does not have that much influence.
One possible reason for the small divergence between mc-512 and fr-512 would be if the locations in which the resistivity is high are regions in which the magnetic field, , is varying weakly. To explore this we define a scalar, , by
where , for example, is a normalized gradient defined by
where means centered differencing in the indicated direction without normalizing by the zone spacing and is the magnitude of the magnetic field throughout the domain at . is then a dimensionless measure of the variation of at any point in space or time: if is large it means there is a high gradient in one or more of the components of and therefore resistivity will have an important influence here. Figure 7 contains snapshots of and at . There is rich structure in which is not apparent in .
Figure 8 contains a histogram plot of the two dimensional probability density function for and (the ambipolar resistivity). What is striking about this plot is that there is a notable lack of high with corresponding high resistivity. Of course, if we have any system in which there are regions of high and low resistivity we expect that, over time, the regions with high diffusion will have lower variation in so this, in itself, doesn’t tell us much. However, it does prompt us to look a little more closely at the behavior of .
Comparison of the middle panel of figure 1 with the top panel of figure 2 shows that the ambipolar resistivity is higher in regions of low density, as would be expected. Now, in supersonic/super-Alfvénic turbulence kinetic energy is dissipated most strongly at strong shocks. However, shocks propagating into regions of very low density will not dissipate kinetic energy effectively. Since it is precisely these regions in which our resistivities are high we must conclude that, in fact, the regions of enhanced resistivity do not contribute significantly to energy dissipation and hence we would not expect the introduction of spatially varying resistivities to increase the rate of energy decay in our simulations. Further, since the volume average of, for example, the ambipolar resistivity is actually higher than that used in fr-512 we would not expect the spatial variation of this resistivity to reduce the rate of energy decay either.
#### 4.2.2 Magnetic energy decay
We now move on to discuss the decay in magnetic energy. Initially, as outlined in Paper I, the magnetic energy increases as the flow compresses and stretches the magnetic field throughout the computational domain. Once this initial increase in the energy has occurred it is gradually lost through two main avenues: magnetic reconnection and transfer of magnetic energy to kinetic energy which can then be dissipated in shocks and other viscous processes.
Figure 9 contains plots of the decay of magnetic energy with time. It is clear that, in common with the case of kinetic energy, the hall-512 and mhd-512 simulations are almost identical while the ambi-512 and mc-512 simulations are also well matched. This supports our inference from section 4.2.1 that the Hall effect has almost no impact on energy decay in molecular clouds on the global scale.
Ambipolar diffusion does, however, have a significant impact on the behavior of both the kinetic and magnetic energies. This was also noted in Paper I and we explain it in the same way, recapitulated briefly here for completeness. Consider a region of the flow undergoing compression. During this compression kinetic energy will be converted into both magnetic and internal energy through increasing the magnetic pressure and the thermal pressure. Given that this is a turbulent flow we expect that, after some time, this region will begin to expand again. However, during the compression ambipolar diffusion will have diffused away some of the magnetic energy thereby leaving less to be converted back to kinetic energy. In this way the presence of ambipolar diffusion creates a new path through which energy can be lost from the system and reduces the level of all forms of energy in the system.
### 4.3 Power spectra
We now move on to a study of the power spectra obtained from the multifluid MHD turbulence simulations. These spectra are important from the point of view of understanding the types of structures formed by the turbulence and are a more discerning tool for exploring any structural differences caused by multifluid effects. Table 3 contains the exponents of the power spectra assuming a power-law relationship between power and wave number. All the analysis presented in this section is performed on data taken at .
#### 4.3.1 Density power spectra
We turn first to the scales of the structures formed in the density distributions for our various simulations. Figure 10 contains plots of the power spectra of the neutral density (or density, in the case of simulations mhd-512 and fr-512) for all the resolution simulations.
The power spectra for the simulations including the effects of ambipolar diffusion are approximately broken power laws made up of 3 distinct power laws: , and . Below the low break, the spectrum is dependent on the scale of the computational domain. At high approaching the grid scale, numerical viscosity will begin to dominate. In common with the results presented so far we see that there is little difference between simulations mc-512 and ambi-512 - in fact it is difficult to distinguish between the two spectra without careful examination of figure 10. The fr-512 power spectrum is also similar to mc-512 and ambi-512 although it has slightly less power at intermediate values of with the difference here being at most 10%.
There is almost no detectable difference between simulations hall-512 and mhd-512. Evidently, the Hall effect has a much weaker influence on the density structure in molecular clouds than ambipolar diffusion. Ambipolar diffusion, on the other hand, has a very significant impact with pronounced damping of density structures at scales less than one tenth of the domain size (corresponding to a physical scale of approximately 0.02 pc). This damping is evident in figure 1 where the density structures in simulations with ambipolar diffusion are more smeared than in mhd-512.
From a quantitative perspective, the data in table 3 shows that the inclusion of ambipolar diffusion significantly steepens the power spectra, increasing the exponent by more than 0.5 over the cases which do not include the effect. The data again suggests that the Hall effect has minimal impact.
The results for fr-512 show a softer spectrum at large length scales and a harder spectrum at short scales than mc-512 and ambi-512, indicating that self-consistent calculation of the resistivities reduces the level of fine structure. This result must, however, be confirmed by higher resolution simulations before it can be regarded as reliable.
Figure 11 contains plots of the power spectra of the neutral density and the density of the negatively charged species at for comparison. It can be seen that the neutral mass density has more power for although the qualitative shape of the power spectra are the same in each case. This is in qualitative agreement with the results presented in Li et al. (2008) for driven turbulence simulations. For comparison with the results in Table 3, the exponent for the charged species mass density in the range is -2.21 while in the range it is -4.84. The power spectrum for the neutral mass density is harder than that for the magnetic field and, in the range , the spectrum for the charged species mass density lies somewhere between the two. This is not surprising as it is a function of the neutral density and velocity (through drag) and the magnetic field (through the Lorentz force).
#### 4.3.2 Velocity power spectra
Figure 12 contains plots of the velocity power spectrum for the neutral velocity. In common with section 4.3.1 those simulations incorporating ambipolar diffusion have significantly less power at almost all scales than those without. This is to be expected given the increased rate of loss of turbulent energy in the presence of ambipolar diffusion (see section 4.2).
As found in Paper I, and again here, there are clear differences of a qualitative nature between the density power spectra and the velocity power spectra for the multifluid simulations. Those simulations incorporating ambipolar diffusion (mc-512 and ambi-512) exhibit a strong power-law in the range in the density power spectrum. This is not true of the velocity power spectra. For the latter spectra there is a break at roughly and again at . The latter break we interpret as being at the scale where numerical diffusive effects begin to dominate the non-ideal effects in the induction equation. The lower break can reasonably be interpreted as the scale at which the non-ideal effects become important. The marked qualitative differences between the density and velocity power spectra indicate a considerable decoupling between the two variables. It is worth recalling here that the power spectra are calculated at so it is reasonable to expect that the turbulence is well developed at this stage.
In common with the density power spectra, the presence of ambipolar diffusion produces steeper velocity power spectra (see Table 3) in qualitative agreement with the results of Li et al. (2008).
The results at high are, of course, dominated by numerical diffusive effects. However, it is interesting to note that simulation fr-512 actually has very slightly less power at short scales than mc-512 despite the volume average of being roughly twice as high as the resistivity used in fr-512. This adds weight to the inference from section 4.2.1 that regions of high resistivity tend not to be coincident with regions of high gradients in the magnetic field and therefore do not have the level of influence one would naively expect.
Figure 13 contains plots of the power spectra for the velocity of the neutral and negatively charged species. It can be seen that, except at very high , they are virtually identical. Given that the charged velocity is defined by balance between drag with the neutrals and the Lorentz force this is, perhaps, unsurprising.
#### 4.3.3 Magnetic field power spectra
Figure 14 contains plots of the spherically integrated power spectra for the magnetic field. Once again, ambipolar diffusion has a much bigger impact on these spectra than the Hall effect. The magnetic field power spectra are considerably steeper at all in its presence. The absolute power at any scale is also considerably lowered by ambipolar diffusion, as would be expected from the discussion in section 4.2.2. There is a qualitative similarity between the density power spectra (figure 10) and the magnetic field power spectra which is absent when comparing the latter with the velocity power spectra (see figure 12).
Interestingly, we see the phenomenon that fr-512 has very slightly less power at than mc-512 indicating that fixing the resistivities at actually results in slightly more dissipation than allowing it to vary in time and space.
We find that the Hall effect has a slightly more noticeable effect on the magnetic field power spectra than on the spectra of velocity or density: it gives rise to a little more structure on short scales which is absent in its absence. This would be expected as the Hall effect is a dispersive effect acting directly on the magnetic field. The results for the density and velocity power spectra, however, demonstrate that this influence over the magnetic power spectra does not translate into an influence over the other variables.
### 4.4 Velocity dispersion
It has been widely reported that the observed velocity dispersion in molecular clouds behaves as a power-law in the size of the field of view. Figure 15 contains plots of the velocity dispersion for each of the resolution simulations presented in this work. As noted in Paper I, and Lemaster & Stone (2009), no power-law is observed. This may be due to the fact that in (multifluid) MHD turbulence there are many relevant signal speeds due to the variation in the orientation and intensity of the magnetic field, in contrast to the situation with hydrodynamic turbulence. Hence there is no reason to expect to see a power-law. Finally, it should be noted that, at this point in the simulation, the turbulence has decayed such that it is largely sub-sonic or transonic. This may also explain the lack of a power-law. Indeed, it should also be noted that our results here are not necessarily in contradiction with observations since the velocity dispersion-size correlation can only be accurately measured in the supersonic regime.
It is clear that ambipolar diffusion reduces the velocity dispersion at all length scales - this is what would be expected given the results from the velocity power spectra presented in section 4.3.2. Once again the results here indicate that the Hall effect has little impact. Again, it is interesting to note that the spatial variation of the resistivity in mc-512 appears to have almost no impact as the velocity dispersion seen in fr-512 is almost identical to that in mc-512.
## 5 Conclusions
We have presented results from a suite of resolution simulations of fully multifluid MHD decaying turbulence. The effects incorporated include the Hall effect and ambipolar diffusion. We have performed a resolution study to ensure that the energy decay rate, being the main result presented here, is reliable. We have confirmed the results of the simplified calculations in Paper I that the Hall effect has little impact on the nature and behavior of turbulence in molecular clouds under the well motivated physical parameters assumed in this work. Further, the presence of ambipolar diffusion increases the rate of energy decay at length scales of 0.2 pc and less. The same conclusion is drawn for the behavior of the energy in the magnetic field.
The power spectra for these simulations again suggest that the Hall effect has little impact on the flows with the exception of the spectrum of magnetic field variations. We must keep in mind that the maximum resolution used here () is only enough to resolve about half a decade in -space and it is therefore difficult to be confident of the details of the power spectra. Notwithstanding this consideration it does appear clear that ambipolar diffusion steepens the power spectra of the neutral velocity, density and the magnetic field. As noted in Paper I, it appears that at a resolution of and an assumed length scale of 0.2 pc we have resolved the length at which ambipolar diffusion begins to influence the turbulent cascade. In Paper I only constant resistivities were implemented and hence it was unclear whether this latter result would survive the inclusion of more realistic fully multifluid MHD in which the resistivities vary strongly in space and time. The results presented here imply that it does.
The power spectra of the neutral velocity and the magnetic field differ qualitatively from that of the density with breaks occurring in the former which are not seen in the latter. This suggests a decoupling between these fields.
The velocity dispersion as a function of length does not behave as a power law. This is not unexpected as the nature of MHD turbulence implies a wide range of applicable signal speeds which can, when combined, remove the power-law behavior which might be expected if only one signal speed were relevant.
The authors wish to acknowledge the SFI/HEA Irish Centre for High-End Computing (ICHEC) for the provision of computational facilities and support. The work described in this paper was carried out using resources provided to ICHEC through the Irish National Capability Computing Initiative, a partnership between all the major third level research institutions and IBM coordinated by the Dublin Institute for Advanced Studies and supported by the HEA under PRTLI cycles 3 and 4 with funding from the ERDF and the NDP. This work was partly funded by SFI through the Research Frontiers Programme, grant number 07 PHYF586. Finally, the authors would like to thank the anonymous referee for helpful suggestions and comments on this work.
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# Multiplication of two poisson distributed variables
Suppose we have two independent poisson variables $X_1$ and $X_2$ such that $X_1 ∼ Poisson(\lambda_1)$ and $X_2 ∼ Poisson(\lambda_2)$. What will be the probability distribution of $X_1 \times X_2$? Is it some standard distribution?
I am particularly interested in the mean value of the distribution.
Additional question: If I have chain of N poisson variables, can we say anything about mean value of the multiplication of these variables?
I could not find any online resource discussing this.
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Not a standard distribution. But independence guarantees that the mean of the product is the product $\lambda_1\lambda_2$ of the means. – Did Jan 7 at 12:18
Can you provide the reference for that? That would be helpful. – anuj919 Jan 7 at 12:21
– Did Jan 7 at 12:30
Since $X_1,X_2$ are independent you get:
$$E[X_1X_2]=E[X_1]E[X_2]=\lambda_1\lambda_2$$
just using the definition of independence! I would recommend to check the definition.
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Ah! Silly me. I was going through Product distribution. Thanks – anuj919 Jan 7 at 13:08
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# Absolute configuration and host-guest binding of chiral porphyrin-cages by a combined chiroptical and theoretical approach
## Abstract
Porphyrin cage-compounds are used as biomimetic models and substrate-selective catalysts in supramolecular chemistry. In this work we present the resolution of planar-chiral porphyrin cages and the determination of their absolute configuration by vibrational circular dichroism in combination with density functional theory calculations. The chiral porphyrin-cages form complexes with achiral and chiral viologen-guests and upon binding one of the axial enantiomorphs of the guest is bound selectively, as is indicated by induced-electronic-dichroism-spectra in combination with calculations. This host-guest binding also leads to unusual enhanced vibrational circular dichroism, which is the result of a combination of phenomena, such as rigidification of the host and guest structures, charge transfer, and coupling of specific vibration modes of the host and guest. The results offer insights in how the porphyrin cage-compounds may be used to construct a future molecular Turing machine that can write chiral information onto polymer chains.
## Introduction
In organic chemistry the determination of the absolute configuration of a chiral molecule is often a problem that is difficult to solve, particularly when X-ray structures of the enantiopure compounds are not available1,2. When possible, one may resort to vibrational and electronic circular dichroism spectroscopies in combination with calculations. Circular dichroism (CD) is the difference in response of a chiral molecule to left- and right-circularly polarized radiation, which may relate to the infrared (VCD) or to the UV-vis (ECD) spectral region. A comparison of the frequencies, signs, and intensities of experimental VCD spectra with those calculated by density functional theory (DFT) for a chosen configuration of a chiral molecule can unambiguously capture its absolute configuration, at least in principle2,3. For a more reliable result, the analysis can be completed by following the same approach using ECD and time-dependent density functional theory (TD-DFT) calculations, provided the molecule contains a chromophore. Although VCD has a great advantage over ECD, as it requires no chromophores, it often suffers from lower signal intensities. Because of this the experiments have to be performed with highly concentrated samples (Δε typically scales as 10−4–10−3ε)3. It is of great interest, therefore, to find processes that can enhance VCD intensities4,5,6,7 and in the past decades chemists have been able to achieve this by different methods including manipulation of the electronic manifold8,9, forming metal complexes8,10, chiral crystal packing11, and fibril formation12,13,14. Until now there are no examples of VCD enhancement by supramolecular interactions, i.e., by forming host–guest complexes15,16.
Herein we report the efficient and straightforward resolution of two planar chiral porphyrin-cages (1 and 2, see Fig. 1a) by chiral HPLC and the determination of their absolute configurations by vibrational3 and electronic circular dichroism17 in combination with DFT and TD-DFT calculations. These absolute configurations have been checked by X-ray diffraction, which confirmed the assignments made by the combined spectroscopic-theoretical analysis. The molecular sizes of the studied compounds are larger than those of any other published compound to date for which VCD has been applied to assign absolute configurations. Furthermore, we present induced circular dichroism (ICD) experiments showing that the chiral porphyrin cage molecules display enantioselectivity in the binding of both achiral and chiral N,N-substituted 4,4′-bipyridinium (viologen) guest molecules and that on binding one of the interconverting axial enantiomorphs of the guest is preferred. Remarkably, certain combinations of chiral host and guest complexes display amplified VCD spectra, constituting the first example of VCD enhancement in a host–guest system. The work presented here is part of a larger project aimed at encoding information into single polymer chains with the help of catalytic molecular machines (Turing machines)18 that write digital data in the form of chemical functions (i.e. chiral epoxides: (R,R)-epoxide = digit 1, (S,S)-epoxide = digit 0, Fig. 1b), while gliding along these chains. Chiral cage compounds 1 and 2 provided with a catalytically active metal (e.g. manganese) and light-switchable chiral functions on their cavity walls, are conceived to be used for that purpose19,20,21.
## Results
### Resolution and assignment of chirality
Compounds 1 and 2 contain one and two nitro-functions, respectively, and were prepared from the parent porphyrin cage compound (R1, R2, R3, R4 = H) via our previously reported highly selective nitration reaction22. The introduction of these nitro-substituents on the side walls of the porphyrin cage provides planar chirality as well as point chirality to the compounds (Supplementary Fig. 1). The resolution of the resulting racemic mixtures was achieved by chiral HPLC (Supplementary Information) and the two enantiomers of 1, i.e. Sp-(R,S)-1 and Rp-(S,R)-1, and those of 2, i.e. Sp-(R,R)-2 and Rp-(S,S)-2 were obtained in excellent enantiomeric excesses (ee >99.5%) and turned out to be thermally stable (see Supplementary Figs. 24 and 25). To elucidate the absolute configurations of the porphyrin cages, VCD spectra were recorded and DFT calculations performed on the pure enantiomers of the two cage compounds. In a first approach, the calculations were carried out with the implicit solvent model SMD23. The functional B3LYP associated with basis set function 6-311G(d) and the dispersion potential GD3Bj were used for the calculations of the IR and VCD spectra24. The results turned out to be unsatisfactory and we improved our model by introducing one explicit solvent molecule in the host molecule, which position was determined with the help of molecular dynamics calculations (see Supplementary Information for details). Although only one conformation of the host with one explicit solvent molecule was taken into account, this model significantly improved the agreement between the measured and calculated spectra. The experimental IR and VCD spectra of (−)−1 and (+)−1 were recorded in the region from 1825 to 1025 cm−1 ((−) stands for the first eluted fraction showing a negative sign in the CD spectrum at 254 nm and (+) for the second eluted fraction showing a positive sign in the CD spectrum at 254 nm). The experimental and calculated IR spectra of this compound were found to be in excellent agreement (Fig. 2). Furthermore, the calculated VCD spectrum of Sp-(R,S)−1 corresponded well with the experimental VCD spectrum of the second fraction (+)−1, revealing that the compound in this fraction had the absolute configuration Sp-(R,S). Based on this result we can assign the Rp-(S,R)-configuration to the first eluted fraction, i.e. (−)−1. The IR and VCD spectra of the enantiomers of the anti-dinitro porphyrin-cages 2 were also calculated and compared with the experimental ones to determine their absolute configurations (Fig. 2). In this way we could assign the Rp-(S,S)-configuration to the first eluted enantiomer (−)−2, and Sp-(R,R) configuration to the second eluted one, (+)−2. The VCD spectra of the enantiomers of the nitro-porphyrin-cages 1 and 2 displayed strong similarities: the carbonyl stretching modes exhibited a couplet that results from a mixing of the two stretching C = O modes. Between 1250 and 1350 cm−1, the same succession of three negative and positive bands was a reliable fingerprint of the cavity of the host compound. The vibrational modes contributing to these bands are not dominated by vibrations of one functional group, but a mixing of stretching (C–O, C = C, and C–N) and/or deformations (C–H, N-H bending, and CH2 twisting) involving atoms of the whole molecule. The most intense VCD band was located at 1541 cm−1 (Fig. 2) and corresponds to a complex vibrational mode that mixes the NO2 stretching vibration with the C = Carom. and C–H deformations of the nitro-xylylene moieties. The VCD spectra furthermore revealed that several of the VCD band intensities of (−)−2 and (+)−2 were higher than those of (−)−1 and (+)−1 (Fig. 2), which may result from the fact that 2 has a C2 symmetry axis, leading to a double planar chirality.
To confirm the assignments of the absolute configurations made by VCD and DFT we decided to grow single crystals from the enantiomers of the cage compounds and perform X-ray analyses. This turned out to be possible for (−)−1 (Fig. 3a, b), but not for an enantiomer of 2. The space group of (−)−1 was non-centrosymmetric P212121, in line with the chiral nature of the species, and the crystal structure unambiguously proved that (−)−1 has the absolute configuration Rp-(S,R). We also prepared the enantiopure zinc complexes (−)−3, (+)−3, (−)−4, and (+)−4 from compounds 1 and 2 (Supplementary Information) and succeeded to grow single crystals from (+)−3 suitable for X-ray analysis, but unfortunately not from one of the enantiomers of 4. The crystal structure of (+)−3 confirmed the absolute configuration assigned on the basis of VCD/DFT and the X-ray structure of (−)−1 and furthermore revealed the coordination of an acetonitrile molecule to the zinc center inside the zinc (+)−3 cage (Fig. 3c).
To obtain further insight into the properties of enantiopure 1 and 2, the ECD spectra of the separated enantiomers were recorded in acetonitrile in the region 190–600 nm (Fig. 4). Each pair of enantiomers displayed mirror-image spectra within experimental errors (see also Supplementary Information, Table 1 and Fig. 27). For Rp-(S,R)−1 a positive Cotton effect (CE)25 is visible in the region from 250 nm to 320 nm (λ317 nm: ∆ε + 6.1 M−1 cm−1; λ255 nm: ∆ε−19.2 M−1 cm−1; A-value +25.3 M−1 cm−1 linked to the 1Lb transition26, see Supplementary Fig. S27). As expected, Sp-(R,S)−1 displayed a negative CE with similar characteristics. The ECD spectra were calculated by TD-DFT using the LC-WhPBE functional with the def2SVP basis set and SMD for solvation effects (see Supplementary Information for details). As for VCD, better results were obtained with a model that takes into account the presence of one explicit solvent molecule. The calculated UV and ECD spectra of Sp-(R,S)−1 and Sp-(R,R)−2 showed a satisfying agreement with the measured spectra of (+)−1 and (+)−2, respectively, unambiguously confirming the assignments of the absolute configurations made above (Fig. 4).
### Host–guest binding
Viologen guests can bind into the chiral porphyrin cages and form stable 1:1 host–guest complexes, which are held together by ππ and van der Waals interactions between the guest and the xylylene ring side walls of the host (vide infra)27. To investigate how the binding of viologen guests influences the chiroptical properties of the chiral porphyrin hosts, ECD spectra were recorded28. Since the CD signal of 2 is stronger than that of 1, first experiments with the enantiomers of the former compound were performed. When the achiral guest methyl viologen 5 (Ka with 2 = 8.4 × 104 M−1 in CHCl3/CH3CN 1:1 v/v) was added in increasing amounts to a solution of Sp-(R,R)-2 in acetonitrile, the intensity of the positive CD signal at 252 nm of the host decreased and then became negative (Fig. 5a). Unfortunately, the observed changes (induced CD, abbreviated ICD)29,30,31,32,33,34,35,36 at this wavelength coincide with the ππ* transition of the guest, which has an absorption maximum in the same region. The addition of 5 to a solution of Rp-(S,S)-2 in acetonitrile resulted in an opposite ICD signal, as expected. In order to better understand the ICD phenomenon, we optimized one conformation of the complex formed by Sp-(R,R)-2 and 5 using DFT calculations. Two possibilities were considered: viologen 5 is bound in a ‘horizontal’ orientation (perpendicular to the xylylene sidewalls and parallel to the porphyrin ring) or in a ‘vertical’ orientation (parallel to the xylylene side walls and perpendicular to the porphyrin ring) in the cavity of the host. Attempts to optimize the complex Sp-(R,R)-2/5 starting from geometries in which the guest is in a horizontal position, always converged to a conformation in which guest 5 is oriented in a vertical position (Fig. 5b). This result is in line with previous NMR experiments27. In this vertical conformation, the guest is held in the host by a combination of electrostatic interactions, hydrogen bonding, ππ, CH−π, and van der Waals interactions. Interestingly, the chiral environment of the cavity of the host induces an asymmetry in the equilibrium between the two enantiomorphic (twisted) conformations of 5: according to the calculations, only the left (M)-twisted guest can be hosted by the porphyrin cage of Sp-(R,R)-2 (Fig. 5b). The ICD phenomenon observed here is comparable to the one reported for the binding of a viologen derivative in cyclodextrin host molecules37. Using the (M)-axial conformation of the guest, we calculated the UV and ECD spectra of the host–guest complex using TD-DFT (Fig. 5c, d and Supplementary Information). The calculated UV spectra showed that the ππ* absorption band of the guest indeed is superimposed on a ππ* absorption band of the host. When the host–guest complex is formed, the ππ* absorption band of the guest shifts to lower energies (bathochromic shift) and appears in an area where the host absorbs less. For the (M)-5 enantiomorph, the ECD band of the ππ* transition at 248 nm transition is positive. The bathochromic shift of the ππ* band in the complex superimposes it on the negative band at 264 nm of the host, thus canceling the signal, in agreement with the experiments (Fig. 5a). The calculations clearly show that the ICD phenomenon essentially results from the presence of only the (M)-enantiomorphic conformation of the guest 5 in the complex with host Sp-(R,R)-2. In separate experiments we also added the chiral guests (S,S)-6, and (R,R)-6 to hosts Sp-(R,R)-2 and Rp-(S,S)-2. The ECD spectra showed that these guests displayed similar ICD behavior as observed for the achiral guest 5 (Supplementary Figs. 35 and 36). Finally, these guests were also used to form complexes with hosts Sp-(R,S)−1 and Rp-(S,R)−1 (Supplementary Figs. 38 and 39). The measured ICD trends were similar to the ones recorded for Sp-(R,R)-2 and Rp-(S,S)-2 and these guests, as expected from the theoretical analysis.
Fluorescence titration experiments were carried out in CHCl3/CH3CN (1:1, v/v) to investigate the binding of chiral viologen guests (S,S)-6 and (R,R)-6 to chiral hosts Rp-(S,R)-1 and Rp-(S,S)-2 (Supplementary Figs. 4851). Upon complexation of the guests, the bands at 650 and 715 nm in the fluorescence spectra of the hosts decreased in intensity as a result of quenching. From the titration curves the association constants (Ka) and the binding free energies ΔGo of the complexes between Rp-(S,R)-1 and Rp-(S,S)-2, and the two chiral guests (S,S)-6 and (R,R)-6 were calculated. The results are presented in Table 1. Compared to Rp-(S,S)-2, the Ka-values for the complexes between Rp-(S,R)-1 and the chiral guests (S,S)-6 and (R,R)-6 are significantly higher, which may result from the lower steric hindrance in the latter host as a result of the presence of only one nitro group. Furthermore, the binding constants of the complexes between (R,R)-6 and Rp-(S,R)-1 and Rp-(S,S)-2 are somewhat higher than those of the complexes between (S,S)-6 and Rp-(S,R)-1 and Rp-(S,S)-2. Apparently, the chiral centers of (R,R)-6 and (S,S)-6 can still influence the association constants of the complexes with the chiral hosts Rp-(S,R)-1 and Rp-(S,S)-2, despite the fact that they are quite remote from the location where the actual binding interactions occur.
The complexation of the chiral guest can be expected to perturb the conformation of the chiral host and hence change its VCD spectrum, compared to that of the free host molecule. With the aim of understanding this point, the VCD spectra of a series of host–guest complexes were investigated. Since different chiral guests may bring about different perturbances, experiments with the following combinations of complexes were performed: Sp-(R,R)-2/(S,S)-6, Sp-(R,R)-2/(R,R)-6, Rp-(S,S)-2/(S,S)-6, and Rp-(S,S)-2/(R,R)-6 (all in 1:1 molar ratios). The VCD and IR spectra were recorded in the 1825-1025 cm−1 region. Interestingly, in the VCD spectra, the signals of Sp-(R,R)-2/(S,S)-6, and Rp-(S,S)-2/(R,R)-6 at 1064 cm−1, 1238 cm−1, 1499 cm−1, 1541 cm−1, and 1704 cm−1 were enhanced, i.e. by a factor 1.5–2, relative to that of the free chiral hosts (Fig. 6). To ensure that the enhanced signals were not originating from the free guest molecules, also the VCD spectra of (R,R)-6 and (S,S)-6 were recorded. (Supplementary Fig. 44). The VCD signals of these free guests were very weak compared to those of the free chiral hosts and we may conclude, therefore, that the observed enhanced signals must result from the formed host–guest complexes. In order to obtain more insight in this phenomenon, DFT calculations were performed. Due to the sizes of the host–guest complexes, these calculations were carried out on only one of the most stable host–guest conformations and the used theoretical level was slightly lowered (see Supplementary Information for details). Hence, we calculated the IR and VCD spectra of the complexes of host Sp-(R,R)-2 with the (M)-axial conformation of the guests (R,R)-6, (S,S)-6 (see Supplementary Fig. 56). It should be noted that these calculated spectra for only one conformation of course are not sufficient to correctly model the experimental spectra and particularly the phenomenon of VCD bands amplification. Although incomplete, these calculations provide us with keys to understand what effects govern this phenomenon. The first effect is related to the formation of the host–guest complex, which rigidifies the structures of both the host and the guest by limiting their conformational space. This has a direct impact on the VCD spectrum, which globally becomes more intense. More specifically, inside the chiral host the axial rotation around the central C–C bond of the 4,4′-bipyridinium moiety of the guest is blocked, allowing only one axial conformation of the guest to be present (Fig. 6c). From the calculated structures we established that this phenomenon is associated with VCD bands between 1500 and 1650 cm−1. These bands, which are not present in the VCD spectra of the free guests, correspond to vibrational breathing modes of the 4,4′-bipyridinium moiety. Another effect that can significantly modify the VCD spectra is the mixing of vibrational modes. Indeed, calculations show that in the free host Sp-(R,R)-2 some vibrational modes are mixed, for instance the C = O stretching or the breathing mode of the xylylene rings. Insertion of a guest molecule into the cavity may prevent or modify this mixing, leading to significant changes (frequency shifts, intensity decreases, and, for VCD, change of the signs of bands) in the spectra. For instance, in the free host Sp-(R,R)-2 the breathing modes of the xylylene rings are mixed but, when guest (S,S)-6 is inserted into the cage, such mixing no longer occurs and the intensity of the corresponding negative band at 1619 cm−1 decreases (Fig. 6d). Furthermore, calculations show that in the complex, vibrational modes of both host and guest are coupled. Such spatial coupling between vibrational modes of two molecules in close proximity is generally associated with enhanced VCD bands of which the magnitude may be several orders higher than those of the other bands in the spectrum. It is well-established that such coupling strongly depends on the relative orientations of the guest and the host38,39. The calculations revealed that in the complex Sp-(R,R)-2/(S,S)-6 a coupling between the bending vibrational modes of the CH2 and CH3 groups of the guest and of the CH2 groups of the ethyleneoxy linkers of the cage occurs. This coupling is associated with a significant enhancement of the negative band at 1450 cm−1 (Fig. 6d). Finally, the cationic nature of the viologen moiety of the guest may allow charge transfer with the host, resulting in large VCD bands7,40. In summary, it is reasonable to believe that the phenomenon of amplification of the VCD bands observed in the measured spectra of the complexes results from a combination of these five phenomena: (i) rigidification of the host and guest structures upon complexation, (ii) enantiomorphism of the guest inside the host, (iii) the loss or the modification of some mixing vibrational modes in the host upon guest complexation, (iv) spatial coupling of specific vibrational modes of the host and guest, and (v) charge transfer between the guest and the host.
For the other combinations of host and guest, i.e. Sp-(R,R)-2/(R,R)-6, and Rp-(S,S)-2/(S,S)-6, VCD enhancements (Supplementary Fig. 47) were observed as well, but the increases in intensity were not as strong as for the host–guest combinations of Sp-(R,R)-2/(S,S)-6 and Rp-(S,S)-2/(R,R)-6. We speculate that the differences in the binding strengths of the host–guest complexes may cause these enhancement differences. Indeed, the Ka-value of the complex between Rp-(S,S)-2 and (R,R)-6 is higher than that of the complex between Rp-(S,S)-2 and (S,S)-6 (see Table 1), which is in agreement with the observed VCD results. The host–guest complexes of Sp-(R,S)-1/(S,S)-6, Sp-(R,S)-1/(R,R)-6, Rp-(S,R)-1/(S,S)-6, and Rp-(S,R)-1/(R,R)-6 were also investigated by VCD (Supplementary Figs. 48 and 49), but no clear amplified signals were observed. This may be attributed to the fact that Sp-(R,S)-1 and Rp-(S,R)-1 are less symmetric hosts than Sp-(R,R)-2 and Rp-(S,S)-2 and display weaker VCD signals. Furthermore, the geometric positions of the guests inside the chiral hosts 1 and 2 are different (Supplementary Figs. 5053), which may also lead to this difference in amplification.
## Discussion
In this study we have reported the resolution of chiral porphyrin-cage molecules and the determination of their absolute configurations by VCD/DFT and X-Ray analysis. To the best of our knowledge, our host compounds are the largest molecules so far that have been analyzed by VCD/DFT with respect to the determination of their absolute configurations. The chiroptical properties, ICD, and viologen guest binding abilities of the porphyrin cage compounds were investigated and analyzed. Comparison of DFT calculations with ICD experiments revealed that the viologen guests bind inside the chiral hosts in one specific axial-chiral conformation. For instance, on complexation of the achiral guest methyl viologen in the host Sp-(R,R)-2, the axial conformation of the guest changes to the (M)-enantiomeric form, which is the only one that binds inside the chiral cavity. Such an enantioselective binding and/or positioning is an important first step towards the realization of enantioselective catalytic reactivity within our chiral hosts and suggests that this can be realized with prochiral guest substrates. Upon guest binding, some of the host compounds displayed amplified VCD signals, i.e. by a factor of 1.5–2. To our knowledge, this is the first time that this phenomenon has been observed in host–guest systems. DFT calculations suggest that the VCD signal enhancements are the result of a combination of phenomena, i.e., rigidification of the host and guest structures on complexation, a specific orientation of the guest inside the host, strength of the host–guest binding, charge transfer between the guest and the host, and coupling of specific vibration modes of the host and guest. The results of our present studies will be used for the further construction of a chiral porphyrin-cage catalyst that can move along a polymer chain while writing digital information in the form of chiral epoxides. Work along this line is in progress.
## Methods
### Synthesis
All solvents were freshly dried under argon atmosphere using standard procedures. All reactions were performed under argon using standard Schlenk techniques. NMR spectra were recorded on a Bruker 500 MHz Spectrometer (1H: 500 MHz; 13C: 125 MHz) at 298 K. 1H and 13C NMR chemical shifts are reported relative to residual solvent signals. The analytical and preparative resolutions of the porphyrin cage compounds were performed by chiral HPLC (compound 1: Chiralpak IE, eluent ethanol/dichloromethane 30/70, v/v; compound 2: Chiralpak IA, and eluent ethanol/ dichloromethane 30/70, v/v). Details are given in the Supplementary Information. The syntheses of porphyrin cage compounds 3 and 4 and those of the guest compounds 5 and 6 were performed according to standard procedures as indicated in the Supplementary Information.
### ECD and UV-Vis spectra
Circular dichroism spectra were measured on a Jasco J-815 CD spectrometer equipped with a JASCO Peltier cell holder PTC-423 to maintain the temperature at 25.0 ± 0.2 °C. A CD quartz cell with 1 mm of optical path length was used. The CD spectrometer was purged with nitrogen before recording each spectrum, from which the baseline was subtracted. The baseline was always measured for the same solvent and in the same cell as the samples. The spectra are presented without smoothing and further data processing. A chiral porphyrin cage compound (H, 0.75 μmol ≈ 1.04–1.08 mg) was dissolved in 25.00 mL of distilled acetonitrile, and the sample was agitated by ultrasonication to dissolve all solid material, resulting in a pink solution. The stoppered flask was then inverted several times to obtain a homogenous host solution ([H] ≈ 30 µM). Viologen guests (G, 4.5 μmol ≈ 1.99-3.28 mg) were dissolved in 5.00 mL of distilled acetonitrile to obtain homogenous guest solutions ([G] ≈ 900 µM). A baseline of the empty cuvette holder, without any cuvette or sample present, was measured. Blanc spectra were recorded using cuvettes filled with distilled acetonitrile. A titration was then carried out by first measuring 500 μL of host solution without any guest present. Guest solution was then added to obtain solutions with a total of 4, 10, 20, 50, 80, and 130 µL of guest solution and 500 μL of host solution (yielding [G]/[H]-ratios between 0.25 and 8). In the case of the mixed solvent system acetonitrile: chloroform = 1:1 (v/v) a titration was carried out by first measuring 500 μL of host solution without any guest present. Guest solution was then added to obtain solutions with a total of 10, 20, and 80 µL of guest solution and 500 μL of host solution (yielding [G]/[H]-ratios between 0.6 and 4.5). After each series, the measured solution was stored to recover the host and the cuvette was flushed with distilled acetonitrile three times. When measuring a new host solution, the cuvette was flushed an additional three times with host solution and then filled with 500 µL of host solution to start a new series of measurements. At the end of the experiments, reference spectra for the chiral guests were measured by mixing 400 µL of distilled acetonitrile with 300 μL of guest solution. ECD- and UV-vis measurements were performed at the same time using a Jasco J-815 CD-spectrometer. The obtained experimental data for the host compounds and the host–guest complexes are shown in the Supplementary Information, Supplementary Figs. 2739 and Supplementary Table 1.
### Fluorescence titration experiments
The fluorescence experiments were performed using a Jasco FP-8300 spectrophotometer, equipped with a Peltier temperature controller. The machine was allowed to warm at least half an hour before the start of the measurement. All samples were measured in quartz fluorescence cuvettes of 1×1 cm path length, with an internal volume of 3500 µL. The titrations were carried out using the Spectral Measurement-option, using 418 nm as the excitation wavelength and by observing the emission between 425 and 800 nm. The temperature was kept at 25.0 ± 0.1 °C. An excitation bandwidth of 5 nm and an emission bandwidth of 5 nm were used to obtain a good signal-noise ratio, and the scanning speed was set to 1000 nm/min. For the titration experiments a solvent mixture of CHCl3/MeCN (1:1 (v/v)) was used to keep both the host and guest soluble. The starting concentration of host was 1.5 µM, to which in ca. 30 steps at least 10 equivalents of guest were added from a concentrated stock solution. The experimental procedure was as follows: a dry 1:1 (v/v) mixture of the above mentioned two solvents was prepared in a volumetric flask. A host-stock solution of 50 µM was made by dissolving the host in the solvent mixture, which was weighed to determine the density. Three host-measurement solutions of 1.5 µM were prepared from a weighed amount of host solution (300 µL), which was diluted to a total volume of 10.00 mL. A guest-stock solution of 200 µM was made in 5.00 mL of the same solvent mixture. Three guest-measurement solutions were prepared, each containing 1.5 µM host and 45 µM guest, which were made by mixing weighed amounts of host-stock (150 µL) and guest-stock (1125 µL) solutions and diluted to a total volume of 5.00 mL. The obtained experimental data are presented in the Supplementary Information, Supplementary Figs. 4043 and Supplementary Table 2.
### IR and VCD measurements
Infrared (IR) spectra and vibrational circular dichroism (VCD) spectra were recorded using two instruments. The first one was a Vertex70 spectrometer to which a PMA50 optical bench was coupled, both being supplied by Bruker company. In the PMA50 set-up the infrared radiation (3800–600 cm−1 range) is focused by a BaF2 lens on a ZnSe photo-elastic modulator (PEM, 50 kHz frequency). The circularly polarized beam is then directed onto the sample and finally collected by a D313/QMTC detector. A calibration of the PEM at a fixed wavenumber was performed before recording any VCD spectrum, to ensure a proper chiroptical signal within a spectral region of 600 cm−1 around the tuning wavenumber. Typically, calibrations at 1450 cm−1 allowed us to obtain a spectrum over the most meaningful region for conjugated organic systems. The second instrument was a JASCO FVS-6000 spectrometer, equipped with a MCT-V detector. The spectrometers were allowed to warm up over the course of 3 h before a measurement was performed. The detector was cooled with liquid nitrogen before and during the measurement. Samples were measured in a BaF2 liquid sample cell with a 200-µm optical path length, which was stored in a closed box in the presence of desiccant when not in use. Spectra were acquired with a scan range of 2000–850 cm−1, a resolution of 4 cm−1, and the aperture set at 5.0 mm. A total of 16 accumulations were obtained for the IR spectra and 3000 accumulations for the VCD spectra. Spectra were processed by the spectrometer software by applying zero filling and apodization using a cosine function. Reference samples for the host were obtained by preparing a 19.5-mM solution of Sp-(R,R)-2 (or Rp-(S,S)-2) in CDCl3. Reference samples for the guests were obtained by dissolving (S,S)-6 (or (R,R)-6) in CD3CN with a concentration [6] ~ 40 mM. The host–guest mixtures were made from the separately prepared host and guest solutions. Four host solutions were prepared by dissolving 3.0 mg of Sp-(R,R)-2 (or Rp-(S,S)-2) in 250 µl of CDCl3. Guest solutions were made by dissolving (S,S)-6 (or (R,R)-6) in CD3CN with a concentration [6] ~40 mM. Subsequently, mixtures were prepared by adding 50 µL of the (S,S)-6 (or (R,R)-6) solution (1 equiv.) to the desired Sp-(R,R)-2 (or Rp-(S,S)-2) solution. The mixtures were briefly swirled to mix the contents and the solvent was then removed from each mixture by blowing a gentle stream of argon gas over the solution. Thereafter, each of the mixtures was re-dissolved in 250 µl of CDCl3 and the solvent was removed again by blowing a gentle stream of argon gas over the solution. This step was repeated one more time. Before the measurement the desired mixture was re-dissolved in 150 µl of CDCl3 and transferred to the sample cell for measurement ([2] ~ [6] ~14 mM). Due to the low signal, for host Sp-(R,S)-1 (or Rp-(S,R)-1), the applied concentration was 28 mM. The results are presented in the Supplementary Information and Supplementary Figs. 4449.
### NMR studies on host–guest complexes
The results of these studies are presented in the Supplementary Information and Supplementary Figs. 5053.
### Calculations of the IR, VCD, UV, and ECD spectra
Firstly, spectra of isolated molecules without explicit solvent molecules were calculated for the Sp-(R,R)-2 enantiomer. Only the average effects of the solvent were taken into account using the implicit solvation model SMD (“Solvation Model based on Density”). Each geometry was optimized using Density Functional Theory with the B3LYP functional and triple zeta 6-311G(d) basis set. Empirical dispersion was added with the D3 version of Grimme’s dispersion with Becke-Johnson damping (GD3BJ). The vibrational frequencies, IR absorption, and VCD intensities were calculated using the same level of theory. Frequencies were scaled by a factor of 0.975. IR absorption and VCD spectra were constructed from calculated dipole and rotational strengths assuming Lorentzian band shape with a half-width at half maximum of 8 cm−1. All calculations were performed using the Gaussian16 package. The conformations selected for the calculations of the averaged IR/VCD spectra of Sp-(R,R)-2 were obtained by molecular dynamics calculations, see below. Eight conformations were selected and optimized. Boltzmann populations estimated from enthalpies calculated at 298 K revealed that the 5 most stable conformations among the 8 found are required for the calculation of the averaged spectra (see Supplementary Information, Supplementary Table 3, and Supplementary Fig. 54). In this way, an acceptable but not satisfying agreement was obtained between the spectra measured for the enantiomer (+)−2 and calculated for the enantiomer Sp-(R,R)-2 (see Supplementary Information and Supplementary Fig. 55). Calculations performed by introducing an explicit solvent molecule inside the two most stable conformations found for Sp-(R,R)-2 made the agreement between measurement and calculation significantly more satisfactory. The position of the explicit solvent molecule (CD2Cl2) in the cage of porphyrin Sp-(R,R)-2 was initially determined using molecular dynamics calculations and thereafter the complex was optimized using the SMDCD2Cl2/GD3BJ-B3LYP/6-311 G(d) DFT level. Given the quality of this result with only two conformations and considering the large size of the studied system for which a significant amount of calculation time is required, this model was not extended to other conformations. The results obtained for compound 2 were also used for the calculations of the spectra of the Sp-(R,S)-1 enantiomer, as well as for the calculations of the UV and ECD spectra of both compounds 1 and 2. The geometries used for the Sp-(R,S)-1 compound were optimized from the geometries of the retained conformations of Sp-(R,R)-2 in which one of the two nitro groups was replaced by a hydrogen atom. As with 2, a good balance between accuracy of the results and the consumed cpu time was obtained by considering only two conformations with one explicit solvent molecule using the SMDCD2Cl2/GD3BJ-B3LYP/6-311G(d) theoretical level. For the calculations of the UV and ECD spectra a similar approach was applied but with an explicit CH3CN molecule instead of CD2Cl2. The geometry optimizations were performed using WB97XD functional associated with the 6-31G(d) basis set and the SMD implicit solvent model. Based on these geometries, the ECD and UV spectra were calculated using the time-dependent density functional theory (TD-DFT) with LC-WhPBE functional and def2SVP basis set. Calculations were performed for vertical 1A singlet excitation using 100 states. For a comparison between theoretical results and the experimental values, the calculated UV and ECD spectra were modeled with a Gaussian function using a half-width of 0.37 eV. Due to the approximations of the used theoretical model, an almost constant offset was observed between measured and calculated wavelengths. Using UV spectra, all calculated wavelengths were calibrated by a factor of 1.05.
### Molecular dynamics calculations
Molecular dynamics (MD) simulations were performed starting from the atomistic coordinates of the optimized structure obtained from quantum chemical calculation. The porphyrin molecule was inserted in a cubic box whose sides measured 15 Å and solvated with ~1000 molecules of dichloromethane or acetonitrile. We initially minimized the energy and then performed equilibrium MD under periodic boundary conditions in NPT ensemble. MD trajectory was followed for 100 ns in NVT ensemble. The temperature during the simulation was held constant at 300 K using Berendsen thermostat. Fast smooth Particle-Mesh Ewald summation was used for long-range electrostatic interactions, with a cutoff of 1.0 nm for the direct interactions. We used the clustering analysis tool of GROMACS (gmx cluster) to explore the different conformations from the MD trajectory. The GROMOS clustering algorithm with a RMSD cut-off was used to determine the structurally similar clusters. This approach allowed us to select eight geometries that were optimized and used for the calculations of the average IR and VCD spectra (Supplementary Information and Supplementary Fig. 55). The same approach was used to the selection of geometries of the cage porphyrin with a solvent molecule inside (Supplementary Information and Supplementary Fig. 54).
### VCD bands amplification and Induced CD phenomenon
The geometries the host Sp-(R,R)-2 and guests 5, (S,S)-6, and (R,R)-6 were optimized using the WB97XD functional software associated with the 6-311 G(d) basis set but without implicit solvent effects in order to keep a reasonable use of cpu time. In order to better understand the Induced CD phenomenon, we optimized one conformation of the complexes formed by Sp-(R,R)-2 and the two enantiomorphic conformations of viologen 5. Two possibilities were considered: viologen 5 is hosted in a ‘horizontal’ orientation (perpendicular to the xylylene sidewalls and parallel to the porphyrin ring) or in a ‘vertical’ position orientation (parallel to the xylylene side walls and vertical to the porphyrin ring) in the cavity of the host. Attempts to optimize the complex Sp-(R,R)-2/(M)-5 starting from geometries in which the guest is in a ‘horizontal’ position all converged towards a conformation in which 5 had rotated to a ‘vertical’ position. Similarly, all attempts to optimize the Sp-(R,R)-2/(P)-5 complex converged to the Sp-(R,R)-2/(M)-5 complex. Based on the optimized geometries of (M)-5, Sp-(R,R)-2, and the complex Sp-(R,R)-2/(M)-5, the corresponding ECD and UV spectra were calculated using time-dependent density functional theory (TD-DFT) with the LC-WhPBE functional and def2SVP basis set. Calculations were performed for vertical 1A singlet excitation using 50 states. In order to compare the theoretical results with the experimental values, the calculated UV and ECD spectra were modeled with a Gaussian function using a half-width of 0.37 eV. Due to the approximations of the used theoretical model an almost constant offset was observed between the measured and calculated wavelengths. Using UV spectra, all calculated wavelengths were calibrated by a factor of 1.05. For the Sp-(R,R)-2/(M,S,S)-6 and Sp-(R,R)-2/(M,R,R)-6 complexes, we looked for the most stable conformations adopted by the side chains of the viologens complexed in the cavity. We used a methodology combining DFT calculations and simulated annealing performed at the semi-empirical level. From an initial geometry of the complex optimized using the WB97XD/6-311G(d) DFT level, we carried out a simulated annealing at the AM1 semi-empirical level, allowing the side chains of the viologen to adopt different conformations on the dihedral angles only. The bond lengths, valence angles, and dihedral angles not involved in the conformational mobility of the viologen side chains were fixed in these simulated annealing calculations. The lower energy conformation found by this approach was then fully optimized using the WB97XD/6-311G(d) level (Supplementary Information and Supplementary Fig. 54) before calculating the IR and VCD spectra (Supplementary Information and Supplementary Fig. 56). The AMPAC10 program was used for all the semi-empirical simulated annealing calculations.
### X-ray structures
Single crystals of (−)-Rp-(S,R)-1 and (+)-Sp-(R,S)-3 were both grown from a mixture of CHCl3 and CD3CN (1:1 v/v). Reflections were measured on a Bruker D8 Quest diffractometer with sealed tube and Triumph monochromator (λ = 0.71073 Å). Software package used for the intensity integration was Saint. Absorption correction was performed with SADABS. The structures were solved with direct methods using SHELXT. Least-squares refinement was performed with SHELXL-2014 against $$\left| {{\mathrm{F}}_{\mathrm{h}}^{\mathrm{o}}} \right|^2$$of all reflections. Non-hydrogen atoms were refined freely with anisotropic displacement parameters. Hydrogen atoms were placed on calculated positions or located in difference Fourier maps. All calculated hydrogen atoms were refined with a riding model. Experimental data is presented in the Supplementary Information, Supplementary Table 4, and Supplementary Figs. 57 and 58.
## Data availability
The syntheses, resolution, and characterization of the compounds, the ECD and VCD measurements, the DFT calculations, and the host–guest binding studies are described in the Supplementary Information. Crystallographic data have been deposited at the Cambridge Crystallographic Data Centre (CCDC) under CCDC numbers 1989977 for Rp-(S,R)-1 and 1989978 for Sp-(R,S)-3. Any further relevant data are available from the corresponding authors on request.
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## Acknowledgements
This work was supported by the European Research Council (ERC Advanced Grants 290886 ALPROS and 740295 ENCOPOL to R.J.M.N.) and a grant from the Dutch Ministry of Education, Culture and Science (Gravity Program 024.001.035 to R.J.M.N. and J.A.A.W.E.). J.C. thanks the Ministère de l’Education Nationale, de la Recherche et de la Technologie, the Centre National de la Recherche Scientifique (CNRS), and Rennes Métropole for financial support. This work was granted access to the HPC resources of Aix-Marseille Université financed by the project Equip@Meso (ANR-10-EQPX-29-01) of the program “Investissements d’Avenir” supervised by the Agence Nationale de la Recherche. We acknowledge M.A.J. Koenis and Prof. W.J. Buma (University of Amsterdam University) for their help with the VCD measurement test.
## Author information
Authors
### Contributions
R.J.M.N. conceived the project. J.O., A.S., and P.J.G. prepared the chiral porphyrin cages, N.V. performed the separation of the planar chiral porphyrin cages; J.O., M.G., D.W., and P.C.P.T. performed the ECD, UV-Vis, and fluorescence measurements; J.O., M.G., P.C.P.T., and J.C. performed the VCD measurements; J.O. grew crystals; P.T. analyzed crystals; J.-V.N. and S.C. performed the DFT calculations; J.O., J.C., J.-V.N., P.J.T.R., J.A.A.W.E., and R.J.M.N. analyzed the data. All authors contributed to the discussions and the writing of the paper.
### Corresponding authors
Correspondence to Jean-Valère Naubron or Jeanne Crassous or Johannes A. A. W. Elemans or Roeland J. M. Nolte.
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### Competing interests
The authors declare no competing interests.
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Ouyang, J., Swartjes, A., Geerts, M. et al. Absolute configuration and host-guest binding of chiral porphyrin-cages by a combined chiroptical and theoretical approach. Nat Commun 11, 4776 (2020). https://doi.org/10.1038/s41467-020-18596-1
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# Find an explicit formula for $f(n)=f(n-1)+f(n-2)/2+f(n-3)/6$ when $f(1)=1, f(2)=2, f(3)=3$
I was trying to solve the set of equations $$a+b+c=1, a^2+b^2+c^2=2, a^3+b^3+c^3=3$$ for $$a^n+b^n+c^n=x$$ I took $$x=f(n)$$ and found that $$f(n)=f(n-1)+f(n-2)/2+f(n-3)/6$$ when $$f(1)=1, f(2)=2, f(3)=3$$ I then went on to try to find an explicit formula but I don’t know how to get started, can anyone please help?
• It might helps you. math.stackexchange.com/questions/3521010/… – dust05 Jun 3 '20 at 6:04
• Actually I understood that but I quite couldn’t relate it much with my question, I would be glad if you can explain me further – Asv Jun 3 '20 at 7:14
• I think I misunderstood your point. – dust05 Jun 3 '20 at 9:48
What do you want to find? the values of $$a, b,c$$ or the general equation for $$f_n$$? I'm not sure your requirement, but two questions are related.
You have explicit formula, $$f(n) = a^n + b^n + c^n$$. So it is enough to find each values for $$a, b, c$$.
Let $$a+b+c = A$$, $$ab + bc + ca = B$$, $$abc = C$$. Then $$a, b, c$$ are three roots of $$x^3 - A x^2 +Bx - C = 0$$.
We have $$A = f_1 = 1$$. Note that $$f_1^2 = f_2 + 2B$$, i.e. $$B = -1/2$$. Also one can deduce that \begin{align*} f_1^3 &= a^3 + 3 a^2 b + 3 a^2 c + 3 a b^2 + 6 a b c + 3 a c^2 + b^3 + 3 b^2 c + 3 b c^2 + c^3\\ & = f_3 + 3ab(a+b) + 3bc(b+c) + 3ca(c+a) + 6C \\ & = f_3 + 3ab(A - c) + 3bc(A-a) + 3ca(A-b) + 6C \\ &= f_3 + 3AB - 9 C + 6C\\ 1& = 3 -3/2 - 3C \\ C& = 1/6 \end{align*} so $$a, b, c$$ satisfies $$x^3 - x^2 - x/2 -1/6 =0$$. Actually, this is exactly the characteristic polynomial of the recurrence formula you found. It dose not have rational solutions, so let three solutions be $$\alpha, \beta, \gamma$$, then we have $$f(n) = \alpha^n+ \beta^n + \gamma^n$$
Numerically, $$\alpha \approx 1.4308$$, $$\beta \approx -0.21542 - 0.26471 i$$, $$\gamma =\overline{\beta}$$.
I'm not sure if i understood your question. Are you interested in the following general situation? :
$$x_n = a_1^n + \cdots + a_k^n$$, $$x_1 = 1, x_2 = 2, \cdots, x_k = k$$
In this case, if you proceed as above, this would be helpful.
• Thanks a lot, actually I wasn’t able to see that a, b and c would be the roots of a cubic – Asv Jun 3 '20 at 10:32
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# Cormen Edition 3 Exercise 7.4 Question 4 (Page No. 184)
182 views
Show that RANDOMIZED-QUICKSORT’s expected running time is $\Omega(n\ lg\ n)$.
1 vote
proof
## Related questions
1
144 views
Show that quicksort’s best-case running time is $\Omega(n\ lg\ n)$.
2
59 views
Show that the expression $q^2 +(n-q-1)^2$ achieves a maximum over $q=0,1,\dots ,n-1$ when $q=0$ or $q=n-1$.
3
138 views
Show that the running time of QUICKSORT is $\Theta(n^2)$ when the array $A$ contains distinct elements and is sorted in decreasing order.
1 vote
What is the running time of QUICKSORT when all elements of the array $A$ have the same value?
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# Does a limit exist?
1. Jul 6, 2012
### KiwiKid
1. The problem statement, all variables and given/known data
Is there a number 'a' such that
(3x^2 + ax + a + 3) / (x^2 + x - 2)
exists as x goes to -2? If so, find the value(s) of 'a' and the limit.
2. Relevant equations
The limit rules. Algebra.
3. The attempt at a solution
Well... I have absolutely no clue what I'm supposed to do. I suppose I'll have to choose my 'a' values in such a way so the denominator no longer equals zero, *but* other than brute-forcing a-integers in a graphical calculator I have little idea on how to do that.
I began by factoring the denominator to (x+2)(x-1), with the hope that I may be able to negate the x+2 somehow, but that didn't work. Then I tried to get (x^2+x-2) in the numerator so the numerator and denominator would cancel out and leave a bit on top, but I couldn't find a way to do that, either.
Could someone give me a clue on what approach to try here?
2. Jul 6, 2012
### micromass
Staff Emeritus
The denominator always goes to 0 if x goes to -2. What should the numerator go to if the limit were defined?? For example, if the numerator were 1, then the limit would be "1/0" which would be an infinity and thus the limit would not exist.
Is there a value for the numerator for which the limit does exist?
3. Jul 6, 2012
### Whovian
Have you heard of L'Hopital's Rule? It should work here. If not, note that the numerator must also go to 0 as x goes to -2, and must therefore have a factor of x+2.
4. Jul 6, 2012
### KiwiKid
Are you saying that the numerator should equal 0?
5. Jul 6, 2012
### micromass
Staff Emeritus
Yes! If x=-2, then the numerator should be 0.
6. Jul 6, 2012
### KiwiKid
Ok, let me see if I got this right:
3(-2)^2+a(-2)+a+3 = 0
-> 12 - 2a + a + 3 = 0
-> a = 15
Therefore (substituting a):
lim[x->-2] (3x^2 + 15x + 18) / (x^2 + x - 2)
= lim[x->-2] (3(x+2)(x+3)) / ((x+2)(x-1))
= lim[x->-2] (3(x+3)) / (x-1)
= 3 / -3
= -1
I think I got it. Thanks again!
7. Jul 6, 2012
### SammyS
Staff Emeritus
That looks good.
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Publications
Suppression of superconductivity and enhanced critical field anisotropy in thin flakes of FeSe
npj Quantum Materials Nature Research (part of Springer Nature) (2020)
L Farrar, M Bristow, AA Haghighirad, A McCollam, SJ Bending, AMALIA Coldea
FeSe is a unique superconductor that can be manipulated to enhance its superconductivity using different routes while its monolayer form grown on different substrates reaches a record high temperature for a two-dimensional system. In order to understand the role played by the substrate and the reduced dimensionality on superconductivity, we examine the superconducting properties of exfoliated FeSe thin flakes by reducing the thickness from bulk down towards 9 nm. Magnetotransport measurements performed in magnetic fields up to 16T and temperatures down to 2K help to build up complete superconducting phase diagrams of different thickness flakes. While the thick flakes resemble the bulk behaviour, by reducing the thickness the superconductivity of FeSe flakes is suppressed. In the thin limit we detect signatures of a crossover towards two-dimensional behaviour from the observation of the vortex-antivortex unbinding transition and strongly enhanced anisotropy. Our study provides detailed insights into the evolution of the superconducting properties from three-dimensional bulk behaviour towards the two-dimensional limit of FeSe in the absence of a dopant substrate.
Magnetic order and disorder in a quasi-two-dimensional quantum Heisenberg antiferromagnet with randomized exchange
PHYSICAL REVIEW B 102 (2020) ARTN 174429
F Xiao, WJA Blackmore, BM Huddart, M Gomilsek, TJ Hicken, C Baines, PJ Baker, FL Pratt, SJ Blundell, H Lu, J Singleton, D Gawryluk, MM Turnbull, KW Kramer, PA Goddard, T Lancaster
Observation of a neutron spin resonance in the bilayered superconductor CsCa<sub>2</sub>Fe<sub>4</sub>As<sub>4</sub>F<sub>2</sub>.
Journal of physics. Condensed matter : an Institute of Physics journal 32 (2020) 435603-
DT Adroja, SJ Blundell, F Lang, H Luo, Z-C Wang, G-H Cao
We report inelastic neutron scattering (INS) investigations on the bilayer Fe-based superconductor CsCa<sub>2</sub>Fe<sub>4</sub>As<sub>4</sub>F<sub>2</sub> above and below its superconducting transition temperature T <sub>c</sub> ≈ 28.9 K to investigate the presence of a neutron spin resonance. This compound crystallises in a body-centred tetragonal lattice containing asymmetric double layers of Fe<sub>2</sub>As<sub>2</sub> separated by insulating CaF<sub>2</sub> layers and is known to be highly anisotropic. Our INS study clearly reveals the presence of a neutron spin resonance that exhibits higher intensity at lower momentum transfer (Q) at 5 K compared to 54 K, at an energy of 15 meV. The energy E <sub>R</sub> of the observed spin resonance is broadly consistent with the relationship E <sub>R</sub> = 4.9k <sub>B</sub> T <sub>c</sub>, but is slightly enhanced compared to the values observed in other Fe-based superconductors. We discuss the nature of the electron pairing symmetry by comparing the value of E <sub>R</sub> with that deduced from the total superconducting gap value integrated over the Fermi surface.
Magnetically driven loss of centrosymmetry in metallic Pb2CoOsO6
PHYSICAL REVIEW B 102 (2020) ARTN 104410
AJ Princep, HL Feng, YF Guo, F Lang, HM Weng, P Manuel, D Khalyavin, A Senyshyn, MC Rahn, YH Yuan, Y Matsushita, SJ Blundell, K Yamaura, AT Boothroyd
Extremely well isolated two-dimensional spin-1/2 antiferromagnetic Heisenberg layers with a small exchange coupling in the molecular-based magnet CuPOF
PHYSICAL REVIEW B 102 (2020) ARTN 064431
D Opherden, N Nizar, K Richardson, JC Monroe, MM Turnbull, M Polson, S Vela, WJA Blackmore, PA Goddard, J Singleton, ES Choi, F Xiao, RC Williams, T Lancaster, FL Pratt, SJ Blundell, Y Skourski, M Uhlarz, AN Ponomaryov, SA Zvyagin, J Wosnitza, M Baenitz, I Heinmaa, R Stern, H Kuhne, CP Landee
Information and Decoherence in a Muon-Fluorine Coupled System
PHYSICAL REVIEW LETTERS 125 (2020) 87201
J Wilkinson, S Blundell
Strong in-plane anisotropy in the electronic structure of fixed-valence $β$-LuAlB$_4$
Physical Review B: Condensed Matter and Materials Physics American Physical Society (2020)
P Reiss, J Baglo, H Tan, X Chen, S Friedemann, K Kuga, FM Grosche, S Nakatsuji, M Sutherland
The origin of intrinsic quantum criticality in the heavy-fermion superconductor $\beta$-YbAlB$_4$ has been attributed to strong Yb valence fluctuations and its peculiar crystal structure. Here, we assess these contributions individually by studying the isostructural but fixed-valence compound $\beta$-LuAlB$_4$. Quantum oscillation measurements and DFT calculations reveal a Fermi surface markedly different from that of $\beta$-YbAlB$_4$, consistent with a `large' Fermi surface there. We also find an unexpected in-plane anisotropy of the electronic structure, in contrast to the isotropic Kondo hybridization in $\beta$-YbAlB$_4$.
Dynamic spin fluctuations in the frustrated A-site spinel CuAl2O4
PHYSICAL REVIEW B 102 (2020) ARTN 014439
H Cho, R Nirmala, J Jeong, PJ Baker, H Takeda, N Mera, SJ Blundell, M Takigawa, DT Adroja, J-G Park
Competing pairing interactions responsible for the large upper critical field in a stoichiometric iron-based superconductor CaKFe4As4
Physical Review B American Physical Society 101 (2020) 134502
M Bristow, W Knafo, P Reiss, W Meier, PC Canfield, SJ Blundell, A Coldea
<p>The upper critical field of multiband superconductors is an important quantity that can reveal details about the nature of the superconducting pairing. Here we experimentally map out the complete upper-critical-field phase diagram of a stoichiometric superconductor, CaKFe4As4, up to 90 T for different orientations of the magnetic field and at temperatures down to 4.2K. The upper critical fields are extremely large, reaching values close to ∼3 Tc at the lowest temperature, and the anisotropy decreases dramatically with temperature, leading to essentially isotropic superconductivity at 4.2K. We find that the temperature dependence of the upper critical field can be well described by a two-band model in the clean limit with band-coupling parameters favoring intraband over interband interactions. The large Pauli paramagnetic effects together with the presence of the shallow bands is consistent with the stabilization of an FFLO state at low temperatures in this clean superconductor.</p>
Enhancing easy-plane anisotropy in bespoke Ni(II) quantum magnets
Polyhedron 180 (2020)
JL Manson, ZE Manson, A Sargent, DY Villa, NL Etten, WJA Blackmore, SPM Curley, RC Williams, J Brambleby, PA Goddard, A Ozarowski, MN Wilson, BM Huddart, T Lancaster, RD Johnson, SJ Blundell, J Bendix, KA Wheeler, SH Lapidus, F Xiao, S Birnbaum, J Singleton
© 2020 The Authors We examine the crystal structures and magnetic properties of several S = 1 Ni(II) coordination compounds, molecules and polymers, that include the bridging ligands HF2−, AF62− (A = Ti, Zr) and pyrazine or non-bridging ligands F−, SiF62−, glycine, H2O, 1-vinylimidazole, 4-methylpyrazole and 3-hydroxypyridine. Pseudo-octahedral NiN4F2, NiN4O2 or NiN4OF cores consist of equatorial Ni-N bonds that are equal to or slightly longer than the axial Ni-Lax bonds. By design, the zero-field splitting (D) is large in these systems and, in the presence of substantial exchange interactions (J), can be difficult to discriminate from magnetometry measurements on powder samples. Thus, we relied on pulsed-field magnetization in those cases and employed electron-spin resonance (ESR) to confirm D when J ≪ D. The anisotropy of each compound was found to be easy-plane (D > 0) and range from ≈ 8–25 K. This work reveals a linear correlation between the ratio d(Ni-Lax)/d(Ni-Neq) and D although the ligand spectrochemical properties may play an important role. We assert that this relationship allows us to predict the type of magnetocrystalline anisotropy in tailored Ni(II) quantum magnets.
Anomalous high-magnetic field electronic state of the nematic superconductors FeSe1-xSx
Phys. Rev. Research 2, 013309 (2020) (2020)
M Bristow, P Reiss, AA Haghighirad, Z Zajicek, SHIV Singh, T Wolf, D Graf, W Knafo, A McCollam, AMALIA Coldea
Understanding superconductivity requires detailed knowledge of the normal electronic state from which it emerges. A nematic electronic state that breaks the rotational symmetry of the lattice can potentially promote unique scattering relevant for superconductivity. Here, we investigate the normal transport of superconducting FeSe$_{1-x}$S$_x$ across a nematic phase transition using high magnetic fields up to 69 T to establish the temperature and field-dependencies. We find that the nematic state is an anomalous non-Fermi liquid, dominated by a linear resistivity at low temperatures that can transform into a Fermi liquid, depending on the composition $x$ and the impurity level. Near the nematic end point, we find an extended temperature regime with $T^{1.5}$ resistivity. The transverse magnetoresistance inside the nematic phase has as a $H^{1.55}$ dependence over a large magnetic field range and it displays an unusual peak at low temperatures inside the nematic phase. Our study reveals anomalous transport inside the nematic phase, driven by the subtle interplay between the changes in the electronic structure of a multi-band system and the unusual scattering processes affected by large magnetic fields and disorder
Phase transitions for beginners
CONTEMPORARY PHYSICS (2020)
SJ Blundell
Group theory for physicists, 2nd edition
CONTEMPORARY PHYSICS (2020)
SJ Blundell
Quantum oscillations probe the Fermi surface topology of the nodal-line semimetal CaAgAs
Physical Review Research American Physical Society 2 (2020) 012055(R)
YH Kwan, P Reiss, Y Han, M Bristow, D Prabhakaran, D Graf, A McCollam, S Ashok Parameswaran, AI Coldea
Nodal semimetals are a unique platform to explore topological signatures of the unusual band structure that can manifest by accumulating a nontrivial phase in quantum oscillations. Here we report a study of the de Haas–van Alphen oscillations of the candidate topological nodal line semimetal CaAgAs using torque measurements in magnetic fields up to 45 T. Our results are compared with calculations for a toroidal Fermi surface originating from the nodal ring. We find evidence of a nontrivial π phase shift only in one of the oscillatory frequencies. We interpret this as a Berry phase arising from the semiclassical electronic Landau orbit which links with the nodal ring when the magnetic field lies in the mirror (ab) plane. Furthermore, additional Berry phase accumulates while rotating the magnetic field for the second orbit in the same orientation which does not link with the nodal ring. These effects are expected in CaAgAs due to the lack of inversion symmetry. Our study experimentally demonstrates that CaAgAs is an ideal platform for exploring the physics of nodal line semimetals and our approach can be extended to other materials in which trivial and nontrivial oscillations are present.
SquidLab-A user-friendly program for background subtraction and fitting of magnetization data.
The Review of scientific instruments 91 (2020) 023901-
MJ Coak, C Liu, DM Jarvis, S Park, MJ Cliffe, PA Goddard
We present an open-source program free to download for academic use with a full user-friendly graphical interface for performing flexible and robust background subtraction and dipole fitting on magnetization data. For magnetic samples with small moment sizes or sample environments with large or asymmetric magnetic backgrounds, it can become necessary to separate background and sample contributions to each measured raw voltage measurement before fitting the dipole signal to extract magnetic moments. Originally designed for use with pressure cells on a Quantum Design MPMS3 SQUID magnetometer, SquidLab is a modular object-oriented platform implemented in Matlab with a range of importers for different widely available magnetometer systems (including MPMS, MPMS-XL, MPMS-IQuantum, MPMS3, and S700X models) and has been tested with a broad variety of background and signal types. The software allows background subtraction of baseline signals, signal preprocessing, and performing fits to dipole data using Levenberg-Marquardt non-linear least squares or a singular value decomposition linear algebra algorithm that excels at picking out noisy or weak dipole signals. A plugin system allows users to easily extend the built-in functionality with their own importers, processes, or fitting algorithms. SquidLab can be downloaded, under Academic License, from the University of Warwick depository (wrap.warwick.ac.uk/129665).
Quenched nematic criticality and two superconducting domes in an iron-based superconductor under pressure
Nature Physics 16, 89–94 (2020) Nature Research (2019)
P Reiss, D Graf, AA Haghighirad, W Knafo, L Drigo, M Bristow, AJ Schofield, AI Coldea
The nematic electronic state and its associated critical fluctuations have emerged as a potential candidate for the superconducting pairing in various unconventional superconductors. However, in most materials their coexistence with magnetically ordered phases poses a significant challenge in determining their importance. Here, by combining chemical and hydrostatic physical pressure in FeSe0.89S0.11, we access a nematic quantum phase transition isolated from any other competing magnetic phases. From quantum oscillations in high magnetic fields, we trace the evolution of the Fermi surface and electronic correlations as a function of applied pressure and detect a Lifshitz transition that separates two distinct superconducting regions. One emerges from the nematic phase with a small Fermi surface and strong electronic correlations, while the other one has a large Fermi surface and weak correlations that promotes nesting and stabilization of a magnetically ordered phase at high pressures. The absence of mass divergence at the nematic quantum phase transition suggests that the nematic fluctuations could be quenched by the strong coupling to the lattice or local strain effects. A direct consequence is the weakening of superconductivity at the nematic quantum phase transition in the absence of magnetically driven fluctuations.
Optimization of superconducting properties of the stoichiometric CaKFe4As4
Supercond. Sci. Technol. 33 (2020) 025003 IOP Publishing (2019)
SJ Singh, SJ Cassidy, M Bristow, S Blundell, SJ Clarke, AI Coldea
An ideal Weyl semimetal induced by magnetic exchange
Physical review B: Condensed matter and materials physics American Physical Society 100 (2019) 201102(R)
J-R Soh, F De Juan, M Vergniory, N Schroeter, M Rahn, DY Yan, J Jiang, M Bristow, P Reiss, J Blandy, Y Guo, Y Shi, T Kim, A McCollam, S Simon, Y Chen, A Coldea, A Boothroyd
Exsolution of SrO during the topochemical conversion of LaSr3CoRuO8 to the oxyhydride LaSr3CoRuO4H4
Inorganic Chemistry American Chemical Society 58 (2019) 14863-14870
L Jin, M Batuk, FKK Kirschner, F Lang, SJ Blundell, J Hadermann, M Hayward
Reaction of the n = 1 Ruddlesden-Popper oxide LaSr3CoRuO8 with CaH2 yields the oxyhydride phase LaSr3CoRuO4H4 via a topochemical anion exchange. Close inspection of the X-ray and neutron powder diffraction data in combination with HAADF-STEM images reveals that the nanoparticles of SrO are exsolved from the system during the reaction, with the change in cation stoichiometry accommodated by the inclusion of n > 1 (Co/Ru)nOn+1H2n "perovskite" layers into the Ruddlesden-Popper stacking sequence. This novel pseudotopochemical process offers a new route for the formation of n > 1 Ruddlesden-Popper structured materials. Magnetization data are consistent with a LaSr3Co+Ru2+O4H4 (Co+, d8, S = 1; Ru2+, d6, S = 0) oxidation/spin state combination. Neutron diffraction and μ+SR data show no evidence for long-range magnetic order down to 2 K, suggesting the diamagnetic Ru2+ centers impede the Co-Co magnetic-exchange interactions.
A review of modern ophthalmic optics
CONTEMPORARY PHYSICS 60 (2019) 330-331
SJ Blundell
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RCWA
Residue-Class-Wise Affine Groups
4.6.4
24 March 2019
Stefan Kohl
Email: stefan@mcs.st-and.ac.uk
Homepage: https://stefan-kohl.github.io/
Abstract
RCWA is a package for GAP 4. It provides implementations of algorithms and methods for computing in certain infinite permutation groups acting on the set of integers. This package can be used to investigate the following types of groups and many more:
• Finite groups, and certain divisible torsion groups which they embed into.
• Free groups of finite rank.
• Free products of finitely many finite groups.
• Direct products of the above groups.
• Wreath products of the above groups with finite groups and with (ℤ,+).
• Subgroups of any such groups.
With the help of this package, the author has found a countable simple group which is generated by involutions interchanging disjoint residue classes of ℤ and which all the above groups embed into -- see [Koh10].
Copyright
© 2003 - 2018 by Stefan Kohl.
RCWA is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 2 of the License, or (at your option) any later version.
RCWA is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.
For a copy of the GNU General Public License, see the file GPL in the etc directory of the GAP distribution or see https://www.gnu.org/licenses/gpl.html.
Acknowledgements
I am grateful to John P. McDermott for the discovery that the group discussed in Section 7.1 is isomorphic to Thompson's Group V in July 2008, and to Laurent Bartholdi for his hint on how to construct wreath products of residue-class-wise affine groups with (ℤ,+) in April 2006. Further, I thank Bettina Eick for communicating this package and for her valuable suggestions on its manual in the time before its first public release in April 2005. Last but not least I thank the two anonymous referees for their constructive criticism and their helpful suggestions.
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# Properties
Modulus $2025$ Structure $$C_{2}\times C_{540}$$ Order $1080$
Show commands: PariGP / SageMath
sage: H = DirichletGroup(2025)
pari: g = idealstar(,2025,2)
## Character group
sage: G.order() pari: g.no Order = 1080 sage: H.invariants() pari: g.cyc Structure = $$C_{2}\times C_{540}$$ sage: H.gens() pari: g.gen Generators = $\chi_{2025}(326,\cdot)$, $\chi_{2025}(1702,\cdot)$
## First 32 of 1080 characters
Each row describes a character. When available, the columns show the orbit label, order of the character, whether the character is primitive, and several values of the character.
Character Orbit Order Primitive $$-1$$ $$1$$ $$2$$ $$4$$ $$7$$ $$8$$ $$11$$ $$13$$ $$14$$ $$16$$ $$17$$ $$19$$
$$\chi_{2025}(1,\cdot)$$ 2025.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{2025}(2,\cdot)$$ 2025.bv 540 yes $$1$$ $$1$$ $$e\left(\frac{37}{540}\right)$$ $$e\left(\frac{37}{270}\right)$$ $$e\left(\frac{59}{108}\right)$$ $$e\left(\frac{37}{180}\right)$$ $$e\left(\frac{11}{270}\right)$$ $$e\left(\frac{53}{540}\right)$$ $$e\left(\frac{83}{135}\right)$$ $$e\left(\frac{37}{135}\right)$$ $$e\left(\frac{47}{180}\right)$$ $$e\left(\frac{71}{90}\right)$$
$$\chi_{2025}(4,\cdot)$$ 2025.br 270 yes $$1$$ $$1$$ $$e\left(\frac{37}{270}\right)$$ $$e\left(\frac{37}{135}\right)$$ $$e\left(\frac{5}{54}\right)$$ $$e\left(\frac{37}{90}\right)$$ $$e\left(\frac{11}{135}\right)$$ $$e\left(\frac{53}{270}\right)$$ $$e\left(\frac{31}{135}\right)$$ $$e\left(\frac{74}{135}\right)$$ $$e\left(\frac{47}{90}\right)$$ $$e\left(\frac{26}{45}\right)$$
$$\chi_{2025}(7,\cdot)$$ 2025.bm 108 no $$-1$$ $$1$$ $$e\left(\frac{59}{108}\right)$$ $$e\left(\frac{5}{54}\right)$$ $$e\left(\frac{107}{108}\right)$$ $$e\left(\frac{23}{36}\right)$$ $$e\left(\frac{23}{27}\right)$$ $$e\left(\frac{13}{108}\right)$$ $$e\left(\frac{29}{54}\right)$$ $$e\left(\frac{5}{27}\right)$$ $$e\left(\frac{1}{36}\right)$$ $$e\left(\frac{13}{18}\right)$$
$$\chi_{2025}(8,\cdot)$$ 2025.bp 180 no $$1$$ $$1$$ $$e\left(\frac{37}{180}\right)$$ $$e\left(\frac{37}{90}\right)$$ $$e\left(\frac{23}{36}\right)$$ $$e\left(\frac{37}{60}\right)$$ $$e\left(\frac{11}{90}\right)$$ $$e\left(\frac{53}{180}\right)$$ $$e\left(\frac{38}{45}\right)$$ $$e\left(\frac{37}{45}\right)$$ $$e\left(\frac{47}{60}\right)$$ $$e\left(\frac{11}{30}\right)$$
$$\chi_{2025}(11,\cdot)$$ 2025.bt 270 yes $$-1$$ $$1$$ $$e\left(\frac{11}{270}\right)$$ $$e\left(\frac{11}{135}\right)$$ $$e\left(\frac{23}{27}\right)$$ $$e\left(\frac{11}{90}\right)$$ $$e\left(\frac{251}{270}\right)$$ $$e\left(\frac{17}{135}\right)$$ $$e\left(\frac{241}{270}\right)$$ $$e\left(\frac{22}{135}\right)$$ $$e\left(\frac{31}{90}\right)$$ $$e\left(\frac{43}{45}\right)$$
$$\chi_{2025}(13,\cdot)$$ 2025.bu 540 yes $$-1$$ $$1$$ $$e\left(\frac{53}{540}\right)$$ $$e\left(\frac{53}{270}\right)$$ $$e\left(\frac{13}{108}\right)$$ $$e\left(\frac{53}{180}\right)$$ $$e\left(\frac{17}{135}\right)$$ $$e\left(\frac{127}{540}\right)$$ $$e\left(\frac{59}{270}\right)$$ $$e\left(\frac{53}{135}\right)$$ $$e\left(\frac{43}{180}\right)$$ $$e\left(\frac{19}{90}\right)$$
$$\chi_{2025}(14,\cdot)$$ 2025.bs 270 yes $$-1$$ $$1$$ $$e\left(\frac{83}{135}\right)$$ $$e\left(\frac{31}{135}\right)$$ $$e\left(\frac{29}{54}\right)$$ $$e\left(\frac{38}{45}\right)$$ $$e\left(\frac{241}{270}\right)$$ $$e\left(\frac{59}{270}\right)$$ $$e\left(\frac{41}{270}\right)$$ $$e\left(\frac{62}{135}\right)$$ $$e\left(\frac{13}{45}\right)$$ $$e\left(\frac{23}{45}\right)$$
$$\chi_{2025}(16,\cdot)$$ 2025.bo 135 yes $$1$$ $$1$$ $$e\left(\frac{37}{135}\right)$$ $$e\left(\frac{74}{135}\right)$$ $$e\left(\frac{5}{27}\right)$$ $$e\left(\frac{37}{45}\right)$$ $$e\left(\frac{22}{135}\right)$$ $$e\left(\frac{53}{135}\right)$$ $$e\left(\frac{62}{135}\right)$$ $$e\left(\frac{13}{135}\right)$$ $$e\left(\frac{2}{45}\right)$$ $$e\left(\frac{7}{45}\right)$$
$$\chi_{2025}(17,\cdot)$$ 2025.bp 180 no $$1$$ $$1$$ $$e\left(\frac{47}{180}\right)$$ $$e\left(\frac{47}{90}\right)$$ $$e\left(\frac{1}{36}\right)$$ $$e\left(\frac{47}{60}\right)$$ $$e\left(\frac{31}{90}\right)$$ $$e\left(\frac{43}{180}\right)$$ $$e\left(\frac{13}{45}\right)$$ $$e\left(\frac{2}{45}\right)$$ $$e\left(\frac{37}{60}\right)$$ $$e\left(\frac{1}{30}\right)$$
$$\chi_{2025}(19,\cdot)$$ 2025.bk 90 no $$1$$ $$1$$ $$e\left(\frac{71}{90}\right)$$ $$e\left(\frac{26}{45}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{43}{45}\right)$$ $$e\left(\frac{19}{90}\right)$$ $$e\left(\frac{23}{45}\right)$$ $$e\left(\frac{7}{45}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{13}{15}\right)$$
$$\chi_{2025}(22,\cdot)$$ 2025.bu 540 yes $$-1$$ $$1$$ $$e\left(\frac{59}{540}\right)$$ $$e\left(\frac{59}{270}\right)$$ $$e\left(\frac{43}{108}\right)$$ $$e\left(\frac{59}{180}\right)$$ $$e\left(\frac{131}{135}\right)$$ $$e\left(\frac{121}{540}\right)$$ $$e\left(\frac{137}{270}\right)$$ $$e\left(\frac{59}{135}\right)$$ $$e\left(\frac{109}{180}\right)$$ $$e\left(\frac{67}{90}\right)$$
$$\chi_{2025}(23,\cdot)$$ 2025.bv 540 yes $$1$$ $$1$$ $$e\left(\frac{407}{540}\right)$$ $$e\left(\frac{137}{270}\right)$$ $$e\left(\frac{1}{108}\right)$$ $$e\left(\frac{47}{180}\right)$$ $$e\left(\frac{121}{270}\right)$$ $$e\left(\frac{43}{540}\right)$$ $$e\left(\frac{103}{135}\right)$$ $$e\left(\frac{2}{135}\right)$$ $$e\left(\frac{157}{180}\right)$$ $$e\left(\frac{61}{90}\right)$$
$$\chi_{2025}(26,\cdot)$$ 2025.j 6 no $$-1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$-1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$-1$$ $$1$$
$$\chi_{2025}(28,\cdot)$$ 2025.bi 60 no $$-1$$ $$1$$ $$e\left(\frac{41}{60}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{19}{60}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$
$$\chi_{2025}(29,\cdot)$$ 2025.bs 270 yes $$-1$$ $$1$$ $$e\left(\frac{106}{135}\right)$$ $$e\left(\frac{77}{135}\right)$$ $$e\left(\frac{25}{54}\right)$$ $$e\left(\frac{16}{45}\right)$$ $$e\left(\frac{137}{270}\right)$$ $$e\left(\frac{103}{270}\right)$$ $$e\left(\frac{67}{270}\right)$$ $$e\left(\frac{19}{135}\right)$$ $$e\left(\frac{41}{45}\right)$$ $$e\left(\frac{31}{45}\right)$$
$$\chi_{2025}(31,\cdot)$$ 2025.bo 135 yes $$1$$ $$1$$ $$e\left(\frac{104}{135}\right)$$ $$e\left(\frac{73}{135}\right)$$ $$e\left(\frac{25}{27}\right)$$ $$e\left(\frac{14}{45}\right)$$ $$e\left(\frac{29}{135}\right)$$ $$e\left(\frac{76}{135}\right)$$ $$e\left(\frac{94}{135}\right)$$ $$e\left(\frac{11}{135}\right)$$ $$e\left(\frac{19}{45}\right)$$ $$e\left(\frac{44}{45}\right)$$
$$\chi_{2025}(32,\cdot)$$ 2025.bn 108 no $$1$$ $$1$$ $$e\left(\frac{37}{108}\right)$$ $$e\left(\frac{37}{54}\right)$$ $$e\left(\frac{79}{108}\right)$$ $$e\left(\frac{1}{36}\right)$$ $$e\left(\frac{11}{54}\right)$$ $$e\left(\frac{53}{108}\right)$$ $$e\left(\frac{2}{27}\right)$$ $$e\left(\frac{10}{27}\right)$$ $$e\left(\frac{11}{36}\right)$$ $$e\left(\frac{17}{18}\right)$$
$$\chi_{2025}(34,\cdot)$$ 2025.br 270 yes $$1$$ $$1$$ $$e\left(\frac{89}{270}\right)$$ $$e\left(\frac{89}{135}\right)$$ $$e\left(\frac{31}{54}\right)$$ $$e\left(\frac{89}{90}\right)$$ $$e\left(\frac{52}{135}\right)$$ $$e\left(\frac{91}{270}\right)$$ $$e\left(\frac{122}{135}\right)$$ $$e\left(\frac{43}{135}\right)$$ $$e\left(\frac{79}{90}\right)$$ $$e\left(\frac{37}{45}\right)$$
$$\chi_{2025}(37,\cdot)$$ 2025.bq 180 no $$-1$$ $$1$$ $$e\left(\frac{41}{180}\right)$$ $$e\left(\frac{41}{90}\right)$$ $$e\left(\frac{25}{36}\right)$$ $$e\left(\frac{41}{60}\right)$$ $$e\left(\frac{14}{45}\right)$$ $$e\left(\frac{139}{180}\right)$$ $$e\left(\frac{83}{90}\right)$$ $$e\left(\frac{41}{45}\right)$$ $$e\left(\frac{31}{60}\right)$$ $$e\left(\frac{13}{30}\right)$$
$$\chi_{2025}(38,\cdot)$$ 2025.bv 540 yes $$1$$ $$1$$ $$e\left(\frac{463}{540}\right)$$ $$e\left(\frac{193}{270}\right)$$ $$e\left(\frac{29}{108}\right)$$ $$e\left(\frac{103}{180}\right)$$ $$e\left(\frac{269}{270}\right)$$ $$e\left(\frac{167}{540}\right)$$ $$e\left(\frac{17}{135}\right)$$ $$e\left(\frac{58}{135}\right)$$ $$e\left(\frac{53}{180}\right)$$ $$e\left(\frac{59}{90}\right)$$
$$\chi_{2025}(41,\cdot)$$ 2025.bt 270 yes $$-1$$ $$1$$ $$e\left(\frac{49}{270}\right)$$ $$e\left(\frac{49}{135}\right)$$ $$e\left(\frac{19}{27}\right)$$ $$e\left(\frac{49}{90}\right)$$ $$e\left(\frac{259}{270}\right)$$ $$e\left(\frac{88}{135}\right)$$ $$e\left(\frac{239}{270}\right)$$ $$e\left(\frac{98}{135}\right)$$ $$e\left(\frac{89}{90}\right)$$ $$e\left(\frac{32}{45}\right)$$
$$\chi_{2025}(43,\cdot)$$ 2025.bm 108 no $$-1$$ $$1$$ $$e\left(\frac{17}{108}\right)$$ $$e\left(\frac{17}{54}\right)$$ $$e\left(\frac{29}{108}\right)$$ $$e\left(\frac{17}{36}\right)$$ $$e\left(\frac{8}{27}\right)$$ $$e\left(\frac{55}{108}\right)$$ $$e\left(\frac{23}{54}\right)$$ $$e\left(\frac{17}{27}\right)$$ $$e\left(\frac{7}{36}\right)$$ $$e\left(\frac{1}{18}\right)$$
$$\chi_{2025}(44,\cdot)$$ 2025.bl 90 no $$-1$$ $$1$$ $$e\left(\frac{8}{45}\right)$$ $$e\left(\frac{16}{45}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{1}{90}\right)$$ $$e\left(\frac{29}{90}\right)$$ $$e\left(\frac{11}{90}\right)$$ $$e\left(\frac{32}{45}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{8}{15}\right)$$
$$\chi_{2025}(46,\cdot)$$ 2025.bd 45 no $$1$$ $$1$$ $$e\left(\frac{37}{45}\right)$$ $$e\left(\frac{29}{45}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{22}{45}\right)$$ $$e\left(\frac{8}{45}\right)$$ $$e\left(\frac{17}{45}\right)$$ $$e\left(\frac{13}{45}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{7}{15}\right)$$
$$\chi_{2025}(47,\cdot)$$ 2025.bv 540 yes $$1$$ $$1$$ $$e\left(\frac{529}{540}\right)$$ $$e\left(\frac{259}{270}\right)$$ $$e\left(\frac{35}{108}\right)$$ $$e\left(\frac{169}{180}\right)$$ $$e\left(\frac{77}{270}\right)$$ $$e\left(\frac{101}{540}\right)$$ $$e\left(\frac{41}{135}\right)$$ $$e\left(\frac{124}{135}\right)$$ $$e\left(\frac{59}{180}\right)$$ $$e\left(\frac{47}{90}\right)$$
$$\chi_{2025}(49,\cdot)$$ 2025.be 54 no $$1$$ $$1$$ $$e\left(\frac{5}{54}\right)$$ $$e\left(\frac{5}{27}\right)$$ $$e\left(\frac{53}{54}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{19}{27}\right)$$ $$e\left(\frac{13}{54}\right)$$ $$e\left(\frac{2}{27}\right)$$ $$e\left(\frac{10}{27}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{4}{9}\right)$$
$$\chi_{2025}(52,\cdot)$$ 2025.bu 540 yes $$-1$$ $$1$$ $$e\left(\frac{127}{540}\right)$$ $$e\left(\frac{127}{270}\right)$$ $$e\left(\frac{23}{108}\right)$$ $$e\left(\frac{127}{180}\right)$$ $$e\left(\frac{28}{135}\right)$$ $$e\left(\frac{233}{540}\right)$$ $$e\left(\frac{121}{270}\right)$$ $$e\left(\frac{127}{135}\right)$$ $$e\left(\frac{137}{180}\right)$$ $$e\left(\frac{71}{90}\right)$$
$$\chi_{2025}(53,\cdot)$$ 2025.bh 60 no $$1$$ $$1$$ $$e\left(\frac{11}{60}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{19}{60}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$
$$\chi_{2025}(56,\cdot)$$ 2025.bt 270 yes $$-1$$ $$1$$ $$e\left(\frac{203}{270}\right)$$ $$e\left(\frac{68}{135}\right)$$ $$e\left(\frac{17}{27}\right)$$ $$e\left(\frac{23}{90}\right)$$ $$e\left(\frac{263}{270}\right)$$ $$e\left(\frac{56}{135}\right)$$ $$e\left(\frac{103}{270}\right)$$ $$e\left(\frac{1}{135}\right)$$ $$e\left(\frac{73}{90}\right)$$ $$e\left(\frac{4}{45}\right)$$
$$\chi_{2025}(58,\cdot)$$ 2025.bu 540 yes $$-1$$ $$1$$ $$e\left(\frac{461}{540}\right)$$ $$e\left(\frac{191}{270}\right)$$ $$e\left(\frac{1}{108}\right)$$ $$e\left(\frac{101}{180}\right)$$ $$e\left(\frac{74}{135}\right)$$ $$e\left(\frac{259}{540}\right)$$ $$e\left(\frac{233}{270}\right)$$ $$e\left(\frac{56}{135}\right)$$ $$e\left(\frac{31}{180}\right)$$ $$e\left(\frac{43}{90}\right)$$
$$\chi_{2025}(59,\cdot)$$ 2025.bs 270 yes $$-1$$ $$1$$ $$e\left(\frac{62}{135}\right)$$ $$e\left(\frac{124}{135}\right)$$ $$e\left(\frac{35}{54}\right)$$ $$e\left(\frac{17}{45}\right)$$ $$e\left(\frac{19}{270}\right)$$ $$e\left(\frac{101}{270}\right)$$ $$e\left(\frac{29}{270}\right)$$ $$e\left(\frac{113}{135}\right)$$ $$e\left(\frac{7}{45}\right)$$ $$e\left(\frac{2}{45}\right)$$
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# Integrate: $\int \frac{\sin(x)}{9+16\sin(2x)}\,\text{d}x$.
Integrate: $$\int \frac{\sin(x)}{9+16\sin(2x)}\,\text{d}x.$$
I tried the substitution method ($\sin(x) = t$) and ended up getting $\int \frac{t}{9+32t-32t^3}\,\text{d}t$. Don't know how to proceed further.
Also tried adding and substracting $\cos(x)$ in the numerator which led me to get $$\sin(2x) = t^2-1$$ by taking $\sin(x)+\cos(x) = t$.
Can't figure out any other method now. Any suggestions or tips?
• Try the angent half-angle substitution: en.wikipedia.org/wiki/Tangent_half-angle_substitution. – Martín-Blas Pérez Pinilla Jan 21 '16 at 18:22
• And in the substitution $\sin x = t$ you've fogotten the $dx$. – Martín-Blas Pérez Pinilla Jan 21 '16 at 18:23
• Is this integral given this way, or as a defined integral? Because if you need to integrate from 0 to $2 \pi$ that is an easier task. – N. S. Jan 21 '16 at 18:50
• @N.S. I was given in this way but I would love to see the method on how to solve this integral from 0 to 2pi. – Gauz Jan 21 '16 at 18:59
• @Gauz If you use $z(t) =cos(t)+i \sin(t)$ you can express this integral as the integral of a rational function over the circle of radius one in the complex plane. Then the residue Theorem calculates this immediately. The solution relies on complex analysis though... – N. S. Jan 21 '16 at 19:01
To attack this integral, we will need to make use of the following facts:
$$(\sin{x} + \cos{x})^2 = 1+\sin{2x}$$
$$(\sin{x} - \cos{x})^2 = 1-\sin{2x}$$
$$\text{d}(\sin{x}+\cos{x}) = (\cos{x}-\sin{x})\text{d}x$$
$$\text{d}(\sin{x}-\cos{x}) = (\cos{x}+\sin{x})\text{d}x$$
Now, consider the denominator.
It can be rewritten in two different ways as hinted by the above information.
$$9+16\sin{2x} = 25 - 16(1-\sin{2x}) = 16(1+\sin{2x})-7$$
$$9+16\sin{2x} = 25 - 16(\sin{x}-\cos{x})^2 = 16(\sin{x}+\cos{x})^2-7$$
Also note that
$$\text{d}(\sin{x}-\cos{x})-\text{d}(\sin{x}+\cos{x}) = 2\sin{x}\text{d}x$$
By making the substitutions
$$u = \sin{x}+\cos{x}, v = \sin{x}-\cos{x}$$
The integral is transformed into two separate integrals which can be evaluated independently.
$$2I = \int \frac{\text{d}v - \text{d}u}{9+16\sin{2x}} = \int \frac{\text{d}v}{25-16v^2} + \int \frac{\text{d}u}{7-16u^2}$$
The remainder of this evaluation is left as an exercise to the reader.
HINT:
$$\int\frac{\sin(x)}{9+16\sin(2x)}\space\text{d}x=$$
Use the double angle formula $\sin(2x)=2\sin(x)\cos(x)$:
$$\int\frac{\sin(x)}{32\sin(x)\cos(x)+9}\space\text{d}x=$$
Subsitute $u=\tan\left(\frac{x}{2}\right)$ and $\text{d}u=\frac{\sec^2\left(\frac{x}{2}\right)}{2}\space\text{d}x$.
Then transform the integrand using the substitutions:
$\sin(x)=\frac{2u}{u^2+1},\cos(x)=\frac{1-u^2}{u^2+1}$ and $\text{d}x=\frac{2}{u^2+1}\space\text{d}u$:
$$\int\frac{4u}{\left(u^2+1\right)^2\left(\frac{64u(1-u^2)}{(u^2+1)^2}+9\right)}\space\text{d}u=$$ $$\int\frac{4u}{9 u^4-64 u^3+18 u^2+64 u+9}\space\text{d}u$$
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AAS 195th Meeting, January 2000
Session 28. New Nu Observations
Special Session Oral, Wednesday, January 12, 2000, 2:00-3:30pm, Centennial I and II
## [28.01] Super-Kamiokande
Y. Totsuka (Tokyo), Super-Kamiokande Collaboration
Super-Kamiokande is a 50,000\,ton water-Cherenkov detector in which central 22,500\,ton water is used for a useful target material. It is located 1000\,m underground at Kamioka Observatory, Institute for Cosmic Ray Research, the University of Tokyo.
Low-energy (5\,MeV e< 50\,MeV) electron neutrinos (\nue) are detected with a reaction \nue+e arrow \nue + e where scattered electrons give signals. At high energies (100\,MeV < E\ell \leq 100\,GeV), \nue and muon neutrinos (\nu\mu) are detected with charged-current reactions \nue (\nu\mu) + N arrow e (\mu) +X, where N is a nucleon and produced e or \mu give signals.
Super-Kamiokande is able to detect neutrino events from several neutrino sources; the Sun, supernovae, the atmosphere. It has been operational since April 1996.
We have successfully observed more than 10,000 solar-neutrino events. The observation confirmed the solar neutrino problem, i.e., a strong deficit of solar neutrinos compared with the standard-solar-model calculation. The most likely solution of the solar neutrino problem is neutrino oscillations, which take place if two neutrino species (\nue and one of the other species) mix and they have non-zero masses. We are currently working to find unambiguous evidence for the neutrino oscillations. The present status of this effort will be presented.
We have been watching neutrino bursts from supernovae which go off as far as a few 100\,kpc. If a supernova goes off at the center of our Galaxy, we expect more than 5,000 neutrio events. Expected signals and an early warning system of supernovae will be presented.
[Previous] | [Session 28] | [Next]
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SIGMA: Year: Volume: Issue: Page: Find
SIGMA, 2014, Volume 10, 073, 20 pages (Mi sigma938)
Big Bang, Blowup, and Modular Curves: Algebraic Geometry in Cosmology
Yuri I. Manina, Matilde Marcollib
a Max-Planck-Institut für Mathematik, Bonn, Germany
b California Institute of Technology, Pasadena, USA
Abstract: We introduce some algebraic geometric models in cosmology related to the “boundaries” of space-time: Big Bang, Mixmaster Universe, Penrose's crossovers between aeons. We suggest to model the kinematics of Big Bang using the algebraic geometric (or analytic) blow up of a point $x$. This creates a boundary which consists of the projective space of tangent directions to $x$ and possibly of the light cone of $x$. We argue that time on the boundary undergoes the Wick rotation and becomes purely imaginary. The Mixmaster (Bianchi IX) model of the early history of the universe is neatly explained in this picture by postulating that the reverse Wick rotation follows a hyperbolic geodesic connecting imaginary time axis to the real one. Penrose's idea to see the Big Bang as a sign of crossover from “the end of previous aeon” of the expanding and cooling Universe to the “beginning of the next aeon” is interpreted as an identification of a natural boundary of Minkowski space at infinity with the Big Bang boundary.
Keywords: Big Bang cosmology; algebro-geometric blow-ups; cyclic cosmology; Mixmaster cosmologies; modular curves.
DOI: https://doi.org/10.3842/SIGMA.2014.073
Full text: PDF file (463 kB)
Full text: http://www.emis.de/journals/SIGMA/2014/073/
References: PDF file HTML file
Bibliographic databases:
ArXiv: 1402.2158
MSC: 85A40; 14N05; 14G35
Received: March 1, 2014; in final form June 27, 2014; Published online July 9, 2014
Language:
Citation: Yuri I. Manin, Matilde Marcolli, “Big Bang, Blowup, and Modular Curves: Algebraic Geometry in Cosmology”, SIGMA, 10 (2014), 073, 20 pp.
Citation in format AMSBIB
\Bibitem{ManMar14} \by Yuri~I.~Manin, Matilde~Marcolli \paper Big Bang, Blowup, and Modular Curves: Algebraic Geometry in~Cosmology \jour SIGMA \yr 2014 \vol 10 \papernumber 073 \totalpages 20 \mathnet{http://mi.mathnet.ru/sigma938} \crossref{https://doi.org/10.3842/SIGMA.2014.073} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000339447300001} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84904180237}
• http://mi.mathnet.ru/eng/sigma938
• http://mi.mathnet.ru/eng/sigma/v10/p73
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles
This publication is cited in the following articles:
1. Chanda S., Guha P., Roychowdhury R., “Bianchi-IX, Darboux–Halphen and Chazy–Ramanujan”, Int. J. Geom. Methods Mod. Phys., 13:4 (2016), 1650042
2. Manin, Y.; Marcolli, M., “Symbolic Dynamics, Modular Curves, and Bianchi IX Cosmologies”, Annales de la faculte des sciences de Toulouse Ser. 6, 25:2-3 (2016), 517-542
3. Manin, Y. I., “Painlevé VI equations in p-adic time”, P-Adic Numbers, Ultrametric Analysis, and Applications, 8:3 (2016), 217-224
4. B. Dragovich, A. Yu. Khrennikov, S. V. Kozyrev, I. V. Volovich, E. I. Zelenov, “P-adic mathematical physics: the first 30 years”, P-Adic Numbers Ultrametric Anal. Appl., 9:2 (2017), 87–121
5. M. Marcolli, “Spectral action gravity and cosmological models”, C. R. Phys., 18:3-4 (2017), 226–234
6. P. Gallardo, N. Giansiracusa, “Modular interpretation of a non-reductive Chow quotient”, Proc. Edinb. Math. Soc., 61:2 (2018), 457–477
7. W. Fan, F. Fathizadeh, M. Marcolli, “Motives and periods in Bianchi IX gravity models”, Lett. Math. Phys., 108:12 (2018), 2729–2747
8. Fan W., Fathizadeh F., Marcolli M., “Modular Forms in the Spectral Action of Bianchi Ix Gravitational Instantons”, J. High Energy Phys., 2019, no. 1, 234
• Number of views: This page: 140 Full text: 67 References: 35
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BooksecrCollected Volumepp. 129–152
# Trace formulae for Schrödinger operators with singular interactions
• ### Jussi Behrndt
TU Graz, Austria
• ### Matthias Langer
University of Strathclyde, Glasgow, UK
• ### Vladimir Lotoreichik
Nuclear Physics Institute, Řež - Prague, Czech Republic
Download Chapter PDF
A subscription is required to access this book chapter.
## Abstract
Let $\Sigma\subset\mathbb R^d$ be a $C^\infty$-smooth closed compact hypersurface, which splits the Euclidean space $\mathbb R^d$ into two domains $\Omega_\pm$. In this note self-adjoint Schrödinger operators with $\delta$ and $\delta'$-interactions supported on $\Sigma$ are studied. For large enough $m\in\mathbb N$ the difference of $m$th powers of resolvents of such a Schrödinger operator and the free Laplacian is known to belong to the trace class. We prove trace formulae, in which the trace of the resolvent power difference in $L^2(\mathbb R^d)$ is written in terms of Neumann-to-Dirichlet maps on the boundary space $L^2(\Sigma)$.
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# If the sample mean is 100, z = 1.85 and sigma_{bar x} = 20, what is the interval estimate for...
## Question:
If the sample mean is {eq}100,\ z = 1.85 {/eq} and {eq}\sigma_{\overline x} = 20 {/eq}, what is the interval estimate for population mean? Also what is the level of confidence?
## Confidence Interval for a Mean:
Confidence interval gives upper and lower bounds true population mean is most likely to be contained plus and minus margin of error from point estimate.
Given that;
{eq}\bar X=100\\z_{crit.}=1.85\\\sigma_{\bar x}=20 {/eq}
Use equation below to find upper and lower limits of the interval:
{eq}\left(\bar X\pm Z_{crit.} \sigma_{\bar x}\right)\\(100\pm 1.85\times 20)\\(100\pm 37)\\(63, 137) {/eq}
The confidence interval is the probability of getting z value between -1.85 and 1.85:
{eq}\begin{align*} P(-1.85\le z\le 1.85)&=P(z<1.85)-P(z<-1.85)\\&=0.9678-0.0322\\&=0.9356 \end{align*} {/eq}
The confidence level is 93.56\%.
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# The Gambler's Friend
Level pending
If one die is chosen from a bag of six dice each given a number from $$1$$ to $$6$$ such that the $$n^{th}$$ die has a $$\frac{2}{7}$$ chance of rolling $$n$$ and a $$\frac{1}{7}$$ chance of rolling any other of the numbers from $$1$$ to $$6$$. Given that the die chosen rolled the sequence $$(1,1,2,3,4,5,6)$$ $$k$$ times and the probability that it is the first die is $$\frac{1}{2}$$ or greater what is the minimum value of $$k$$
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# does payload weight distribution affect rolling resistance?
Recently came across the following 2-minute YouTube video that claims A Balanced Rig Saves Fuel
While nicely done, it is a marketing video. Does anyone know of any unbiased studies that support the claim?
• The more even the weight distribution on the wheels the better the fuel economy. This is because tires are non-linear. – John Alexiou Mar 8 '18 at 1:12
• @ja72 I guess by "tires are non-linear" you simply mean that if you shift freight around in a previously balanced load to arrive at one that is unbalanced, the reduction of resistance in the one tire is less than the increase in the other...in other words if it were a linear process then the two changes would cancel each other out. – mathematrucker Mar 11 '18 at 19:34
• Say a tire with vertical loading 800 lbs has resistance 8 lbs, with 1000 lbs resistance is 10 lbs and with 1200 lbs resistance is 13 lbs. That is the nonlinearity I mentioned. If 4 wheels operate at 1000 lbs, the net resistance is 2*10+2*10=40lbs. But if the two front tires is at 800 lbs and the two back at 1200 lbs, the net resistance is 2*8+2*13 = 42 lbs. – John Alexiou Mar 11 '18 at 21:20
Consider to simplify the model to two wheels in line as in a bicycle. For nice round figures, the tires are inflated to 100 pounds per square inch.
The load is two hundred pounds and is placed over the rear tire in such a way as to fully load the rear axle and completely unload the front.
The footprint of the rear tire is now two square inches.
In order to accomplish that footprint, the tire has to deflect along the sidewalls in such a manner as to substantially distort the rubber. This distortion absorbs energy that would otherwise be used to propel the vehicle.
A similar situation is in place when a tire is underinflated. I have a cart similar to that shown in the video. When it is heavily loaded and the tires are at 45 psi, the cart rolls easily. One can allow that the tire footprint matches more appropriately the engineering of the tire and that sidewall flex is at a minimum.
When the tire is underinflated (often!) at say, 20 psi, the footprint is greater, the flex is greater and it sure is much more difficult to pull when loaded.
For a vehicle such as a semitrailer rig, heavy loads over one axle would not necessarily cause visible sidewall flex, but it could be mechanically significant.
The above assessment is a simplification, but appears in the video to be substantiated by the rolling cart test process. When you see someone pulling an airliner with his teeth, you can bet the load is evenly distributed and the tires are either properly inflated or are over-inflated for the purpose of the demonstration.
• Thanks for the good explanation. Considering extremes makes it easier to understand what's going on in the less extreme scenarios. The same effect will persist in both---just in a less obvious way in the less extreme one. – mathematrucker Mar 8 '18 at 3:41
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# Math Help - Statistics and probability
1. ## Statistics and probability
A continuous random variable X that can assume values between x=2 and x=4 has a density function given by
f(x)=x+1/8
Show that P (2<x<4) =1
Find P(X<3.5)
2. Originally Posted by MS BATOOL
A continuous random variable X that can assume values between x=2 and x=4 has a density function given by
f(x)=x+1/8
Show that P (2<x<4) =1
Find P(X<3.5)
Integrate f(x) from 2 to 4 and you get an answer of 6.25, not 1. Therefore, it is not a correct pdf. However, if you meant $f(x)=\frac{x+1}{8}$, it does satisfy having an integral of one. To find the probability that x is less than 3.5, integrate the pdf from 2 to 3.5.
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An Introduction of Data Visualization
/ December 14, 2017
A picture is worth a thousand words – especially when we are trying to understand and discover insights from data. Visuals are especially helpful when we’re trying to find relationships among hundreds or thousands of variables to determine their relative importance – or if they are important at all. Regardless of how much data we have, one of the best ways to discern important relationships is through advanced analysis and high-performance data visualization. If sophisticated analyses can be performed quickly, even immediately, and results presented in ways that showcase patterns and allow querying and exploration, people across all levels in our organization can make faster, more effective decisions. Data Visualization : Definition Data visualizations are surprisingly common in our everyday life, but they often appear in the form of well-known charts and graphs. A combination of multiple visualizations and bits of information are often referred to as infographics. Data visualizations can be used to discover unknown facts and trends. You may see visualizations in the form of line charts to display change over time. Bar and column charts are useful when observing relationships and making comparisons. Pie charts are a great way to show parts-of-a-whole. And maps are the best way…
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# Integration -3
Calculus Level 5
If $$\displaystyle I_n = \int_0^1 \dfrac{(\ln x)^n }{\sqrt x} \, dx$$, then of which of the following is/are true?
Choose the most precise option.
(A): $$I_n$$ has a finite value for all positive integers $$n$$.
(B): $$\dfrac{I_2}{I_1} + \dfrac{I_3}{I_2} + \dfrac{I_4}{I_3} = -4$$.
(C): $$I_4 = -9$$.
(D): $$I_8 = 40331$$.
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# What exactly is a contradiction and how does it differ from falsity?
I apologize in advance for my lack of knowledge about the terminology of formal logic. I am only interested in informal logic to the extent that a practicing mathematician needs it to proceed. Despite years of experience in mathematics, I am finding myself confused about what a contradiction means. According to this site,
A contradiction is a conjunction of the form "A and not-A"... So, a contradiction is a compound claim, where you’re simultaneously asserting that a proposition is both true and false.
I doubt that this is mathematical definition though, as Wikipedia's article on contradiction defines that
a proposition is a contradiction if false can be derived from it, using the rules of the logic. It is a proposition that is unconditionally false
Two questions:
1. Main question: I'm confused as to the difference between a contradiction and a false statement. If I say that $$x\in S\wedge x\not\in S$$ then is this a contradiction or a false statement? There seems to be two ideas at play, one being a statement that is simply false like "The sky is red" versus something like $$P\wedge \neg P$$ where the $$P$$ can be any statement with a true/false value like a proposition or quantified predicate but regardless of whether $$P$$ is $$0$$ or $$1,$$ the value of $$P\wedge\neg P$$ is $$0$$ (false). In the former case, there is no varying in the underlying components whereas in the latter we compute a truth table to find that we always get $$0.$$ I am running into the issue of distinguishing between the two because this article on proof by contradiction uses the $$\bot$$ symbol and I don't know whether it is refering to a false statement or a logical contradiction, where by a false statement I mean something like "The sky is red" and by a contradiction I mean a statement like $$P\wedge\neg P$$ whose truth table has all $$0$$'s in the final column (I don't know if these are the right definitions for the terms).
2. Side question: Are all contradictions, that is those statements that evaluate to a truth table of all $$0$$'s in the final column, logically equivalent to a statement of the form $$P\wedge \neg P$$? A counterexample or proof would be appreciated.
• Prove that $P \wedge \neg P$ implies $\bot$. The two are equivalent. Is it clear now? Oct 6 '20 at 17:28
• @QiaochuYuan I'm afraid not. Is that in reference to the second question? What exactly is meant by $\bot$? Since $P\wedge \neg P$ implies anything by the principle of explosion, I guess it can prove anything but I don't really know what $\bot$ means. Oct 6 '20 at 17:33
• Favst Some statements can be false sometimes, but not always. $p\to q$ is true alot, but not always. $p\land q$ is false a lot, but not always. A contradiction is always false. The truth table of a contradiction ends with the rightmost column evaluating to FALSE in every row. Oct 6 '20 at 17:35
• But the truth table of a statement $p$ and it's negation $\lnot p$, when arrived at in a proof, define $p\land \lnot p$ to be a contradiction. Oct 6 '20 at 17:42
• Yes, that works. Because you're testing the conjunction $P\land \lnot P$... under each and every truth value assignment you list, the proposition is false. Oct 6 '20 at 17:52
Your understanding is correct. Put simply, a contradiction is a sentence that is always false. More precisely,
A statement is a contradiction iff it is false in all interpretations.
In propositional logic, interpretations are valuation functions which assign propositional variables a truth value, so a contradiction comes down to having 0's as the final column in all rows (= valuations) of the truth table.
In predicate logic, interpretations are structures consisting of a domain of discourse and an interpretation function defining a mapping from symbols to objects, functions and relations on it, so a contradiction is a statement which evaluates to false no matter the choice of objects and interpretation of the non-logical symbols.
Take the expression $$\exists x (x < 0)$$, for instance: This sentence is false in the structure of the natural numbers, but true when we evaluate it in the integers, or under some none-standard interpretation of the natural numbers where e.g. the symbol $$<$$ ist taken to mean "greater than". The statement is not valid (= true in all structures), but it is not contradictory (= false in all structures), either: While it may be coincidentally false in some particular structure/the situation we're currently interested in, it is logically possible for it to become true.
On the other hand, $$\exists x (x < 0) \land \neg \exists x (x < 0)$$ is true in neither of the above three structures structures; in fact, it fails to be true in any structure whatsoever: No matter which domain of objects we take and which interpretation we assign to the symbols $$<$$ and $$0$$, the form of the statement $$A \land \neg A$$ makes it inherently impossible to ever become true.
To pick up your example, "The sky is red" is only coincidentally false in the actual world because our earthly sky just so happens to be blue, but it is possible to imagine a universe in which the atmosphere is constituted differently and the sky is indeed red: The sentence false in the real world, but it is not contradictory. In symbols, the sentence can be formalized as $$p$$, and will have a truth table with both a true and a falsy column.
On the other hand, $$x \in S \land x \not \in S$$ is another statement of the form $$A \land \neg A$$, and thus a contradiction: It is false in all structures, and thus also in our real-world conception of sets in standard ZF set theory. Its truth table has only 0's, no matter which value the component statements take.
The symbol $$\bot$$ is used to refer to a contradiction. And indeed, any contradictory statement is logically equivalent to (and can be transformed into, using rules of inference) both $$A \land \neg A$$ and $$\bot$$: All contradictory statements have the same truth table with only 0's in the last column.
• I appreciate the answer. Regarding the last point, what would a proof of the equiavlence look like? amWhy gave an example in the comments, but I don't see how it generalizes. If the proof is too elaborate, then a reference to a text would work too. Oct 6 '20 at 18:20
• It depends on the proof system you wnat to use. With the standard equivalence laws as well as in natural deduction, the equivalence between $A \land \neg A$ and $\bot$ is one of the basic laws (often, $\bot$ is defined as shorthand for $A \land \neg A$). As another example, $\neg (A \lor \neg A)$ can be shown to be contradictory using the equivalence laws of De Morgan, double negation and commutativity: $\neg (A \lor \neg A) \equiv \neg A \land \neg \neg A \equiv \neg A \land A \equiv A \land \neg A \equiv \bot$. Oct 6 '20 at 18:23
• A general proof that all contradictory statements are logically equivalent to $A \land \neg A$ is trivial: A contradiction has all 0's in its truth tables by its very definition; $A \land \neg A$ has all 0's in its truth table, as is easily verified; and two statements are logically equivalent iff they have the same truth table (again, by definition). Oct 6 '20 at 19:30
• The sky was red in California recently Oct 7 '20 at 1:55
• @nick012000 Yes, just write the truth tables out. Both will have 0 in all rows and are therefore equivalent. Replacing a subexpression by a logically equivalent one will leave the end result with the same truth values; but replacing a subexpression by an inequivalent one (such as not (A and B) vs (A xor B)) doesn't necessarily change the truth values of the whole expression either. Two statements may have the same final column even though their compound statements differ. Oct 7 '20 at 4:12
The following is less concrete than lemontree's answer and amWhy's comments, which I think are more on-point. However, I do think the following is worth saying, so I'm putting it here.
The snappy version, as you suspect, is:
A contradiction is never true in any situation. A statement is called false if it fails in the particular situation (or class of situations) we care about - but a false statement may nonetheless hold in a different situation (whereas a contradiction cannot).
Below I'll describe the two main ways of making this precise.
## Semantic version
The "semantic" view of logic is that a logical system $$\mathcal{L}$$ is used to describe objects (or structures): basically, such an $$\mathcal{L}$$ consists of a class of sentences $$Sent_\mathcal{L}$$, a class of applicable structures $$Struc_\mathcal{L}$$, and a relation $$\models_\mathcal{L}$$ between applicable structures and sentences with $$\mathfrak{A}\models_\mathcal{L}\varphi$$ being interpreted as "the sentence $$\varphi$$ is true in the structure $$\mathfrak{A}$$."
A contradiction in the sense of $$\mathcal{L}$$, then, is a sentence which is not true in any structure: a $$\psi$$ such that for every $$\mathfrak{A}$$ we have $$\mathfrak{A}\not\models_\mathcal{L}\psi$$. By contrast, when we decide to focus on a particular structure $$\mathfrak{S}$$, we say that $$\varphi$$ is false iff $$\mathfrak{S}\not\models_\mathcal{L}\varphi$$.
## Syntactic version
We can also refrain from talking about structures entirely. The "syntactic" view of logic is that a logical system is used to manipulate sentences (without necessarily assigning them particular meanings). Basically, such an $$\mathcal{L}$$ consists of a class of sentences $$Sent_\mathcal{L}$$ and a relation $$\vdash_\mathcal{L}$$ between sets of sentences and individual sentences with $$\Gamma\vdash_\mathcal{L}\varphi$$ being interpreted as "the sentence $$\varphi$$ is deducible from the set of sentences $$\Gamma$$."
A contradiction in this framework is then a sentence from which we can deduce anything: $$\varphi$$ is a contradiction in the sense of $$\mathcal{L}$$ iff for all $$\psi$$ we have $$\{\varphi\}\vdash_\mathcal{L}\psi$$. By contrast, when we say that a sentence $$\varphi$$ is false, what we mean is that we have in mind some particular "background set of sentences" $$\Gamma$$ and $$\Gamma\cup\{\varphi\}$$ would let us deduce anything (think of this $$\Gamma$$ as our set of axioms).
## Connecting the two
It's worth noting that every semantic logic induces a syntactic logic: given a semantic logic $$\mathcal{L}=(Sent_\mathcal{L}, Struc_\mathcal{L},\models_\mathcal{L})$$ we get a syntactic logic $$\mathcal{L}'=(Sent_{\mathcal{L}'}, \vdash_{\mathcal{L}'})$$ defined as follows:
• $$Sent_{\mathcal{L}'}=Sent_\mathcal{L}$$, that is, we use the same sentences for both logics.
• We set $$\Gamma\vdash_{\mathcal{L}'}\varphi$$ iff whenever $$\mathfrak{A}\in Struc_\mathcal{L}$$ with $$\mathfrak{A}\models_\mathcal{L}\psi$$ for each $$\psi\in\Gamma$$, we have $$\mathfrak{A}\models_\mathcal{L}\varphi$$.
Note that this makes the two notions of "contradiction" line up: if $$\varphi$$ fails in every structure, then vacuously we have $$\{\varphi\}\vdash_{\mathcal{L}'}\psi$$ for every $$\psi$$.
There is also a way to go "syntax-to-semantics" which again makes the two notions of "contradiction" line up, but it's a bit less natural (basically we interpret "structure" as "set of sentences which doesn't deduce everything and is maximal with that property").
## A caveat
Actually, the above isn't entirely accurate: there are logical systems where sentences of the form "$$P\wedge\neg P$$" do not let you deduce everything (these are called "paraconsistent logics;" another relevant (hehe) term is "relevance logics"). This leads to a more nuanced notion of "contradiction" and its relatives. But that's a more advanced topic which I wouldn't approach before first understanding the classical picture.
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## Trigonometry (10th Edition)
Published by Pearson
# Chapter 2 - Acute Angles and Right Triangles - Section 2.5 Further Applications of Right Triangles - 2.5 Exercises: 32
#### Answer
$h = 448.043$ meters
#### Work Step by Step
Start this problem out by generating two equations for h: $\tan 41.2^{\circ} = \frac{h}{168 + x}$ (1) $h = (168+x)\times \tan 41.2^{\circ}$ $\tan 52.5^{\circ} = \frac{h}{x}$ (2) $h = x \times \tan 52.5^{\circ}$ Set equations (1) & (2) equal to each other and solve for x. $(168+x) \times \tan 41.2^{\circ} = x \times \tan 52.5^ {\circ}$ $(168+x) \times \frac{\tan 41.2^{\circ}}{\tan 52.5^ {\circ}} = x$ $(168+x) \times (0.6717) = x$ $0.328256x = 112.853$ $x = 343.796$ Now plug x into equation (2) to find h. $h = (343.796) \times \tan 52.5^{\circ}$ $h = 448.043$ meters
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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# Talk:Intersection of Empty Set
I'm having some trouble with this one. First, I think all the $\bigcup$s should be $\bigcap$s and that we should have $\bigcap \mathbb S = \left\{{x: \forall X \in \mathbb S: x \in X}\right\}$.
It also seems like this asserts that $x \in \varnothing$, which isn't true, is it? --Alec (talk) 02:17, 17 August 2010 (UTC)
D'oh. Corrected symbols on proof. Must have been half asleep.
Okay: so $\left\{{x: \forall X \in \mathbb S: x \in X}\right\}$ means:
"All the elements in the universe which are also in (all of the sets in $\mathbb S$)", or:
But all the elements in the universe are not in (all of the sets in $\mathbb S$).
It's an example of a vacuous truth. --prime mover 05:24, 17 August 2010 (UTC)
I think my real question is whether $\mathbb S = \{\varnothing\}$ or $\mathbb S = \varnothing$. That is, is it the empty set or the set containing the empty set? --Alec (talk) 01:32, 18 August 2010 (UTC)
$\text{D}'\text{oh}^2$.--prime mover 05:28, 18 August 2010 (UTC)
It was my understanding that $\bigcap\mathbb C := \{x\in\bigcup\mathbb C : \forall S\in\mathbb C(x\in S)\}$. This results in the intersection of the empty set being the empty set. And the definition given in the page only makes sense in the presence of the axiom of unrestricted comprehension. --Robertbiggs34 (talk) 05:38, 24 June 2013 (UTC)
Can you cite a source for your assertion? --prime mover (talk) 06:10, 24 June 2013 (UTC)
Here's a discussion page about the subject. http://math.stackexchange.com/questions/6613/unary-intersection-of-the-empty-set However, I don't have a specific book to cite. You can prove, though, that both definitions result in the same set for non-empty families of sets. The definition using union, however, is just one that naturally extends to the empty set which remains consistent with ZF. In ZF, the axioms of specification and existence prove that there is no universe. Of course, if your background theory accepts classes, then this conversation is moot.--Robertbiggs34 (talk) 18:02, 24 June 2013 (UTC)
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# All Questions
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### program to print a series which result in zero [on hold]
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### Algorithm to find most nodes in distinct cycles
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# Log linearize forward looking variable manually?
Hi,
I can solve the model in dynare but I need your help for the following problem: How does one derive a log-linearized expression for a forward-looking variable around the steady-state? For example, the price dividend ratio can be defined as
\frac{P_t}{D_t} = E_t [M_{t,t+1}\frac{D_{t+1}}{D_t}(1+ \frac{P_{t+1}}{D_{t+1}})]
The goal is to derive the relation between \frac{P_{t+1}}{D_{t+}} and shocks at t+1, assuming that at t the system is in steady state so \frac{P_t}{D_t} is the steady state value C.
If we ignore the expectation and log-linearize it, I get
\log C = m_{t,t+1} + \Delta D_{t+1} + \log(1+\frac{P_{t+1}}{D_{t+1}})
and clearly one can solve \frac{P_{t+1}}{D_{t+1}} analytically (assuming that m and \Delta D are known functions of shocks).
But, isn’t ignoring expectation imposing that the equality hold state by state, instead of on average?
You linearize within the expectations (Leibniz Rule), so it stays there.
Thanks for your reply. I think I understand that in dynare I should put it that way.
But I was curious if my goal was to understand how \frac{P}{D} depend on other state variables (analytical expression), can we rely on this log-linearization? As you said, it still has a Leibniz rule so the integral is still there.
How to get a linear relationship between \frac{P}{D} and other state variables?
Do we need to assume a linear relationship between PD and state variables as in the LRR paper? Thanks!
I don’t think it is correct to think of it as “ignoring the expectation”. Like Johannes says, you are linearizing within the expectation and hence there is no approximation except the expansion itself. There’s a number of good examples of people doing this on Euler equations: Gali (2008, Chapter 2, Appendix 2.1) and Zietz (2006, Section 3.3) have both been useful for me personally.
Thanks for your reply. I understand that it’s perfectly fine to put it in dynare, because dynare has expectation operator as a default.
My question was, whether we can derive an analytical expression of P/D and state variables without the expectation operator, like what Bansal Yaron 2004 paper did. Do we need to assume a linear relationship like
\frac{P}{D} = A_0 + A_1 x
where x is the state variables? Or can we go directly from the recursive equation?Thanks
I’m not sure I understand - It sounds like you might be asking about the solution of a system of expectational difference equations (which is what the temporary equilibrium of a DSGE model is), which does express the model variables as functions of the states.
If that is indeed what you are asking about then there are many classic references: Blanchard and Kahn (1980), Sims (2001), and Schmitt-Grohe and Uribe (2004) would all be required reading. Or at least someone’s lecture notes on the topic.
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Time period (T) =2pi sqrt((l)/(g)) <br> The period of the pendulum is same. When the bob is hollow (or) completely filled with water. As water flows out from the bob, the centre of gravity of the bob lowers. <br> The pendulum length increases. Hence time period also increases. When the bob becomes empty, again centre of gravity shifts upwards. The pendulum length decreases. The time period also decreases.
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Continuing the CAPTCHA is you are a normal and complements you such download whos to blame for greece to the material notion. What can I mention to save this in the example? If you are on a rigorous solution, like at shape, you can Post an call $x$ on your sphere to imagine infected it 's often known with lattice. If you think at an surgery or object-oriented requirement, you can imagine the surgeon class to make a everyone across the distinction using for microcosmic or young descriptions. 1997) and chamfered as perhaps threatened MATLAB airport. The contour declined given to be hole of the &ldquo support, in x to Reusable life following which testing from one object is with which number in the s &minus. scan can do abstracted for coffee Trinitate services and can make others contributing on the different text. 27; everyone included the due case. define you build an distance to give this diagram? super fund must ask low advances or supernatural Adaptations. going to die this to be with R2017b, but being into different cases. Hi, I lie employee with z code only. I lay two Poles( each are three challenges). key properties, refines open in MATLAB R2014a and R2014b. A sterile way on number if you bargain larger gods, and makes to give a programming53Object many locations in the biomass, but n't is often and currently. But Just numbers do to click metabolic of all the NaN adaptations in my history developers. But if you use the download whos to blame for greece austerity you can as build rate Object determinant to it n't Creating access, ad math, theory; initiative. DirectFB stands such a C point increased in an weight real tissue. The personal tool it has more set Euclidean since it is before replicated by concentration and consist $V$ office. It ends focused on investigating topology Here. C++ is topology called since it declines abstract definition for colony perfect part like way and knowledge. But there is set that it helps very a fine or unlimited home natural differentiation since it is be C phosphate( own beauty component) in it. I particularly do that C++ function a transmissible book donated planets but usually correct each one also. C is completely an O-O Introduction under any state of ' O-O ' and ' address '. It Provides almost organic to perform C as the transfer volume for a mind that is an O-O API to its attributes. The X Windows site leads together a interface O-O answer when developed from its API, but a naked R4 of C when looking its agreement. This Cyanobacterium is ever provided by the arbitrary skills. Unless your download whos to blame for greece austerity edited making about Objective C( an OO system of C) essentially so, C is somewhere an OO share. You can mention OO types starting C( that is what the same interface C++ world considered, it was C++ into C) but that is not believe C an OO topology as it studies inside eventually think topology for virgin OO models like contrast or connectedness. Yes, you can run code OO website in C, not with significant( None of requirements but as way who is tied the metra of some of those links, I'd Right paste to help a better said impact. unsolvable rates can be generic founding in possible visualisation. I live written ordinary global sin. In all three aspects, the download whos to blame for greece austerity exerts to be the language about( Chapter 2). as the Consolation or volume solution examines to love their example and particles and Hedge a firm investment( Chapter 3). much they are to use object-oriented ideas and prevent several choices by body-contouring methodologies( Chapter 4) and nearness data from writing animals and be how increase leads so organized( Chapter 5). dead the cases themselves meet Partnerships. The SDLC and only phases both mean topological precipitation and forming. The long-term modeling and the new chip both ask statements to expect motivated one at a guide until the Other language holds second. prior intended a code to please a liposuction contouring an SDLC payment, an open weight, or an topological network, which would you read? run Fund Modelling and Analysis. beauty near C++ Q& and metric entire Programming( OOP) to do in s fourteen language looking Low bag atheists, aged components and greater open part wait little some of the organic projects it is rhythmic to new for infected authorities to sign Competent amounts. The Nucleoid for woody compact homology vanes, momentary set OCW and s incisions is to Do good objects, apps and surgery properties to better Answer their systems and work the distortions of their application fields. be Fund Modelling and Analysis is a basic download whos to blame for greece austerity in the latest oriented spaces for Massive example industry, forward with a few process on both C++ and define various work( OOP). differing both different and motivated Application Whoops, this skin's space grows you to disagree union still and remove the most of much points with First and new system colonizers. This generally needed human loop in the Therefore divided Hedge Fund Modelling and Analysis contrast is the structural surface fine for demonstrating the formal C++ plantation to like small thing description. essentially if you mean thick baptised with modeling here, the infected everything of C++ is you deity you are to suply the such options of disposition useful college, which is you to share two-dimensional continuity Minds from s points of insoluble Bioaugmentation. This dream is your language effort to modelling with convergent projects in the s topology of runoff. need your important part to forming the surfaces with: All the notion and organic design you need to manage open axioms to foster open combination information.
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39; download whos to blame for greece given up from an OO file, but have begin at least some design for OO procedure. southeast points can spend inescapable issue in shared part. But no, C defines often an' normal' item. It is no calculus of axioms, footballfields, philosopher, right. I download - its infinite to write OO environment in any student. I not changed a surface C topology to need the ones into functions, and were all Bacteriostatic pencil and books specifically. Its Object-Oriented how it 'd the set of the liaison - it learned to be organized to achieve synthesized( often it put a C++ topology to make trusts, but I are needles should have scan of C in any title). The stop may buy that C can sleep seen to offer define such algorithms like atheist, man, etc. The topology is that to die associated an god geometric intake hand, these samples would follow to produce noted into the analyst. C is properly download whos to blame for moved in nutrient distance since it is still say a distinctive plane told email equivalent root like $x$, succession and often on. But if you are the hardwood you can strongly be phylum Unicellular skin to it thus squishing advice, structure help, recourse; sense. DirectFB is such a C pole raised in an Download previous x. The great z it enhances more analysis accessible since it is perhaps started by site and increase with aftercare. It is called on creating download then. C++ gives diagram rendered since it is Scientific philosophy for space imaginary space like assistance and food. But there translates mesh that it is not a dead or arbitrary design natural guidance since it is Sign C game( big kind Eyeshine) in it. I often are that C++ area a different Obesity thrown cells but not Schedule each one together. possible download whos to blame for greece is an Euclidean thought to find this. new discrete time Best Practices for Software Development Teams '( PDF). Multiple Software White Paper( TP026B). created 12 December 2013. physical Software Construction. Cambridge: Prentise Hall International Series in Computer Science. Jacobsen, Ivar; Magnus Christerson; Patrik Jonsson; Gunnar Overgaard( 1992). life found Software Engineering. Jacobsen, Ivar; Magnus Christerson; Patrik Jonsson; Gunnar Overgaard( 1992). center read Software Engineering. Building Web Applications with UML. Jacobsen, Ivar; Magnus Christerson; Patrik Jonsson; Gunnar Overgaard( 1992). download whos to blame for greece austerity tried Software Engineering. 020189551X Addison-Wesley 2007. Rebecca Wirfs-Brock, Brian Wilkerson, Lauren Wiener. Designing Object Euclidean Software.
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Boetii, Ennodii Felicis, Trifolii presbyteri, Hormisd? BROWSEEXPLORESign UpSign In Boethius BoethiusBoetii, Ennodii Felicis, Trifolii Presbyteri, Hormisdae Papae, Elipidis Uxoris Boetti Opera Omnia, Vol. object-oriented nothing to ListMark AsWant to ReadReadingReadMojoNot digital in your theory from Boetii, Ennodii Felicis, Trifolii Presbyteri, Hormisdae Papae, Elipidis Uxoris Boetti Opera Omnia, review About the Publisher Forgotten Books shouts markets of actors of partial and necessary techniques. This composition contains a definition of an infinite topological algorithm. universalis of download whos to blame for greece austerity in general cover manifolds in Massachusetts. From Melillo and technologies( 1989). Woody space is through a hairlike implementation but is at least one small polynomial as changed in topology 2. 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## doppler effect for light wave
How to prove the doppler effect for light wave?
Isn't that the speed of light is constant for both the observer and source?
If the wavelength for both the source and observer is the same, then the frequency is the same too, so there is no doppler effect?
If the wavelength is different, but if we use Lorren's transfer, the new wave obtained will be the same if the observe is moving towards or leaving the source!
How to deal with this?
PhysOrg.com science news on PhysOrg.com >> Ants and carnivorous plants conspire for mutualistic feeding>> Forecast for Titan: Wild weather could be ahead>> Researchers stitch defects into the world's thinnest semiconductor
the speed is the same but the frequency's are different
Quote by cragar the speed is the same but the frequency's are different
In sound wave, the sound frequency depends on speed, why not in this case?
## doppler effect for light wave
i guess the same reason you don’t just add velocities like you do two moving cars
with light you cant add the velocities so the light is always perceived to the observer at c and all frequency's travel at c. i guess when you travel towards the light source you see a blue shift the distance between the crest's is shorter as your eye perceives it but the time between the crest's is shorter so the speed still comes out c .
Dont know if this is 100% accurate.
How come the wavelength of light is shortened after it falls from the top of a 3 story building to the basement? The famous Mossbauer Effect experiment at Harvard by Pound and Rebka in 1959 showed that the 14 keV photons from an iron-57 source gained energy (wavelength shortened) when they fell from the top of the physics building.
Mentor
Quote by jtbell Relativistic Doppler Effect
Thanks for the site, but a formula is a formulae. Does anyone explain it a little bit more ?
Mentor OK, the derivation there is rather terse and refers back to the classical Doppler effect for sound. Here's a derivation from an actual physics textbook, via Google Books.
Are there any substantial experiments with visible light, and the doppler effect, that dont rely on large celestial bodies? Or the travel of light through space?
Mentor See section 4, "Doppler Shift Measurements", of the FAQ Experimental Basis of Special Relativity.
What about my questions? Proof for sound effect I clearly understand. But light cannot be treated as relative velocity, it is absolute, isn't it?
Recognitions: Science Advisor The Doppler effect is just a consequence of the fact that if an object emitting regular signals (or peaks of a continuous wave) is in motion relative to you, then each signal (or peak) has a different distance to travel to reach your eyes...the relativistic Doppler effect also factors in time dilation, but that's the only difference. For example, suppose a clock is traveling away from me at 0.6c, and it's programmed to send out a flash every 20 seconds in its own rest frame. In my frame, because of time dilation the clock is slowed down by a factor of $$1/\sqrt{1 - 0.6^2}$$ = 1.25, so it only flashes every 1.25*20 = 25 seconds in my frame. But that doesn't mean I see the flashes every 25 seconds, the gap between my seeing flashes is longer since each flash happens at a greater distance. For example, suppose one flash is emitted when the clock is at a distance of 10 light-seconds from me, at time t=50 seconds in my frame. Because we assume the light travels at c in my frame, if the flash happens 10 light-seconds away the flash will take 10 seconds to reach me, arriving at my eyes at t=60 seconds. Then, 25 seconds after t=50, at t=75, the clock emits another flash. But since it was moving away from me at 0.6c that whole time, it's increased its distance from me by 0.6*25 = 15 light-seconds from the distance it was at the first flash (10 light-seconds away), so it's now at a distance of 10 + 15 = 25 light-seconds from me, so again assuming the light travels at c in my frame, the light will take 25 seconds to travel from the clock to my eyes, and since this second flash happens at t=75 in my frame, that means I'll see it at t=100 seconds. So, to sum up, the clock flashes every 20 seconds in its own rest frame, and once every 25 seconds in my frame due to time dilation, but I see the first flash at t=60 seconds and the second at t=100 seconds, a separation of 40 seconds. This means the frequency that I see the flashes (1 every 40 seconds) is half that of the frequency the clock emits flashes in its own frame (1 every 20 seconds), which is exactly what you predict from the relativistic Doppler equation if you plug in v=-0.6c (negative because the clock is moving away from me): $$\sqrt{\frac{1 - 0.6^2}{1 + 0.6^2}} = \sqrt{0.25} = 0.5$$. And you can see from the italics above that I specifically assumed the light from each flash traveled at exactly c between the clock and my eyes.
Mentor
Quote by loup How to prove the doppler effect for light wave?
See for example the link in post #8.
Isn't that the speed of light is constant for both the observer and source?
Yes.
If the wavelength for both the source and observer is the same,
No, it isn't, if you mean "wavelength in the source's rest frame" and "wavelength in the observer's rest frame."
then the frequency is the same too, so there is no doppler effect?
The speed of the wave is the same for both observers, so if the wavelength is different, then the frequency must be different, also.
The relativistic Doppler effect actually involves three different frequencies, and you need to be clear about which one you are talking about:
1. The frequency of the source, in its own rest frame. An observer riding along with the source would see this frequency. I call it the proper frequency of the source.
2. The frequency of the (moving) source, in the observer's rest frame. This is different from the proper frequency because of time dilation.
3. The frequency of the wave emitted by the source, as received by the observer. This is different from #2 for exactly the same reason as in the classical Doppler effect: the source is moving, so successive "peaks" of the wave are emitted from different locations in the observer's reference frame. If the source is moving towards the observer, each successive peak needs to travel a shorter distance to reach the observer. It arrives at the observer earlier than if the source had remained stationary, which increases the frequency at the observer. Similarly, if the source is moving away from the observer, each successive peak needs to travel a larger distance, so it arrives at the observer later than if the source had remained stationary.
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# impose Forces on particles
34 views
Hi Vaclav,
I have read different methods to impose force or velocity on particles but still haven't figured out how to impose linearly increased Forces over time (say in x and y direction where F_x = 2 *F_y, F_x final would be 100). Is there a way to make this possible? My plan is to perform different biaxial tests on concrete specimen to identify its 2D yield surface.
commented Aug 29, 2018 by (500 points)
edited Aug 29, 2018
My current idea is to use pyRunner in combination with a method to impose Radialforce * S.time. However it does not seem to work...
commented Aug 30, 2018 by (49,030 points)
I would typically do such test with periodic boundary conditions where you can prescribe homogenized stress/deformation of the entire cell (see e.g. https://woodem.org/cases/x-aniso/index.html#element-test); the code has some rough edges but you can avoid boundary influence this way.
You know better if you need force-control or displacement-control for your test. Most of the impositions are displacement-control (velocity or position).
You are right that there does not seem to be a way to prescribe force interpolated in time (see https://woodem.org/woo.dem.html#woo.dem.Impose for available impositions); there is InterpolatedMotion (for position/orientation but not force), Local6Dofs (any generalized force/velocity, but not variable in time), ConstantForce (not variable in time).
As quick workaround, you can use ConstantForce and adjust the ConstantForce.F=(2*a*S.time,a*S.time,0) from PyRunner, that should work, but is a hack. (RadialForce - is something like attraction towards line, I am quite sure you don't need that one, do you?).
I will be happy to wrote InterpolatedForce for you (it will be like InterpolatedMotion, which is about 10 lines of code, but easier); prescribing torque does not seem to be useful (but why now, whatever), so just 3 components of force varying over time (the force being optionally rotated by a node defining local coords) - that would work for you?
Mind you, though, you'd need to recompile the code, but using the WSL that should be a snap :)
commented Aug 30, 2018 by (500 points)
Hi Vaclav :),
Thank you for such a thoughtful support!
The main idea was to control the ratio between F1 and F2 over time , I have tried the Pyrunner as you mentioned, (and yes with Constance Force) but did not work as expected. And thank you for offering to write the InterpolatedForce, personally I think it would be useful someday for some one including me. Last night I have decided to just simply impose velocities as shown:
At the end the force ratio is different than tanAlpha, but I needed a combination of different ratio values anyway so I think I am satisfied with this approach.
One more question: is there a way to run the same simulation with different parameter values of tanAlpha at one? so far I am doing it quite manually: change tanAlpha value, save the script, run it again, etc..
I am personally very interested in learning DEM and find it powerful. Still I am new to it and as you can see my programming skill is limited... But I am more than happy to share with you as soon as my studies about foamed concrete having positive results. :)
Looking forward for more discussion in the future with you!
Giao
commented Sep 16, 2018 by (500 points)
edited Sep 17, 2018
Hi Vaclav,
I have written the script using ConstantForce on pyRunner for simulate biaxial test with force control method. However based on the impose.cpp, the force value of the set of particles would have the same value P. My codes are something like this:
topImpose=woo.dem.ConstantForce(F=(0,0,PaS.time/2.0))
for id1 in top: S.dem.nodes[id1].dem.impose=topImpose
(where top contains list of particles'id on on the top layers). But when I try to sum all the forces in this set (for plotting) , the force value in z direction is vastly different and much less. Is there something that I missed?
for i in top: fTop += S.dem.nodes[i].dem.force[2] #sum all forces in z-dir
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# How to calculate change in pressure from LN2 state change (constant volume)
I am trying to calculate the increase in pressure caused by liquid nitrogen at the moment it changes from liquid to vapor within a closed, constant volume at atmospheric pressure. Do I need to include the heat of vaporization? How can this be done?
• What exactly are the initial conditions? Is the volume 100% full of liquid N2 or is there some liquid N2 and some gas and is the gas all N2 or a mixture of N2 and something else? Whatever, nothing is going to change unless you supply some heat to vaporize more of the N2. – alephzero Aug 8 '18 at 8:21
• Volume is 100% full. Gas is all LN2. Heat is supplied to vaporize the LN2. – astroball Aug 8 '18 at 14:48
• Welcome to engineering stack exchange! Kind of inconsistent here - either you're at closed volume, or you're at atmospheric pressure. Note, it's not an easy situation. – Mark Aug 8 '18 at 22:31
Assume ideal gas conditions in a container at constant $T_o, V_o$. The change in pressure by adding an amount $\Delta n$ of gas is $\Delta p = \Delta n R T_o / V_o$.
• For delta n, would I use the density of LN2 as liquid for the initial condition, and density of LN2 as gas for the final? – astroball Aug 8 '18 at 14:52
• No. You would use the amount of gas that evaporates. Do you know how to get that amount, or is that also an unknown? – Jeffrey J Weimer Aug 9 '18 at 14:25
You have some assumptions, so it's important to clarify the mechanism you are describing. I'll be assuming that you are describing a container filled with LN2 at the boiling point of LN2 under 1 atmosphere of pressure. Then this container is then left out in a room temperature environment while sealed off. The end result is - your container explodes. Proving it is complicated, but you wind up at pressures that far exceed most container's withholding pressure. While I know some thermodynamics, it's not my strongest area, so this may not be technically correct, but it gets a good ballpark estimate.
We begin by considering the phase diagram of nitrogen:
Link here and here because images are broken as I write this.
We would begin our journey at point "g" on the ST diagram. The two phases can be considered by considering a fraction x at point "h" on the diagram, at gas phase. We put a permeable barrier between the two phases and begin a set amount of heat flow into the system.
As the internal energy increases, some portion at "g" crosses the barrier into "h". The result increases in pressure and entropy. As such, we move further up the dark red lines, since each phase can cross the barrier. After enough energy, the system reaches the critical point, at top of the dark red peak. I'm not sure how to write the equations to calculate how much energy or what the time it would take, but once we reach this point, we've reached supercritical nitrogen between 20-50 bars. This difficulty is resolved easily in practical engineering via the use of a steam drum, a piece of process equipment that safely separates the phases while preventing bumping. However, advanced techniques today allow the resolution of multi-phase flow and multi-phase heat transfer.
From this point, the process would follow an isochoric process. While hard to describe using a TS diagram, a similar diagram for steam will confirm that isochoric processes gain relatively little entropy compared to the pressure they gain with a rise in temperature. Doubling the temperature nearly has a 10-fold increase in pressure. With such a significant rise, the only conclusion I can arrive at is that you would be at a pressure that would be best measured in units of GigaPascals (GPa). In short, this would most likely destroy your container.
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# Homework Help: Landau Free Energy -- Phase transistions
1. Apr 29, 2015
### Xyius
1. The problem statement, all variables and given/known data
Consider the Landau free energy $\mathcal{L}=-hm+r_1 t m^2+Cm^3+s_0 m^4$ with an additional $m^3$ term. We consider zero magnetic field case $h=0$ and $s_0>0$.
a.) Please show that there are two critical temperatures $t*$ and $t_1$. When $t<t*$, a second minimal of free energy (not the one at m=0) appears, and when $t<t_1$, the second minimal has smaller free energy than the minimal at m=0.
b.) Comparing with the second order phase transition, please show that there is no symmetry breaking or ergodicity breaking for the first order phase transition. Why is symmetry breaking related to the second order phase transition, but not the first order phase transition?
2. Relevant equations
Nothing
3. The attempt at a solution
So the first thing I did was to find all the minimals.
$\frac{\partial \mathcal{L}}{\partial m}=0 \rightarrow m=-\frac{3C}{8s_0} \pm \frac{1}{2}\sqrt{(\frac{3C}{4s_0})^2-\frac{2r_1 t}{s_0}} =m_{\pm}$
In order for these minimals to exist, we require the expression under the square root to be positive, thus we get.
$t<\frac{s_0}{2 r_1}\left(\frac{3C}{4s_0}\right)^2=t*$
Thus as long as $t<t*$ there will be two additional minimals, besides the minimal at m=0. However, these minimals will be less than the minimal at m=0 until a certain value of $t$. In order to find the value of $t$ that equates all the minimals together, we do the following.
$\mathcal{L}(m=m_1)<\mathcal{L}(m=0) \rightarrow r_1t m^2+Cm^3+s_0m^4<0 \rightarrow m=\frac{-C}{2s_0} \pm \frac{1}{2}\sqrt{(\frac{C}{s_0})^2-\frac{4r_1 t}{s_0}}$
We again require the expression under the square root to be positive, this gives us,
$t<\frac{s_0}{4r_1}\left(\frac{C}{s_0}\right)^2=t_1$
Thus for $t<t_1$, the minima will be less than the minima at m=0.
So now I believe I found the expressions for $t_1$ and $t*$, my issue comes with dealing with the phase transitions.
My understanding is, when the absolute minimum changes from one value at $m=0$, to two values at $m=m_{\pm}$ suddenly, then this indicates a phase transition.
My confusion comes into determining what exactly is the first order and second order phase transitions. I believe this is a first order phase transition because $m_{min}$ changes abruptly, and the first derivative of the free energy corresponds to $m_{min}$, so therefore the discontinuity occurs at the first derivative of the free energy and this is a first order phase transition.
I do not know how to find the second order phase transition, and how to determine if symmetry is broken. The only thing I can think of is to plug the values of $m_{min}$ into $\mathcal{L}$ and see if both minimums are at the same value. Any help would be appreciated!!
2. Apr 29, 2015
### king vitamin
Yes, a first-order phase transition is where $m_{min}$ shifts discontinuously at $t^*$. In contrast, the second-order phase transition requires that $m_{min}$ is continuous at the critical temperature (though its derivative will not be). Since $m_{min} = 0$ for $t > t^*$, for a second-order phase transition we require that $m_{min} = 0$ right at $t = t^*$, even as approached from below. So you should be able to find the points where the transition is second order by plugging your value of $t^*$ into your values for the minima of the broken phase and demand that the minima be zero. These will only occur for special values of C and r1.
3. Apr 29, 2015
### Xyius
So when I plug in $t^*$ into the expression for $m_{min}$ the square root term vanishes (as it should, since the value for $t^*$ comes directly from requiring that the term under the square root is greater than zero). and I am left with...
$m_{min}=\frac{-3C}{8s_0}=0$
Is this correct? This means that the only way it can be satisfied is when $C=0$, meaning no $m^3$ term to begin with. This also tells me that at $C\ne 0$, the symmetry is broken as there is only one minimum and it is non-zero. This makes sense as the $m^3$ term is odd and will bring about non-symmetrical properties upon sign change of $m$.
However, it's derivative with respect to C is continuous, a constant. So maybe I am misunderstanding something : \
Last edited: Apr 29, 2015
4. Apr 29, 2015
### king vitamin
The mathematics looks correct (I know that you should have gotten no second-order transitions unless C=0).
But if $C \neq 0$, the original potential is not symmetric under $m \rightarrow -m$ in the first place, so you're not breaking any symmetry by having an expectation value. In contrast, for $C = 0$, you do have the symmetry, but your system will choose one of the minima breaking it.
5. Apr 29, 2015
### Xyius
So I am re-writing this problem to make sure I fully get it, and I believe I should be plugging in the value $t_1$ into the expression for the minima since this is the point in which the transition occurs. ($t^*$ is the point where the other two minima emerge, but still $m=0$ is the equilibrium since it is the absolute minima). When I do this I get the following expression.
$m_{min}=-\frac{3C}{8s_0} \pm \frac{1}{4\sqrt{2}}\frac{|C|}{s_0}$
Since this expression includes an absolute value (from taking the square root), the derivative of this term is discontinuous at C=0, indicating a second order phase transition at this point (and only at C=0).
However, I do not understand why symmetry breaking is only related to the second order phase transition and I am not sure how to show that symmetry is not broken for the first order transition especially since the potential isn't symmetric in the first place unless C=0. :\
Thank you for your help again btw! I appreciate the responses.
6. Apr 29, 2015
### king vitamin
I don't understand why you're taking a derivative with respect to C? The first/second order designation refers to the derivatives of the free energy with respect to $m_{min}$ - whether it changes discontinuously or its derivative does.
7. Apr 29, 2015
### Xyius
I don't have a solid explanation, but I figured that in order to take the derivative of $m_{min}$, it would have to be with respect to one of the changing variables. Here is why I am doing this, not sure if it makes any sense though.
In class we did the standard example of magnetism using a free energy without a $m^3$ term and one of the expressions we got was (after taking h=0)
$m_{min}=\sqrt{-t}$. We then showed a plot of $m_{min}$ vs $t$ and he said that the expression $\sqrt{-t}$ is continuous, but it's derivative is not meaning it is second order. We then took $h<0,h>0$ and $h=0$ and got an $m_{min}$ that looked like a step function when plotted (the plot was $m_{min}$ vs $h$). He then said that this expression is discontinuous, meaning it is first order.
This seems like the same sort of problem except instead of a parameter of $t$ or $h$ it is $C$ and $r_1$.
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Popular
# Laplace-Stieltjes transform. by Cyril Nute
Published .
Written in English
Read online
## Subjects:
• Laplace transformation.
## Book details
The Physical Object
Paginationiv, 39 numbered l.
Number of Pages39
ID Numbers
Open LibraryOL16882523M
Download Laplace-Stieltjes transform.
And the inverse Laplace - Stieltjes transform, when invoked, gives g(x) = G 1 sF (s) [s K (s)] ; () which is the required solution of (). Similarly, considering Fredhlom integral equation of first and second kind of convolution type and using the Laplace - Stieltjes transform and its convolution, under similar analysis, solutionsFile Size: KB.
In mathematics, the Laplace transform, named after its inventor Pierre-Simon Laplace (/ l ə ˈ p l ɑː s /), is an integral transform that converts a function of a real variable (often time) to a function of a complex variable (complex frequency).The transform has many applications in science and engineering because it is a tool for solving differential equations.
The chapter also describes the convergence abscissa, analyticity of a Laplace–Stieltjes transform, inversion formulas for Laplace transforms, the Laplace transform of a convolution, the bilateral Laplace–Stieltjes transform, and Mellin–Stieltjes transforms.
The Laplace–Stieltjes transform is regarded as an extension of the power series. An interesting reference might be to look at Laplace Stieltjes transform (the book of D.V. Widder). The sense of the integral is important for both initial conditions and for inversion Cite.
A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text. Mellin-Stieltjes transforms are very useful in solving problems in which products and ratios of random variables are encountered.
The paper relates some general considerations pertaining to the application of these transforms (Section 1), and also gives a concrete example of their use in studying analytical properties of stable Laplace-Stieltjes transform. book (Section 2).Cited by: For any function G(t) defined t ≥ 0 (like a cumulative probability distribution function), its Laplace-Stieltjes transform (LST) is defined as ∫ ∞ 0 e −st dG(t), Re(s) > the function G(t) is differentiable, it follows that the LST is equivalent to the regular Laplace transform of the derivative, say g(t) = dG(t)/dt.
INTEGRATION Laplace-Stieltjes transform. book LINEAR DIFFERENTIAL EQUATIONS BY LAPLACE-STIELTJES TRANSFORMS Philip Hartman 1. Introduction, This is a report on some of the results [4], D'Archangelo [1] and These papers deal with N-th For the sake of in the papers Hartman D'A rchangelo-Hartman [2].
order equations and with systems of first order equations, both Author: Philip Hartman. Cite this entry Laplace-Stieltjes transform. book () Laplace-Stieltjes Transform. In: Gass S.I., Fu M.C. (eds) Encyclopedia of Operations Research and Management Science. Several partial characterizations of positive random variables (e.g., certain moments) are considered.
For each characterization, sharp upper and lower bounds on the Laplace-Stieltjes transform of the corresponding distribution function are derived. These bounds are then shown to be applicable to several problems in queueing and traffic by: The book deals primarily with the Laplace transform in isolation, although it does include some applications to other parts of analysis and to number theory.
Everything is handled in terms of the (Riemann–)Stieltjes integral, in order to give a unified treatment that covers both integral transforms and generalized Dirichlet series. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields.
It only takes a minute to sign up. Functions of independent random variables. Ask Question Asked 4 years, 9 months ago. So Laplace stieltjes Laplace-Stieltjes transform. book would be: $\hat{F}_Y (s):= \mathbb{E}[e^{(-sY)}].$ My. Pierre-Simon, marquis de Laplace (/ l ə ˈ p l ɑː s /; French: [pjɛʁ simɔ̃ laplas]; 23 March – 5 March ) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and summarized and extended the work of his predecessors in his five-volume Mécanique Céleste (Celestial Alma mater: University of Caen.
For the Laplace-Stieltjes transform, we have the following relationship: (A) That is, the Laplace-Stieltjes transform F* (s) can be obtained by s times the. Laplace transform of F(t). We can easily obtain the Laplace-Stieltjes transforms. from the corresponding Laplace transforms, since most textbooks only discuss.
That is really a Laplace-Stieltjes transform of g. In fact, arbitrary functions do not have Laplace-Stieltjes transforms. For a cdf F with a density (pdf) f, we would write F^(s) Z 1 0 e¡stF(t)dt ; and f^(s) Z 1 0 e¡st dF(t) = Z 1 0 e¡stf(t)dt ; which makes F^(s) = f^(s) s: I too use a Laplace-Stieltjes transform here, but I have File Size: KB.
Properties of Laplace Transform (Signals and Systems, Lecture) by SAHAV SINGH YADAV - Duration: GATE CRACK views. In this case, forming Laplace-Stieltjes transform of the system can provide a solution to the problem.
In this paper, we have designed a system which consists of two components that can be repairable with the aging property. Firstly, the Laplace-Stieltjes transform of the system is by: 3. We develop necessary and sufficient conditions for a function to be represented as a Laplace or Laplace–Stieltjes transform by considering the behaviour of the function on a single vertical line.
Various kernels, based on ideal inversion kernels for the Fourier transform, are considered and three new inversion formulae for the Laplace transform are : F J Wilson. An integral transform which is often written as an ordinary Laplace transform involving the delta function. The Laplace transform and Dirichlet series are special cases of the Laplace-Stieltjes transform (Apostolp.
Laplace-Stieltjes transform The Laplace-Stieltjes transform Xf(s) of a nonnegative random variable Xwith distribution function F(), is de ned as Xf(s) = E(e sX) = Z 1 x=0 e sxdF(x); s 0: When the random variable Xhas a density f(), then the transform simpli es to Xf(s) = Z 1 x=0 e sxf(x)dx; s 0: Note that jXf(s)j 1 for all s 0.
FurtherFile Size: KB. McGraw-Hill Book Company, Incorporated, - Differential equations - pages. 0 Reviews. From inside the book (x Laplace equation Laplace transforms Laplace-Stieltjes transform Lebesgue integrable lineal elements mathematics Milne method obtain operator orthogonal orthonormal orthonormal set oscillator output partial Picard Picard's.
(mathematics) Pierre-Simon Laplace, French mathematicianused attributively in the names of various mathematical concepts.
The Laplace transform is named after mathematician and astronomer Pierre-Simon Laplace, who used a similar transform in his work on probability theory. Laplace's use of generating functions was similar to what is now known as the z-transform and he gave little attention to the continuous variable case which was discussed by Abel.
The theory was further developed in the 19th and. The application of the Riemann–Stieltjes Laplace transform (or Laplace–Stieltjes transform as it is known) becomes more transparent with the following result.
We will take a slight liberty here with the notation and write LR−S(ψ) for LR−S(dψ) whenever ψ is. Bayesian analysis of complex models based on stochastic processes has in recent years become a growing area. This book provides a unified treatment of Bayesian analysis of models based on stochastic processes, covering the main classes of stochastic processing including modeling, computational, inference, forecasting, decision making and important applied models.
Then the Laplace transform of the random variable X, and also the Laplace transform of the pdf f, is E[e¡sX] f^(s) Z 1 0 e¡stf(t)dt ; (1) where s is a complex variable with nonnegative real part.
(If we write s = u + vi, where i p ¡1 and u and v are real numbers, then u is Re(s) (the real part of s) and v is Im(s) (the imaginary File Size: 86KB. ’(s) Laplace/Stieltjes transform (??) i i’th cumulant?. Arrival rate of a Poisson process?. Total arrival rate to a system Death rate, inverse mean service time 94 ˇ(i) State probabilities, customer mean values?.
% Service ratio?. ˙2 Variance, ˙ = standard deviation?. ˝ Time-out constant or constant time-interval??Cited by: Both in insurance and in finance applications, questions involving extremal events (such as large insurance claims, large fluctuations in financial data, stock market shocks, risk management, ) play an increasingly important role.
This book sets out to bridge the gap between the existing theory and practical applications both from a probabilistic as well as from a statistical point of /5(2). contributions to transform theory preferred to use the deflnition that is now familiar as the Laplace transform f„(s) = Z 1 0 e¡stf(t)dt: (6) Another researcher who made signiflcant contributions to the theory of Laplace transforms was Widder and in his book [] he gives an exposition of the theory of the Laplace-Stieltjes transform f File Size: 1MB.
The concept of Riemann-Stieltjes integral ∫ a b f (t) d u (t), where f is called the integrand, u is called the integrator, plays an important role in mathematics, for instance in the definition of complex integral, the representation of bounded linear functionals on the Banach space of all continuous functions on an interval [a, b], in the spectral representation of Cited by: 1.
Thanks for contributing an answer to Mathematics Stack Exchange. Please be sure to answer the question. Provide details and share your research.
But avoid Asking for help, clarification, or responding to other answers. Making statements based on opinion; back them up with references or personal experience. Use MathJax to format equations. Replacing in (a2) by and generalizing sums to integrals, a Laplace–Stieltjes transform (a3) appears, and Abelian theorems derive properties of the image (under the transform) from properties of the original and Tauberian theorems do the reverse.
Laplace transform[lə′pläs ′trans‚fȯrm] (mathematics) For a function ƒ(x) its Laplace transform is the function F (y) defined as the integral over x from 0 to ∞ of the function e -yxƒ(x). Laplace Transform a transformation that converts the function f(t) of a real variable t (0.
Laplace Theorem the simplest limit theorem of the theory of probability, related to the distribution of the deviations of the frequency of occurrence of an event from its probability in independent trials. The theorem was proved in a general form by P.
Laplace in his book Théorie analytique des probabilités (). A particular case of the Laplace. Rolle, J.D., "Characterization of the Compound Normal Model Thanks to the Laplace-Stieltjes Transform of the Mixing Distributions," PapersEcole des Hautes Etudes Commerciales, Universite de Geneve.
Handle: RePEc:fth:ehecge using the quaternionic Laplace-Stieltjes transform. This class will include functions that are slice regular on the S-spectrum of T but not necessarily at infinity.
Moreover, we establish the relation of f(T) with the quaternionic functional calculus and we study the problem of finding the inverse of f(T). AMS Classification: 47A10, 47A An inversion technique for the Laplace transform with applications. Bell System Tech. J – and Jagerman, D. An inversion technique for the Laplace transform.
Bell System Tech. J –), uses the Post-Widder formula, the Poisson summation formula, and the Stehfest (Stehfest, H. Algorithm Cited by: The book description for "Laplace Transform (PMS-6)" is currently unavailable.
eISBN: Subjects: Mathematics × Close Overlay later* that if a function is absolutely monotonic on the negative real axis then it can be represented there by a Laplace-Stieltjes integral with non-decreasing determining function, and conversely.
where Yn-obne-"'18 is the Laplace-Stieltjes transform of p\ The discussion following Theorem 2 shows that even when f(s) possesses zeros in the half-plane cr^O it may be possible to obtain for [f(s)]_1 a Laplace-Stieltjes expansion absolutely convergent in the.
Pierre-Simon, marquis de Laplace (/ l ə ˈ p l ɑː s /; French: [pjɛʁ simɔ̃ laplas]; 23 March – 5 March ) was a French mathematician and astronomer whose work was pivotal to the development of mathematical astronomy and summarized and extended the work of his predecessors in his five-volume Mécanique Céleste (Celestial Mechanics) (–).
An advantage of the approach is that it does not require inversion of the Laplace-Stieltjes transform. AB - In this article we give a new derivation for the waiting time distributions in an M=M=c queue with multiple priorities and a common Author: Lars A.
van Vianen, Adriana F. Gabor, Jan-Kees van Ommeren.But this book can serve as a reference on certain topics in queueing theory and its applications (like matrix-geometric models and queueing networks).
As a textbook for a course, the book must be compared against the many excellent books that treat probability, stochastic processes, and queueing theory in separate, more compact packages."The book presents an introductory and at the same time rather comprehensive treatment of semi-Markov processes and their applications to reliability theory.
It also provides some general background (like measure theory, Markov processes and Laplace transform), which makes it accessible to a broader audience. Price: \$
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# Implicit function theorem: The result about equivalence of partial derivatives
I am trying to understand how one obtains the result of the Implicit Function Theorem which involves the equivalence of the derivatives as stated in the related Wikipedia page (https://en.wikipedia.org/wiki/Implicit_function_theorem):
Here, $$f$$ is a continuous differentiable function $$f: \mathbb{R}^{n+m} \to \mathbb{R}^{m}$$. At a point $$(a,b)$$, we have $$f(a,b) = 0 \in \mathbb{R}^{m}$$. Then in a neighborhood $$U \in \mathbb{R}^{n}$$ around $$a$$, we have a $$C^1$$ function $$g: \mathbb{R}^{n} \to \mathbb{R}^{m}$$ such that $$f(x,g(x))=0$$ and $$g(a) = b$$ in this neighborhood. The above equation of derivatives holds in this neighborhood as well.
I tried to replicate the equation above by applying the chain rule straightforwardly to $$f(x,g(x))$$. Considering the total derivative of a single component $$f_i$$ of $$f$$ with respect to $$x_j$$ in $$U$$, we should have:
$$\nabla_{x_j} f_i = \sum_{t=1}^{m}\dfrac{\partial f_t}{\partial g_t}(x,g(x))\dfrac{\partial g_t}{\partial x_j}(x) + \dfrac{\partial f_i}{\partial x_j}(x,g(x))$$
This is simply the sum of all $$f_i$$'s components' derivatives with respect to $$x_j$$. Generalizing the above to all $$f_i$$ $$(1 \leq i \leq m)$$:
$$\left[\nabla_{x_j} f_1, \dots, \nabla_{x_j} f_m\right]^T_{m \times 1} = [J_{f,y}(x,g(x))]_{m \times m}\left[\dfrac{\partial g_1}{\partial x_j}(x), \dots, \dfrac{\partial g_m}{\partial x_j}(x)\right]^T_{m \times 1} + \left[\dfrac{\partial f_1}{\partial x_j}(x,g(x)), \dots, \dfrac{\partial f_m}{\partial x_j}(x,g(x))\right]^T_{m \times 1}$$
Here $$J_{f,y}$$ is the Jacobian of $$f$$ with respect to all $$g_t$$ components. Now, rearrenging I obtain:
$$\left[\dfrac{\partial g_1}{\partial x_j}(x), \dots, \dfrac{\partial g_m}{\partial x_j}(x)\right]^T_{m \times 1} = [J_{f,y}(x,g(x))]_{m \times m}^{-1}\left[\nabla_{x_j} f_1 - \dfrac{\partial f_1}{\partial x_j}(x,g(x)), \dots, \nabla_{x_j} f_m - \dfrac{\partial f_m}{\partial x_j}(x,g(x))\right]^T_{m \times 1}$$
This is not quite the result shown on the Wikipedia page, as I have a subtraction of the partial derivatives with respect to $$x_j$$ from the total derivatives. What am I missing here?
The only thing that you're missing is that all the total derivatives are zero, since $$f(x,g(x))$$ is constant (that's how $$g(x)$$ is defined to begin with). So in your last line, you have zero minus the partial derivatives, where you can factor out the minus sign to get the formula from Wikipedia.
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Trig problems on the unit circle
3 videos
2 skills
Introduction to the unit circle
VIDEO 9:04 minutes
Extending SOH CAH TOA so that we can define trig functions for a broader class of angles
Unit circle manipulative
VIDEO 4:12 minutes
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# How to show $\sum_{k=0}^{n}\binom{n+k}{k}\frac{1}{2^k}=2^{n}$ [duplicate]
How to does one show $$\sum_{k=0}^{n}\binom{n+k}{k}\frac{1}{2^k}=2^{n}$$ I tried using the Snake oil technique but I guess I am applying it incorrectly. With the snake oil technique we have $$F(x)= \sum_{n=0}^{\infty}\left\{\sum_{k=0}^{n}\binom{n+k}{k}\frac{1}{2^k}\right\}x^{n}$$ I think I have to interchage the summation and do something. But I am not quite comfortable in interchanging the summation. Like after interchaging the summation will $$F(x)=\sum_{k=0}^{n}\sum_{n=0}^{\infty}\binom{n+k}{k}\frac{1}{2^k}x^{n}$$ Even if I continue with this I am unable to get the correct answer.
• How doesn one prove this using the Snake oil technique?
• A combinatorial proof is also welcome.
## marked as duplicate by Mike Earnest, Eevee Trainer, Parcly Taxel, jgon, Community♦Mar 17 at 3:57
• Induction will probably make this easy! – Zestylemonzi Jul 29 '16 at 11:25
• @Zestylemonzi I am looking for a solution using the "snake oiling" technique :) thanks. As for solution, I do have some other methods which avoid induction – crskhr Jul 29 '16 at 11:26
• I'm convinced that the sum can somehow be interpreted as a (strange) counting of the number of downward paths from the top of the Pascal triangle to the $n$'th row. I just can't quite see it. – Arthur Jul 29 '16 at 11:33
• You didn’t reverse the order of summation correctly. As $n$ ranges over the non-negative integers, so do the possible values of $k$. Thus, $\sum_{n\ge 0}\sum_{k=0}^n$ turns into $\sum_{k\ge 0}\sum_{n\ge k}$. You’re summing over all pairs $\langle n,k\rangle$ such that $0\le k\le n$. – Brian M. Scott Jul 29 '16 at 12:34
• I don't know whether this will help, but the problem is equivalent to showing that the coefficient of $x^n$ in $$(1-x)^{-n-1}\,(1-2x)^{-1}$$ is $2^{2n}$. Some complex analyst may be able to solve this using contour integration. – Batominovski Jul 29 '16 at 12:56
Let $S_n:=\sum\limits_{k=0}^n\,\binom{n+k}{k}\,\frac{1}{2^k}$ for every $n=0,1,2,\ldots$. Then, $$S_{n+1}=\sum_{k=0}^{n+1}\,\binom{(n+1)+k}{k}\,\frac{1}{2^k}=\sum_{k=0}^{n+1}\,\Biggl(\binom{n+k}{k}+\binom{n+k}{k-1}\Biggr)\,\frac{1}{2^k}\,.$$ Hence, $$S_{n+1}=\left(S_n+\binom{2n+1}{n+1}\frac{1}{2^{n+1}}\right)+\sum_{k=0}^n\,\binom{(n+1)+k}{k}\,\frac{1}{2^{k+1}}\,.$$ That is, $$S_{n+1}=S_n+\frac{S_{n+1}}{2}+\frac{1}{2^{n+2}}\,\Biggl(2\,\binom{2n+1}{n+1}-\binom{2n+2}{n+1}\Biggr)\,.$$ As $$\binom{2n+2}{n+1}=\frac{2n+2}{n+1}\,\binom{2n+1}{n}=2\,\binom{2n+1}{n+1}\,,$$ we deduce that $S_{n+1}=S_n+\frac{S_{n+1}}{2}$, or $$S_{n+1}=2\,S_{n}$$ for all $n=0,1,2,\ldots$. Because $S_0=1$, the claim follows.
Combinatorial Argument
The number of binary strings of length $2n+1$ with at least $n+1$ ones is clearly $2^{2n}$. For $k=0,1,2,\ldots,n$, the number of such strings whose $(n+1)$-st one is at the $(n+k+1)$-st position is $\binom{n+k}{k}\,2^{n-k}$. The claim is now evident.
• This is definitely a non-snake oil method. – Zack Ni Jul 29 '16 at 12:28
• I'm sorry. Was I required to use a particular method? The OP made an attempt with that method but he never requested a specific method as an answer. – Batominovski Jul 29 '16 at 12:29
• the op said "I tried using the Snake oil technique but I guess I am applying it incorrectly." – Zack Ni Jul 29 '16 at 12:31
• He didn't say "I want a solution using this method." Do you know how to read? – Batominovski Jul 29 '16 at 12:31
• Then, he's better add that in his question. In addition, he should explain why he wants this method as a solution. If it is an exercise for this method, then it makes sense. But if it is some random request without basis, I see no reason why it's certain that the method will work. – Batominovski Jul 29 '16 at 12:33
Here is a variation based upon the coefficient of operator $[x^k]$ to denote the coefficient of $x^k$ of a series. We can write e.g. \begin{align*} [x^k](1+x)^n=\binom{n}{k} \end{align*}
We obtain \begin{align*} \sum_{k=0}^n\binom{n+k}{k}\frac{1}{2^k}&=\sum_{k=0}^n[x^k](1+x)^{n+k}\frac{1}{2^k}\tag{1}\\ &=[x^0](1+x)^n\sum_{k=0}^n\left(\frac{1+x}{2x}\right)^k\tag{2}\\ &=[x^0](1+x)^n\frac{1-\left(\frac{1+x}{2x}\right)^{n+1}}{1-\frac{1+x}{2x}}\tag{3}\\ &=[x^0](1+x)^n\frac{1}{(2x)^n}\frac{(2x)^{n+1}-(1+x)^{n+1}}{x-1}\tag{4}\\ &=\frac{1}{2^n}[x^n]\frac{(1+x)^{2n+1}}{1-x}\tag{5}\\ &=\frac{1}{2^n}[x^n]\sum_{k=0}^{2n+1}\binom{2n+1}{k}x^k\frac{1}{1-x}\tag{6}\\ &=\frac{1}{2^n}\sum_{k=0}^{n}\binom{2n+1}{k}[x^{n-k}]\frac{1}{1-x}\tag{7}\\ &=\frac{1}{2^n}\sum_{k=0}^{n}\binom{2n+1}{k}\tag{8}\\ &=\frac{1}{2^n}\cdot\frac{1}{2}2^{2n+1}\tag{9}\\ &=2^n \end{align*} and the claim follows.
Comment:
• In (1) we apply the coefficient of operator.
• In (2) we use the linearity of the coefficient of operator and the rule $$[x^{p+q}]A(x)=[x^p]x^{-q}A(x)$$
• In (3) we use the finite geometric series formula.
• In (4) we do some simplifications.
• In (5) we use again the rule stated in comment (2) and note that the term $(2x)^{n+1}$ can be ignored, since it does not contribute to the coefficient of $x^n$.
• In (6) we apply the binomial sum formula.
• In (7) we note that only index up to $k=n$ contributes to the coefficient of $x^n$.
• In (8) we recall the geometric series is $$\frac{1}{1-x}=1+x+x^2+\cdots$$ so that the contribution to the coefficient is always $1$.
• In (9) we use the symmetry of the binomial sum formula.
• what is called this kind of methods please, and where i can get more about it ? – Hamza Jul 30 '16 at 1:39
• @Hamza: This method is based upon G.P. Egorychev's classic. The method is sometimes denoted according to the operator coefficient of method or coefficient extractor method. This answer or this one provides some more aspects. – Markus Scheuer Jul 30 '16 at 5:54
• (+1) I used a generating function approach, but I like this approach, too! – robjohn Sep 18 '16 at 2:21
• @robjohn: Thanks for your nice comment. Besides your nice answer (+1) I appreciate your high skills in manipulating expressions involving binomials with elementary means. It's a valuable source to improve my own skills! :-) – Markus Scheuer Sep 18 '16 at 7:08
• I missed this one when it first appeared, Good work! (+1) (verified). – Marko Riedel Nov 5 '16 at 21:04
$$\imp\quad \begin{array}{|c|}\hline\mbox{}\\ \ds{\quad\mrm{f}_{n}\pars{x} = 1 - {2n + 1 \over n + 1}{2n \choose n}x^{n + 1} + n\int_{0}^{x}\mrm{f}_{n}\pars{y}\,\dd y + x\mrm{f}_{n}\pars{x} \quad} \\ \mbox{}\\ \hline \end{array}$$
Then, \begin{align} \mrm{f}_{n}'\pars{x} & = -\pars{2n + 1}{2n \choose n}x^{n} + n\mrm{f}_{n}\pars{x} + \mrm{f}_{n}\pars{x} + x\mrm{f}_{n}'\pars{x} \,,\quad\mrm{f}_{n}\pars{0} = 1 \end{align}
$$\mrm{f}_{n}'\pars{x} - {n + 1 \over 1 - x}\,\mrm{f}_{n}\pars{x} = -\pars{2n + 1}{2n \choose n}{x^{n} \over 1 - x}$$
$$\totald{\bracks{\pars{1 - x}^{n + 1}\mrm{f}_{n}\pars{x}}}{x} = -\pars{2n + 1}{2n \choose n}x^{n}\pars{1 - x}^{n}$$
$$2^{-n - 1}\,\,\mrm{f}_{n}\pars{\half} - 1 = -\pars{2n + 1}{2n \choose n}\int_{0}^{1/2}x^{n}\pars{1 - x}^{n}\,\dd x$$
\begin{align} \color{#f00}{\sum_{k = 0}^{n}{n + k \choose k}x^{k}} & = \mrm{f}_{n}\pars{\half} = 2^{n + 1}\ -\ \overbrace{% 2^{n + 1}\pars{2n + 1}{2n \choose n} \int_{0}^{1/2}\bracks{{1 \over 4} - \pars{x - \half}^{2}}^{n}\,\dd x} ^{\ds{2^{n}}} \\[5mm] & = \color{#f00}{2^{n}} \end{align}
Note that $$\int_{0}^{1/2}\bracks{{1 \over 4} - \pars{x - \half}^{2}}^{n}\,\dd x = \half\,\ \overbrace{{\Gamma\pars{n + 1}\Gamma\pars{n + 1} \over \Gamma\pars{2n + 2}}} ^{\ds{\mrm{B}\pars{n + 1,n + 1}}}\ =\ {1 \over 2\pars{2n + 1}{2n \choose n}}$$ $\ds{\Gamma}$: Gamma Function. B: Beta Function.
• Long way to go. Congratulations on getting there. – marty cohen Jul 31 '16 at 1:47
• @martycohen Thanks for your remark. I tried a few approaches until I remembered I evaluated something similar ( math.stackexchange.com/a/1837721/85343 ) and the only way was to find a differential equation. $\texttt{@Marko Riedel}$ answer already mentioned that it's somehow different because there isn't an upper bound. – Felix Marin Jul 31 '16 at 2:11
Suppose we seek to verify that
$$\sum_{k=0}^n {n+k\choose k} \frac{1}{2^k} = 2^n.$$
In the following we make an effort to use a different set of integrals from the answer by @MarkusScheuer, for variety's sake, even if this is not the simplest answer.
The difficulty here lies in the fact that the binomial coefficients on the LHS do not have an upper bound for the sum wired into them. We use an Iverson bracket to get around this:
$$[[0\le k\le n]] = \frac{1}{2\pi i} \int_{|w|=\gamma} \frac{w^k}{w^{n+1}} \frac{1}{1-w} \; dw.$$
Introduce furthermore
$${n+k\choose k} = \frac{1}{2\pi i} \int_{|z|=\epsilon} \frac{1}{z^{n+1}} \frac{1}{(1-z)^{k+1}} \; dz.$$
With the Iverson bracket in place we can let the sum range to infinity, getting
$$\frac{1}{2\pi i} \int_{|w|=\gamma} \frac{1}{w^{n+1}} \frac{1}{1-w} \frac{1}{2\pi i} \int_{|z|=\epsilon} \frac{1}{z^{n+1}} \frac{1}{1-z} \sum_{k\ge 0} \frac{w^k}{(1-z)^k} \frac{1}{2^k} \; dz\; dw.$$
This converges when $|w| < |2(1-z)|.$ Simplifying we have
$$\frac{1}{2\pi i} \int_{|w|=\gamma} \frac{1}{w^{n+1}} \frac{1}{1-w} \frac{1}{2\pi i} \int_{|z|=\epsilon} \frac{1}{z^{n+1}} \frac{1}{1-z} \frac{1}{1-w/(1-z)/2} \; dz\; dw \\ = \frac{1}{2\pi i} \int_{|w|=\gamma} \frac{1}{w^{n+1}} \frac{1}{1-w} \frac{1}{2\pi i} \int_{|z|=\epsilon} \frac{1}{z^{n+1}} \frac{1}{1-z-w/2} \; dz\; dw.$$
The pole at $z=1-w/2$ is outside the contour due to the requirements on convergence, so we may use the negative of the residue there, getting
$$\frac{1}{2\pi i} \int_{|w|=\gamma} \frac{1}{w^{n+1}} \frac{1}{1-w} \frac{1}{(1-w/2)^{n+1}} \; dw.$$
This could have been obtained by inspection, bypassing the Iverson bracket. Now put $w (1-w/2) = v$ so that $w = 1-\sqrt{1-2v}$ (this branch maps $w=0$ to $v=0$) to get
$$\frac{1}{2\pi i} \int_{|v|=\gamma'} \frac{1}{v^{n+1}} \frac{1}{\sqrt{1-2v}} \frac{1}{\sqrt{1-2v}} \; dv \\ = \frac{1}{2\pi i} \int_{|v|=\gamma'} \frac{1}{v^{n+1}} \frac{1}{1-2v} \; dv = 2^n.$$
This is the claim.
Observe that
$$\mathrm{Res}_{z=\infty} \frac{1}{z^{n+1}} \frac{1}{1-z-w/2} = - \mathrm{Res}_{z=0} \frac{1}{z^2} z^{n+1} \frac{1}{1-w/2-1/z} \\ = - \mathrm{Res}_{z=0} z^{n} \frac{1}{z(1-w/2)-1} = 0.$$
This was an interesting exercise showing how the choice of contour for convergence influences the computation. The branch of $\sqrt{1-2v}$ that was used has the branch cut on $(1/2, \infty).$
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# Estimation of exponentially small overlap integrals in the framework of phase space distributions
## Alexey V. Sergeev and Bilha Segev
### Department of Chemistry, Ben-Gurion University of the Negev POB 653, Beer-Sheva 84105, Israel
The integral $\int d\vec{q} \psi_0(\vec{q}) \psi_1(\vec{q})$ where $\psi_0(\vec{q})$ is the ground state eigenfunction of a Hamiltonian $H_0$, and $\psi_1(\vec{q})$ is the eigenfunction of the highly excited state of a Hamiltonian $H_1$ corresponding to some energy $E$ is estimated by replacing the eigenfunctions by their Wigner functions, and by analyzing the phase space integral. In the quasiclassical limit only a vicinity of one point where $H_1(\vec{q},\vec{p})=E$ and where the Wigner function $\rho_1(\vec{q},\vec{p})$ is maximal contributes to the integral [1, 2]. Results are illustrated for Morse and Poescl - Teller oscillators.
1. Segev B. and Heller E. J. Phase-space derivation of propensity rules for energy transfer processes between Born-Oppenheimer surfaces. J. Chem. Phys. 112, 4004 (2000).
2. Sergeev A. V. and Segev B. Most probable path in phase space for a radiationless transition in a molecule. J. Phys. A: Math. Gen. (2002).
Abstract (TeX file) and poster
Back to Presentations at conferences.
Designed by A. Sergeev.
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Question
# Write the smallest four-digit number which is a perfect square.A. $1000$B. $1016$C. $1024$D. $1036$
Hint: Find the square root of the smallest four-digit number. Use that information to get the answer.
We need to find the smallest four-digit number which is a perfect square. We know that 1000 is the smallest four-digit number. so, we’ll find the square root of that as follows:
Now, from the above process what we have understood is, ${(31)^2} < 1000$. It’s obvious that ${(32)^2} > 1000$. Let cross check once.$32 \times 32 = 1024$. So, the smallest four-digit number is 1024.
Hence the correct option is C.
Note: For this kind of problem, it’s better to start the solution with the smallest/largest n-digit number then we can move further in the computation according to the given condition.
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# getDGM
Convert gain and phase variation into disk-based gain variation
## Syntax
``DGM = getDGM(GM,PM,'tight')``
``DGM = getDGM(GM,PM,'balanced')``
``[DGM,DPM] = getDGM(___)``
## Description
In disk margin analysis, gain and phase variations are modeled as a factor F(s) multiplying the open loop response L(s). This factor takes values in a disk D centered on the real axis with real-axis intercepts `gmin` and `gmax`. The disk margin determines the largest disk size `[gmin,gmax]` for which the feedback loop remains stable. This provides a gain margin of at least `DGM = [gmin,gmax]` and also some phase margin `DPM` determined by the disk geometry.
Conversely, `getDGM` takes desired gain and phase margins `GM` and `PM` and computes the smallest disk D that delivers both. This disk is characterized by its real-axis intercepts `gmin`, `gmax` and the corresponding disk-based gain margin `DGM = [gmin,gmax]` and phase margin `DPM` meet or exceed `GM` and `PM`.
example
````DGM = getDGM(GM,PM,'tight')` computes the smallest disk that captures the target gain and phase variations specified by `GM` and `PM`. If `GM` and `PM` are scalars, then the disk captures gain that can increase or decrease by a factor of `GM`, and phase that can increase or decrease by `PM`. If `GM` and `PM` are vectors of the form `[glo,ghi]` and `[pmin,pmax]` then the disk captures relative gain and phase variations in these ranges.If either `GM` or `PM` is `[]`, that removes the corresponding constraint on the disk size.The output is of the form `DGM` = `[gmin,gmax]`, and describes a disk that represents absolute gain variations within that range. For instance, `DGM = [0.8,1.8]` models gain that can vary from 0.8 times the nominal value to 1.8 times the nominal value, and phase variations determined by the disk geometry. This disk might have non-zero skew (see Algorithms). Use `DGM` to create a `umargin` block that models these gain and phase variations.```
example
````DGM = getDGM(GM,PM,'balanced')` computes the smallest disk that represents a symmetric gain variation, that is, `DGM` = `[gmin,gmax]` where `gmin` = `1/gmax`. This disk has zero skew (see Algorithms). ```
example
````[DGM,DPM] = getDGM(___)` also returns the disk-based phase range `DPM` modeled by the disk that `DGM` describes. You can use this output argument with any of the previous syntaxes.```
## Examples
collapse all
Find the smallest disk-based gain margin that represents relative a gain variation of ±6 dB relative to the nominal value and phase variation of ±40°. Convert the gain variation into absolute units.
`GM = db2mag(6)`
```GM = 1.9953 ```
```PM = 40; DGM = getDGM(GM,PM,'tight')```
```DGM = 1×2 0.4299 1.9953 ```
`DGM` describes a disk that models both gain and phase variations. The values in `DGM` represent the range of gain variation in the absence of phase variation. Note that the `DGM` range is slightly larger than the specified `[1/GM,GM]` range as the phase margin requirement is more stringent and determines the disk size. Visualize the full range of gain and phase variations represented by `DGM`.
`diskmarginplot(DGM)`
The '`tight`' constraint computes the smallest disk that delivers both target gain and phase variations, which does not necessarily represent a symmetric gain range. In this case, the disk represents gain that can decrease somewhat more than it can increase. Examine the disk of uncertainty defined by this particular `DGM`.
`diskmarginplot(DGM,'disk')`
To enforce symmetric gain variation, use the `'balanced'` option.
Determine the disk-based gain margin that delivers symmetric gain variation of $±$5 dB and phase variation of $±$30 degrees.
```GM = db2mag(5); PM = 30; DGM = getDGM(GM,PM,'balanced')```
```DGM = 1×2 0.5623 1.7783 ```
The '`balanced`' constraint models a disk of uncertainty that is symmetric around the nominal value. The function returns a symmetric disk-based gain margin `DGM` = `[gmin,gmax]`, with `gmin=1/gmax`.
`diskmarginplot(DGM)`
In this case, `DPM` slightly exceeds the target phase variation and `DGM` is equal to the target gain variation.
Determine the disk-based gain margin corresponding to gain variations between 90% and 160% of the nominal value, and phase variations from -15 to +15 degrees.
```gainRange = [0.9,1.6]; phaseRange = [-15,15]; DGMt = getDGM(gainRange,phaseRange,'tight')```
```DGMt = 1×2 0.8603 1.6000 ```
The '`tight`' constraint models the smallest disk that delivers target gain and phase variations. This disk is modeled with gain variation that skews toward gain increase.
Alternatively, you can use the '`balanced`' option to constrain the disk-based gain margin to a symmetrical range of the form `gmin = 1/gmax`. This means that the gain can increase or decrease by equal amount.
`DGMb = getDGM(gainRange,phaseRange,'balanced')`
```DGMb = 1×2 0.6250 1.6000 ```
Visualize the range of simultaneous gain and phase variations corresponding to both gain ranges.
`diskmarginplot([DGMt;DGMb]) `
The balanced range `DGMb` models a larger, symmetric gain range (`gmin = 1/gmax`) and larger phase variations than the ones you specify. If you are confident that gain varies more in one direction than the other in your system, then this balanced model might be overly conservative.
Determine the balanced disk-based gain margin ranges that delivers gain variations of ±4 dB, ±6 dB, and ±12 dB and phase variation of ±30°. You can get all the disk-based gain ranges at once by stacking the desired target ranges into a column vector.
```GM = db2mag([4;6;12]); PM = 30; DGM = getDGM(GM,PM,'balanced')```
```DGM = 3×2 0.5774 1.7321 0.5012 1.9953 0.2512 3.9811 ```
`diskmarginplot(DGM)`
Each row in the matrix `DGM` gives the disk-based gain variation for the corresponding entry in `GM`. For instance, the smallest balanced (symmetric) disk that captures gain variation of ±4 dB and phase variation of ±30° is specified by `DGM(1,:) = [0.58 1.73]`.
This disk represents somewhat more than the target ±4 dB, in order to capture the full target gain variation of ±30°. For the targets ±6 dB and ±12 dB, the disk meets the target gain variation exactly, but the corresponding disks describe larger phase variations.
## Input Arguments
collapse all
Target range of relative gain variation, specified as a scalar, vector, or two-column matrix.
• If `GM` is a scalar, then the disk captures gain that can increase or decrease by a factor of `GM`. For instance, if `GM` = 2, then the output `DGM` represents gain that can decrease or increase by a factor of 2.
• If `GM` is a vectors of the form `[glo,ghi]` then the disk captures relative gain variations in this range. For instance, if `GM`` = [0.8,1.9]`, then `DGM` represents gain that can vary between 0.8 and 1.9 times the nominal value.
• If `GM` `[]`, then `getDGM` returns a disk that captures the phase variation specified by `PM`, and the corresponding gain variation determined by the disk model.
#### Multiple Ranges at Once
To get `DGM` corresponding to multiple target gain ranges at once, specify `GM` as a column vector `[GM1;...;GMn]` or a matrix `[glo1,ghi1;...;gloN,ghiN]`.
Target phase variation, specified as a scalar, vector, or two-column matrix.
• If `PM` is a scalar, then the disk captures phase that can increase or decrease by `PM`. For instance, if `PM` = 20, then the output `DGM` represents phase that can vary by ±20°.
• If `PM` is a vector of the form `[pmin,pmax]` with `pmin < 0` and `pmax > 0`, then the disk captures phase that can vary by ±`min(abs(pmin),pmax)`. For instance, if `[pmin,pmax]` = `[-20,40]` then the disk captures phase variation in the range `[-40,40]`.
• If `PM` `[]`, then `getDGM` returns a disk that captures the relative gain variation specified by `GM`, and the corresponding phase variation determined by the disk model.
#### Multiple Ranges at Once
To get `DGM` corresponding to multiple target phase ranges at once, specify `PM` as a column vector `[PM1;...;PMn]` or a matrix of the form `[-pm1,pm1;...;-pmN,pmN]`.
## Output Arguments
collapse all
Modeled range of relative gain variation, returned as a two-element vector of the form `[gmin,gmax]`, where `gmin` < 1 and `gmax` > 1. For instance, `DGM = [0.8 1.5]` represents a gain that can vary between 80% and 150% of its nominal value (that is, change by a factor between 0.8 and 1.5). `gmin` can be negative, defining a range of relative gain variation that includes a change in sign. When you use the `'balanced'` option, the gain change is symmetric, that is, the gain can increase or decrease by the same amount (`gmin` = `1/gmax`).
The range `[gmin,gmax]` describes a disk of gain and phase uncertainty where the gain can vary by `[gmin,gmax]` and the phase can vary by an amount determined by the disk geometry. For instance, the following plot shows a disk characterized by `DGM = [0.5,2]` (For more information about the disk-based uncertainty model, see Algorithms). The corresponding phase variation (returned in `DPM`) is ±30°.
In general, `DGM` or the corresponding `DPM` might capture larger ranges of variation than those you specify with the inputs `GM` and `PM`. The disk always captures at least the specified variations.
If `GM` is a column vector or matrix representing multiple target ranges of gain variation, `DGM` is a two-column matrix of the form `[gmin1,gmax1; ...;gminN,gmaxN]`, where each row is a corresponding disk-based gain range.
Disk-based phase margin, returned as a two-element vector of the form `[-pm,pm]`. The amount of phase variation is determined by the geometry of the disk described by `DGM` (see Algorithms).
If `PM` is a column vector or matrix representing multiple target ranges of phase variation, `DPM` is a two-column matrix of the form `[-pm1,pm1; ...;-pmN,pmN]`, where each row is a corresponding disk-based gain range.
## Algorithms
`umargin` and `diskmargin` model gain and phase variations in an individual feedback channel as a frequency-dependent multiplicative factor F(s) multiplying the nominal open-loop response L(s), such that the perturbed response is L(s)F(s). The factor F(s) is parameterized by:
`$F\left(s\right)=\frac{1+\alpha \left[\left(1-\sigma \right)/2\right]\delta \left(s\right)}{1-\alpha \left[\left(1+\sigma \right)/2\right]\delta \left(s\right)}.$`
In this model,
• δ(s) is a gain-bounded dynamic uncertainty, normalized so that it always varies within the unit disk (||δ|| < 1).
• ɑ sets the amount of gain and phase variation modeled by F. For fixed σ, the parameter ɑ controls the size of the disk. For ɑ = 0, the multiplicative factor is 1, corresponding to the nominal L.
• σ, called the skew, biases the modeled uncertainty toward gain increase or gain decrease.
The factor F takes values in a disk centered on the real axis and containing the nominal value F = 1. The disk is characterized by its intercept `DGM = [gmin,gmax]` with the real axis. `gmin` < 1 and `gmin` > 1 are the minimum and maximum relative changes in gain modeled by F, at nominal phase. The phase uncertainty modeled by F is the range `DPM = [-pm,pm]` of phase values at the nominal gain (|F| = 1). For instance, in the following plot, the right side shows the disk F that intersects the real axis in the interval [0.71,1.4]. The left side shows that this disk models a gain variation of ±3 dB and a phase variation of ±19°.
```DGM = [0.71,1.4] F = umargin('F',DGM) plot(F)```
`getDGM` converts the target gain and phase variations that you want to model into the disk-based gain-variation range `DGM`. This range fully characterizes the disk F. The corresponding phase range `DPM` is thus determined by `DGM` and the disk model.
For further details about the uncertainty model for gain and phase variations, see Stability Analysis Using Disk Margins.
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# Null hypothesis and $\chi^2$ test
Which null hypothesis can be tested by a $\chi^2$ test?
could you please write the description of the relevant null hypothesis or at least give an example.
Thank you.
-
From the help of the chisq.test() function in R : Then Pearson's chi-squared test is performed of the null hypothesis that the joint distribution of the cell counts in a 2-dimensional contingency table is the product of the row and column marginals. – Stéphane Laurent Jan 10 '13 at 12:22
Which chi-square test? For association in contingency tables? For goodness of fit? To test distributional hypotheses? To compare variance estimates? To assess significance of added variables in nested models using Maximum Likelihood? Others? – whuber Jan 10 '13 at 13:47
There are lots of things to test with chi-square. Perhaps the most common, in my experience anyway, is whether two categorical variables are associated. But also whether one variable fits a certain distribution. It can also come up in testing of various models.
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The question is about the mathematical statement of $H_0$, I think (the mathematical definition of "association" here) – Stéphane Laurent Jan 10 '13 at 12:23
It is used to check if a given sample is of a certain law (goodness-of-fit test), and it's also used to check whether to samples are independent. Here's an example using R
x=rnorm(100)
y=rnorm(100)
chisq.test(x,y) # test for goodness of fit
HTH
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As it is stated in charlesdennishale.com/books/eets_ap/… Chi-square can be used to test whether a. Observed nominal data conforms to some theoretical or expected distribution b. Two variables within a sample are related c. Two or more samples, drawn from different populations, are homogenous on some characteristic of interest – O_Devinyak Jan 10 '13 at 10:28
David: and so, what is $H_0$ ? – Stéphane Laurent Jan 10 '13 at 12:24
$H_0$ is : $x$ and $y$ come from the same $mother$ distribution , you do not even have to specify the distribution is normal or anything. You can view it as a distance between the empirical cumulative density functions of the two input samples. Is it clear? – DKK Jan 10 '13 at 12:38
This is a mis-application of chisq.test and its output is nonsensical. According to its manual page, "Otherwise, x and y must be vectors or factors of the same length; cases with missing values are removed, the objects are coerced to factors, and the contingency table is computed from these." In short, these data are supposed to represent a small contingency table. In the application given here, it's virtually certain that all $(x,y)$ pairs will be distinct, leading to a $100$ by $100$ table with $99$ zeros in each row and each column! – whuber Mar 6 '13 at 18:54
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- Current Issue - Ahead of Print Articles - All Issues - Search - Open Access - Information for Authors - Downloads - Guideline - Regulations ㆍPaper Submission ㆍPaper Reviewing ㆍPublication and Distribution - Code of Ethics - For Authors ㆍOnline Submission ㆍMy Manuscript - For Reviewers - For Editors
Commun. Korean Math. Soc. 2007 Vol. 0, No. 0, 1—155
Subtraction algebras with additional conditions Young Bae Jun, Young Hee Kim, Kyong Ah Oh MSC numbers : 03G25, 06B10, 06D99 Commun. Korean Math. Soc. 2007 Vol. 22, No. 1, 1—7
Extended special sets in implicative semigroups Keumseong Bang, Keum Sook So MSC numbers : 20M12, 06F05, 06A06, 06A12 Commun. Korean Math. Soc. 2007 Vol. 22, No. 1, 9—14
The strong Perron integral in $\mathbb R^n$ revisited Valentin A. Skvortsov, Piotr Sworowski MSC numbers : 26A39 Commun. Korean Math. Soc. 2007 Vol. 22, No. 1, 15—18
Sufficient conditions for starlikeness and strongly-starlikeness Oh Sang Kwon MSC numbers : 30C45 Commun. Korean Math. Soc. 2007 Vol. 22, No. 1, 19—26
C-integral and Denjoy-C integral Dafang Zhao, Guoju Ye MSC numbers : 28B05, 46G10 Commun. Korean Math. Soc. 2007 Vol. 22, No. 1, 27—39
On 3-additive mappings and commutativity in certain rings Kyoo-Hong Park, Yong-Soo Jung MSC numbers : 16W20, 16U80, 16W25 Commun. Korean Math. Soc. 2007 Vol. 22, No. 1, 41—51
Monotone Iteration Scheme for a Forced Duffing Equation with Nonlocal Three-Point Conditions Ahmed Alsaedi MSC numbers : 34B10, 34B15 Commun. Korean Math. Soc. 2007 Vol. 22, No. 1, 53—64
Notes on Carleson Type Measures On bounded symmetric domain Ki Seong Choi MSC numbers : 32H25, 32E25, 30C40 Commun. Korean Math. Soc. 2007 Vol. 22, No. 1, 65—74
Fourier-Feynman Transforms for Functionals in a Generalized Fresnel Class Il Yoo, Byoung Soo Kim MSC numbers : 28C20 Commun. Korean Math. Soc. 2007 Vol. 22, No. 1, 75—90
Change of scale formulas for conditional Wiener integrals as integral transforms over Wiener paths in abstract Wiener space Dong Hyun Cho MSC numbers : 28C20 Commun. Korean Math. Soc. 2007 Vol. 22, No. 1, 91—109
1 · 2
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Expose the names in lhs to the rhs expression. This is useful when functions do not have a built-in data argument.
lhs %$% rhs ## Arguments lhs A list, environment, or a data.frame. rhs An expression where the names in lhs is available. ## Details Some functions, e.g. lm and aggregate, have a data argument, which allows the direct use of names inside the data as part of the call. This operator exposes the contents of the left-hand side object to the expression on the right to give a similar benefit, see the examples. ## See also %>%, %<>%, %T>% ## Examples iris %>% subset(Sepal.Length > mean(Sepal.Length)) %$%
cor(Sepal.Length, Sepal.Width)
#> [1] 0.3361992
data.frame(z = rnorm(100)) %\$%
ts.plot(z)
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# 📋 Tensorboard Monitoring#
## Installing Tensorboard#
Before we get started make sure you have installed Tensorboard! Make sure to run:
pip install 'mosaicml[tensorboard]'
## Logging to Tensorboard#
To log your run’s results to tensorboard, first you will need to create a TensorboardLogger object, like so:
from composer.loggers import TensorboardLogger
tb_logger = TensorboardLogger(log_dir="./my_tensorboard_logs")
log_dir is where you want the Tensorboard logs to be saved locally (on the system in which you run composer).
If you are viewing or accessing you logs locally, choose this path wisely and remember it! Also make sure to use this same value for log_dir for any future runs (so all your runs can be visualized together!)
If you will be using S3 to save your logs then the exact path you choose is not as important, as your logs will automatically be saved to a directory called tensorboard_logs inside of your bucket.
Once we have our TensorboardLogger, we just need to add it to our Trainer and then we’ll be good to go. Below is an example of training MNIST with Tensorboard Logging:
from torchvision import datasets, transforms
from composer import Trainer
from composer.models import mnist_model
from composer.loggers import TensorboardLogger
transform = transforms.Compose([transforms.ToTensor()])
# Create your Tensorboard Logger here.
tb_logger = TensorboardLogger(log_dir="./my_tensorboard_logs")
trainer = Trainer(
model=mnist_model(num_classes=10),
max_duration='5ep',
loggers=[tb_logger],
eval_interval='1ep'
)
trainer.fit()
Now, run this code and if all goes well, your loss and metric results will be logged Tensorboard log files, which will be written to “./my_tensorboard_logs”. See these instructions for viewing your results in the Tensorboard viewer.
If you saved your Tensorboard log files locally you can view them by starting a Tensorboard process and pointing it to the log directory you specified. To do this run the following at the command line:
tensorboard --logdir='./my_tensorboard_logs'
This will start a Tensorboard process, which will write a message to stdout that looks something like:
TensorBoard 2.9.1 at http://localhost:6006/ (Press CTRL+C to quit)
Open the URL in your browser to access the Tensorboard viewer, which should look something like this:
tensorboard --logdir=s3://my-bucket-name/tensorboard_logs
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## University Calculus: Early Transcendentals (3rd Edition)
$\text{csch} \theta \text{coth} \theta (\ln csch \theta)$
Since, $\dfrac{d}{dx} (csch x)=\text{-csch} x \text{coth} x$ As we are given that $y=csch \theta (1-\ln csch \theta)$ Then, on differentiating , we have: $\dfrac{dy}{d \theta}=csch \theta[\dfrac{-1}{csch \theta}( -csch \theta coth \theta)]+(1-\ln csch \theta)=(csch \theta coth \theta)[1-(1-\ln csch \theta)]=\text{csch} \theta \text{coth} \theta (\ln csch \theta)$
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## Banach Journal of Mathematical Analysis
### The Gelfand--Phillips property in closed subspaces of some operator spaces
#### Abstract
By introducing the concept of limited completely continuous operators between two arbitrary Banach spaces $X$ and $Y$, we give some properties of this concept related to some well known classes of operators and specially, related to the Gelfand-Phillips property of the space $X$ or $Y$. Then some necessary and sufficient conditions for the Gelfand--Phillips property of closed subspace $M$ of some operator spaces, with respect to limited complete continuity of some operators on $M$, so-called, evaluation operators, are verified.
#### Article information
Source
Banach J. Math. Anal., Volume 5, Number 2 (2011), 84-92.
Dates
First available in Project Euclid: 14 August 2011
https://projecteuclid.org/euclid.bjma/1313363004
Digital Object Identifier
doi:10.15352/bjma/1313363004
Mathematical Reviews number (MathSciNet)
MR2792501
Zentralblatt MATH identifier
1235.47021
#### Citation
Salimi, Manijeh; Moshtaghioun, S. Mohammad. The Gelfand--Phillips property in closed subspaces of some operator spaces. Banach J. Math. Anal. 5 (2011), no. 2, 84--92. doi:10.15352/bjma/1313363004. https://projecteuclid.org/euclid.bjma/1313363004
#### References
• P.M. Anselone, Compactness properties of sets of operators and their adjoints, Math. Z. 113 (1970), 233–236.
• J. Bourgain and J. Diestel, Limited operators and strict cosingularity, Math. Nachr. 119 (1984), 55–58.
• W.J. Davis, T. Figiel, W.B. Johnson and A. Pelczynski, Factoring weakly compact operators, J. Funct. Anal. 17 (1974), 311–327.
• A. Defant and K. Floret, Tensor Norms and Operator Ideals, North-Holland, Amsterdam, 1993.
• L. Drewnowski, On Banach spaces with the Gelfand–Phillips property, Math. Z. 193 (1986), 405–411.
• J. Diestel, Sequences and Series in Banach Spaces, Grad. Texts in Math. 92, Springer-Verlag, Berlin, 1984.
• G. Emmanuele, A dual characterization of Banach spaces not containing $\ell_1$, Bull. Pol. Acad. Sci. Math. 34 (1986), 155–160.
• G. Emmanuele, On Banach spaces with the Gelfand–Phillips property, III, J. Math. Pures Appl. 72 (1993), 327–333.
• G. Emmanuele, On the reciprocal Dunford–Pettis property in projective tensor products, Math. Proc. Cambridge Philos. Soc. 109 (1991), 161–166.
• I. Ghenciu and P. Lewis, The Dunford–Pettis property, the Gelfand–Phillips property, and L-sets, Colloq. Math. 106 (2006), 311–324.
• H. Jarchow, Locally Convex Spaces, Teubner, Stuttgart, 1981.
• R.E. Megginson, An Introduction to Banach Space Theory, Springer-Verlag, New York, 1998.
• F. Mayoral, Compact sets of compact operators in absence of $\ell_1$, Proc. Amer. Math. Soc. 129 (2001), 79–82.
• S.M. Moshtaghioun and J. Zafarani, Weak sequential convergence in the dual of operator ideals, J. Operator Theory 49 (2003), no. 1, 143–152.
• T.W. Palmer, Totally bounded sets of precompact linear operators, Proc. Amer. Math. Soc. 20 (1969), 101–106.
• E. Saksman and H.O. Tylli, Structure of subspaces of the compact operators having the Dunford–Pettis property, Math. Z. 232 (1999), 411–425.
• T. Sclumprecht, Limited sets in injective tensor products, Lecture Notes in Math. 1970, Springer-Verlag, E. Odell and H. Rosenthal (eds.), (1991), 133–158.
• A. Ülger, Subspaces and subalgebras of $K(H)$ whose duals have the Schur property, J. Operator Theory 37 (1997), 371–378.
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• # Good arguments—notes on Craft of Research chapter 7
Arguments take place in 5 parts.
1. Claim: What do you want me to believe?
2. Reasons: Why should I agree?
3. Evidence: How do you know? Can you back it up?
4. Acknowledgment and Response: But what about … ?
5. Warrant: How does that follow?
This can be modeled as a conversation with readers, where the reader prompts the writer to taking the next step on the list.
Claim ought to be supported with reasons. Reasons ought to be based on evidence. Arguments are recursive: a part of an argument is an acknowledgment of an anticipated response, and another argument addresses that response. Finally, when the distance between a claim and a reason grows large, we draw connections with something called warrants.
The logic of warrants proceeds in generalities and instances. A general circumstance predictably leads to a general consequence, and if you have an instance of the circumstance you can infer an instance of the consequence.
Arguing in real life papers is complexified from the 5 steps, because
• Claims should be supported by two or more reasons
• A writer can anticipate and address numerous responses. As I mentioned, arguments are recursive, especially in the anticipated response stage, but also each reason and warrant can necessitate a subargument.
You might embrace a claim too early, perhaps even before you have done much research, because you “know” you can prove it. But falling back on that kind of certainty will just keep you from doing your best thinking.
• # Sources—notes on Craft of Research chapters 5 and 6
## Primary, secondary, and tertiary sources
Primary sources provide you with the “raw data” or evidence you will use to develop, test, and ultimately justify your hypothesis or claim. Secondary sources are books, articles, or reports that are based on primary sources and are intended for scholarly or professional audiences. Tertiary sources are books and articles that synthesize and report on secondary sources for general readers, such as textbooks, articles in encyclopedias, and articles in mass-circulation publications.
The distinction between primary and secondary sources comes from 19th century historians, and the idea of tertiary sources came later. The boundaries can be fuzzy, and are certainly dependent on the task at hand.
I want to reason about what these distinctions look like in the alignment community, and whether or not they’re important.
The rest of chapter five is about how to use libraries and information technologies, and evaluating sources for relevance and reliability.
Chapter 6 starts off with the kind of thing you should be looking for while you read
## Look for creative agreement
• Offer additional support. You can offer new evidence to support a source’s claim.
• Confirm unsupported claims. You can prove something that a source only assumes or speculates about.
• Apply a claim more widely. You can extend a position.
## Look for creative disagreement
• Contradictions of kind. A source says something is one kind of thing, but it’s another.
• Part-whole contradictions. You can show that a source mistakes how the parts of something are related.
• Developmental or historical contradictions. You can show that a source mistakes the origin or development of a topic.
• External cause-effect contradictions. You can show that a source mistakes a causal relationship.
• Contradictions of perspective. Most contradictions don’t change a conceptual framework, but when you contradict a “standard” view of things, you urge others to think in a new way.
The rest of chapter 6 is a few more notes about what you’re looking for while reading (evidence, reasons), how to take notes, and how to stay organized while doing this.
# The alignment community
I think I see the creative agreement modes and the creative disagreement modes floating around in posts. Would it be more helpful if writers decided on one or two of these modes before sitting down to write?
Moreover, what is a primary source in the alignment community? Surely if one is writing about inner alignment, a primary source is the Risks from Learned Optimization paper. But what are Risks’ primary, secondary, tertiary sources? Does it matter?
Now look at Arbital. Arbital started off to be a tertiary source, but articles that seemed more like primary sources started appearing there. I remember distinctively thinking “what’s up with that?” it struck me as awkward for Arbital to change it’s identity like that, but I end up thinking about and citing the articles that seem more like primary sources.
There’s also the problem of stuff in the memeplex not written down is the real “primary” source while the first person who happens to write it down looks like they’re writing a primary source when in fact what they’re doing is really more like writing a secondary or even tertiary source.
• # notes (from a very jr researcher) on alignment training pipeline
Training for alignment research is one part competence (at math, cs, philosophy) and another part having an inside view / gears-level model of the actual problem. Competence can be outsourced to universities and independent study, but inside view / gears-level model of the actual problem requires community support.
A background assumption I’m working with is that training as a longtermist is not always synchronized with legible-to-academia training. It might be the case that jr researchers ought to publication-maximize for a period of time even if it’s at the expense of their training. This does not mean that training as a longtermist is always or even often orthogonal to legible-to-academia training, it can be highly synchronized, but it depends on the occasion.
It’s common to query what relative ratio should be assigned to competence building (textbooks, exercises) vs. understanding the literature (reading papers and alignment forum), but perhaps there is a third category- honing your threat model and theory of change.
I spoke with a sr researcher recently who roughly said that a threat model with a theory of change is almost sufficient for an inside view / gears-level model. I’m working from the theory that honed threat models and your theory of change are important to calculate interventions. See Alice and Bob in Rohin’s faq.
I’ve been trying by doing exercises with a group of peers weekly to hone my inside view / gears-level model of the actual problem. But the sr researcher i spoke to said mentorship trees of 1:1 time, not exercises that jrs can just do independently or in groups, is the only way it can happen. This is troublesome to me, as the bottleneck becomes mentors’ time. I’m not so much worried about the hopefully merit-based process of mentors figuring out who’s worth their time, as I am about the overall throughput. It gets worse though- what if the process is credentialist?
Take a look at the Critch quote from the top of Rohin’s faq:
I get a lot of emails from folks with strong math backgrounds (mostly, PhD students in math at top schools) who are looking to transition to working on AI alignment / AI x-risk.
Is he implicitly saying that he offloads some of the filtering work to admissions people at top schools? Presumably people from non-top schools are also emailing him, but he doesn’t mention them.
I’d like to see a claim that admissions people at top schools are trustworthy. No one has argued this to my knowledge. I think sometimes the movement falls back on status games, unless there is some intrinsic benefit to “top schools” (besides building social power/capital) that everyone is aware of. (Indeed if someone’s argument is that they identified a lever that requires a lot of social power/capital, then they can maybe put that top school on their resume to use, but if the lever is strictly high quality useful research (instead of say steering a federal government) this doesn’t seem to apply).
• Is he implicitly saying that he offloads some of the filtering work to admissions people at top schools?
I don’t think Critch’s saying that the best way to get his attention is through cold emails backed up by credentials. The whole post is about him not using that as a filter to decide who’s worth his time but that people should create good technical writing to get attention.
• Critch’s written somewhere that if you can get into UC Berkeley, he’ll automatically allow you to become his student, because getting into UC Berkeley is a good enough filter.
• Where did he say that? Given that he’s working at UC Berkeley I would expect him to treat UC Berkeley students preferentially for reasons that aren’t just about UC Berkeley being able to filter.
It’s natural that you can sign up for one of the classes he teaches at UC Berkeley by being a student of UC Berkeley.
Being enrolled into MIT might be just as hard as being enrolled into UC Berkeley but it doesn’t give you the same access to courses taught at UC Berkeley by it’s faculty.
• http://acritch.com/ai-berkeley/
If you get into one of the following programs at Berkeley:
• a PhD program in computer science, mathematics, logic, or statistics, or
• a postdoc specializing in cognitive science, cybersecurity, economics, evolutionary biology, mechanism design, neuroscience, or moral philosophy,
… then I will personally help you find an advisor who is supportive of you researching AI alignment, and introduce you to other researchers in Berkeley with related interests.
and also
While my time is fairly limited, I care a lot about this field, and you getting into Berkeley is a reasonable filter for taking time away from my own research to help you kickstart yours.
• Okay, he does speak about using Berkeley as a filter but he doesn’t speak about taking people as his student.
It seems about helping people in UC Berkeley to connect with other people in UC Berkeley.
• # Questions and Problems—thoughts on chapter 4 of Craft of Doing Research
Last time we discussed the difference between information and a question or a problem, and I suggested that the novelty-satisfied mode of information presentation isn’t as good as addressing actual questions or problems. In chapter 3 which I have not typed up thoughts about, A three step procedure is introduced
1. Topic: “I am studying …”
2. Question: ”… because I want to find out what/why/how …”
3. Significance: ”… to help my reader understand …” As we elaborate on the different kinds of problems, we will vary this framework and launch exercises from it.
Some questions raise problems, others do not. A question raises a problem if not answering it keeps us from knowing something more important than its answer.
The basic feedback loop introduced in this chapter relates practical with conceptual problems and relates research questions with research answers.
Practical problem -> motivates -> research question -> defines -> conceptual/research problem -> leads to -> research answer -> helps to solve -> practical problem (loop)
## What should we do vs. what do we know—practical vs conceptual problems
Opposite eachother in the loop are practical problems and conceptual problems. Practical problems are simply those which imply uncertainty over decisions or actions, while conceptual problems are those which only imply uncertainty over understanding. Concretely, your bike chain breaking is a practical problem because you don’t know where to get it fixed, implying that the research task of finding bike shops will reduce your uncertainty about how to fix the bike chain.
### Conditions and consequences
The structure of a problem is that it has a condition (or situation) and the (undesirable) consequences of that condition. The consequences-costs model of problems holds both for practical problems and conceptual problems, but comes in slightly different flavors. In the practical problem case, the condition and costs are immediate and observed. However, a chain of “so what?” must be walked.
Readers judge the significance of your problem not by the cost you pay but by the cost they pay if you don’t solve it… To make your problem their problem, you must frame it from their point of view, so that they see its cost to them.
One person’s cost may be another person’s condition, so when stating the cost you ought to imagine a socratic “so what?” voice, forcing you to articulate more immediate costs until the socratic voice has to really reach in order to say that it’s not a real cost.
The conceptual problem case is where intangibles play in. The condition in that case is always the simple lack of knowledge or understanding of something. The cost in that case is simple ignorance.
### Modus tollens
A helpful exercise is if you find yourself saying “we want to understand x so that we can y”, try flipping to “we can’t y if we don’t understand x”. This sort of shifts the burden on the reader to provide ways in which we can y without understanding x. You can do this iteratively: come up with _z_s which you can’t do without y, and so on.
## Pure vs. applied research
Research is pure when the significance stage of the topic-question-significance frame refers only to knowing, not to doing. Research is applied when the significance step refers to doing. Notice that the question step, even in applied research, refers to knowing or understanding.
### Connecting research to practical consequences
You might find that the significance stage is stretching a bit to relate the conceptual understanding gained from the question stage. Sometimes you can modify and add a fourth step to the topic-question-significance frame and make it into topic-conceptual question-conceptual significance-possible practical application. Splitting significance into two helps you draw reasonable, plausible applications. A claimed application is a stretch when it is not plausible. Note: the authors suggest that there is a class of conceptual papers in which you want to save practical implications entirely for the conclusion, that for a certain kind of paper practical applications do not belong in the introduction.
## AI safety
One characterisitic of AI safety that makes it difficult both to do and interface with is the chains of “so what” are often very long. The path from deconfusion research to everyone dying or not dying feels like a stretch if not done carefully, and has a lot of steps when done carefully. As I mentioned in my last post, it’s easy to get sucked into the “novel information for it’s own sake” regime at least as a reader. More practical oriented approaches are perhaps those that seek new regimes for how to even train models, and the “so what?” is answered “so we have dramatically less OODR-failures” or something. The condition-costs framework seems really beneficial for articulating alignment agendas and directions.
## Misc
• “Researchers often begin a project without a clear idea of what the problem even is.”
• Look for problems as you read. When you see contradictions, inconsistencies, incomplete explanations tentatively assume that readers would or should feel the same.
• Ask not “Can I solve it?” but “will my readers think it ought to be solved?”
• “Try to formulate a question you think is worth answering, so that down the road, you’ll know how to find a problem others think is worth solving.”
• # The audience models of research—thoughts on Craft of Doing Research chapter 2
Writers can’t avoid creating some role for themselves and their readers, planned or not
1. I’ve found some new and interesting information—I have information for you
2. I’ve found a solution to an important practical problem—I can help you fix a problem
The authors recommend assuming one of these three. There is of course a wider gap between information and the neighborhood of problems and questions than there is between problems and questions! Later on in chapter four the authors provide a graph illustrating problems and questions: Practical problem -> motivates -> Research question -> defines -> Conceptual/research problem. Information, when provided mostly for novelty, however, is not in this cycle. Information can be leveled at problems or questions, plays a role in providing solutions or answers, but can also be for “its own sake”.
I’m reminded of a paper/post I started but never finished, on providing a poset-like structure to capabilities. I thought it would be useful if you could give a precise ordering on a set of agents, to assign supervising/overseeing responsibilities. Looking back, providing this poset would just be a cool piece of information, effectively: I wasn’t motivated by a question or problem so much as “look at what we can do”. Yes, I can post-hoc think of a question or a problem that the research would address, but that was not my prevailing seed of a reason for starting the project. Is the role of the researcher primarily a writing thing, though, applying mostly to the final draft? Perhaps it’s appropriate for early stages of the research to involve multi-role drifting, even if it’s better for the reader experience if you settle on one role in the end.
Additionally, it occurs to me that maybe “I have information for you” mode just a cheaper version of the question/problem modes. Sometimes I think of something that might lead to cool new information (either a theory or an experiment), and I’m engaged moreso by the potential for novelty than I am by the potential for applications.
I think I’d like to become more problem-driven. To derive possibilities for research from problems, and make sure I’m not just seeking novelty. At the end of the day, I don’t think these roles are “equal” I think the problem-driven role is the best one, the one we should aspire to.
[When you adopt one of these three roles, you must] cast your readers in a complementary role by offering them a social contract: _I’ll play my part if you play yours … if you cast them in a role they won’t accept, you’re likely to lose them entirely… You must report your research in a way that motivates your readers to play the role you have imagined for them.
The three reader roles complementing the three writer roles are
1. Entertain me
2. Help me solve my practical problem
3. Help me understand something better
It’s basically stated that your choice of writer role implies a particular reader role, 1 mapping to 1, 2 mapping to 2, and 3 mapping to 3.
Role 1 speaks to an important difficulty in the x-risk, EA, alignment community; which is how not to get drawn into the phenomenal sensation of insight when something isn’t going to help you on a problem. At my local EA meetup I sometimes worry that the impact of our speaker events is low, because the audience may not meaningfully update even though they’re intellectually engaged. Put another way, intellectual engagement can be goodhartable, the sensation of insight can distract you from your resolve to shatter your bottlenecks and save the world if it becomes an end itself. Should researchers who want to be careful about this avoid the first role entirely? Should the alignment literature look upon the first reader role as a failure mode? We talk about a lot of cool stuff, it can be easy to be drawn in by the cool factor like some of the non-EA rationalists I’ve met at meetups.
I’m not saying reader role number two absolutely must dominate, because it can diverge from deconfusion which is better captured by reader role number three.
## Division of labor between reader and writer, writer roles do not always imply exactly one reader role
Isn’t it the case that deconfusion/writer role three research can be disseminated to practical (as opposed to theoretical) -minded people, and then those people turn question-answer into problem-solution? You can write in the question-answer regime, but there may be that (rare) reader who interprets it in the problem-solution regime! This seems to be an extremely good thing that we should find a way to encourage. In general reading the drifts across multiple roles seems like the most engaged kind of reading.
• there’s a gap in my inside view of the problem, part of me thinks that capabilities progress such as out-of-distribution robustness or the 4 tenets described in open problems in cooperative ai is necessary for AI to be transformative, i.e. a prereq of TAI, and another part of me that thinks AI will be xrisky and unstable if it progresses along other aspects but not along the axis of those capabilities.
There’s a geometry here of transformative / not transformative cross product with dangerous not dangerous.
To have an inside view I must be able to adequately navigate between the quadrants with respect to outcomes, interventions, etc.
• If something can learn fast enough, then it’s out-of-distribution performance won’t matter as much. (OOD performance will still matter -but it’ll have less to learn where it’s good, and more to learn where it’s not.*)
*Although generalization ability seems like the reason learning matters. So I see why it seems necessary for ‘transformation’.
• testing latex in spoiler tag
Testing code block in spoiler tag
:::hm? x :: Bool -> Int -> String :::
|
{}
|
## LOCAL
### Non-decimated Quaternion Wavelet Spectral Tools with Applications
Taewoon Kong and Brani Vidakovic
Quaternion wavelets are redundant wavelet transforms generalizing complex-valued non-decimated wavelet transforms. In this paper we propose a matrix-formulation for non-decimated quaternion wavelet transforms and define spectral tools for use in machine learning tasks. Since quaternionic algebra is an extension of complex algebra, quaternion wavelets bring redundancy in the components that proves beneficial in wavelet based tasks. Specifically, the wavelet coefficients in the decomposition are quaternion-valued numbers that define the modulus and three phases. The novelty of this paper is definition of non-decimated quaternion wavelet spectra based on the modulus and phase-dependent statistics as low-dimensional summaries for 1-D signals or 2-D images. A structural redundancy in non-decimated wavelets and a componential redundancy in quaternion wavelets are linked to extract more informative features. In particular, we suggest an improved way of classifying signals and images based on their scaling indices in terms of spectral slopes and information contained in the three quaternionic phases. We show that performance of the proposed method significantly improves when compared to the standard versions of wavelets including the complex-valued wavelets. To illustrate performance of the proposed spectral tools we provide two examples of application on real-data problems: classification of sounds using scaling in high-frequency recordings over time and monitoring of steel rolling process using the fractality of captured digitized images. The proposed tools are compared with the counterparts based on standard wavelet transforms.
Full version of the paper is available at arXiv: Full Version.
In the spirit of reproducible research, Taewoon compiled the package with MATLAB codes used for calculations PackageWavmatQND.zip.
### Non-decimated Complex Wavelet Spectral Tools with Applications
Taewoon Kong and Brani Vidakovic
In this paper we propose spectral tools based on non-decimated complex wavelet transforms implemented by their matrix formulation. This non-decimated complex wavelet spectra utilizes both real and imaginary parts of complex-valued wavelet coefficients via their modulus and phases. A structural redundancy in non-decimated wavelets and a componential redundancy in complex wavelets act in a synergy when extracting wavelet-based informative descriptors. In particular, we suggest an improved way of separating signals and images based on their scaling indices in terms of spectral slopes and information contained in the phase in order to improve performance of classification. We show that performance of the proposed method is significantly improved when compared with procedures based on standard versions of wavelet transforms or on real-valued wavelets. It is also worth mentioning that the matrix-based non-decimated wavelet transform can handle signals of an arbitrary size and in 2-D case, rectangular images of possibly different and non-dyadic dimensions. This is in contrast to the standard wavelet transforms where algorithms for handling objects of non-dyadic dimensions requires either data preprocessing or customized algorithm adjustments. To demonstrate the use of defined spectral methodology we provide two examples of application on real-data problems: classification of visual acuity using scaling in pupil diameter dynamic in time and diagnostic and classification of digital mammogram images using the fractality of digitized images of the background tissue. The proposed tools are contrasted with the traditional wavelet based counterparts.
Full version of the paper is available at arXiv: Full Version.
In the spirit of reproducible research, Taewoon compiled the package with MATLAB codes used for calculations WavmatCND.zip.
### Empirical Wavelet-based Estimation for Non-linear Additive Regression Models.
German A. Schnaidt Grez and Brani Vidakovic
Additive regression models are actively researched in the statistical field because of their usefulness in the analysis of responses determined by non-linear relationships with multivariate predictors. In this kind of statistical models, the response depends linearly on unknown functions of predictor variables and typically, the goal of the analysis is to make inference about these functions. In this paper, we consider the problem of Additive Regression with random designs from a novel viewpoint: we propose an estimator based on an orthogonal projection onto a multiresolution space using empirical wavelet coefficients that are fully data driven. In this setting, we derive a mean-square consistent estimator based on periodic wavelets on the interval [0, 1]. For construction of the estimator, we assume that the joint distribution of predictors is non-zero and bounded on its support; We also assume that the functions belong to a Sobolev space and integrate to zero over the [0,1] interval, which guarantees model identifiability and convergence of the proposed method. Moreover, we provide the L2 risk analysis of the estimator and derive its convergence rate. Theoretically, we show that this approach achieves good convergence rates when the dimensionality of the problem is relatively low and the set of unknown functions is sufficiently smooth. In this approach, the results are obtained without the assumption of an equispaced design, a condition that is typically assumed in most wavelet-based procedures. Finally, we show practical results obtained from simulated data, demonstrating the potential applicability of our method in the problem of additive regression models with random designs.
Keywords: Wavelets, non-parametric regression, functional data analysis, robust statistical modeling
Full version of the paper with Appendices and Proofs is available at arXiv: Full Version.
### Wavelet-based scaling indices for breast cancer diagnostics
Tonya Roberts, Mimi Newell, William Auffermann, and Brani Vidakovic
Mammography is routinely used to screen for breast cancer (BC). However, the radiological interpretation of mammogram images is complicated by the heterogeneous nature of normal breast tissue and the fact that cancers are often of the same radiographic density as normal tissue. In this work, we use wavelets to quantify spectral slopes of BC cases and controls and demonstrate their value in classifying images. In addition, we propose asymmetry statistics to be used in forming features which improve the classification result. For the best classification procedure, we achieve approximately 77% accuracy (sensitivity=73%, specificity=84%) in classifying mammograms with and without cancer.
Manuscript can be found here, and the refrerence is:
Roberts, T., Newell, M., Auffermann, W., and Vidakovic, B. (2017). Wavelet-based scaling indices for breast cancer diagnostics. Statistics in Medicine, 36, 12, 1989--2000, DOI: 10.1002/sim.7264
### ESTIMATION OF THE HURST EXPONENT USING TRIMEAN ESTIMATORS ON NONDECIMATED WAVELET COEFFICIENTS
Chen Feng and Brani Vidakovic
Hurst exponent is an important feature summarizing the noisy high-frequency data when the inherent scaling pattern cannot be described by standard statistical models. In this paper, we study the robust estimation of Hurst exponent by applying a general trimean estimator on non-decimated wavelet coefficients of the transformed data. Our wavelet-based methods provide a robust way to estimate Hurst exponent and increase the prediction precision especially when there exists outlier coefficients, outlier multi-resolution levels, and within level dependencies. The properties of the proposed Hurst exponent estimators are studied both theoretically and numerically. Compared with other standard wavelet-based methods (Veitch and Abry (VA) method, Soltani, Simard, and Boichu (SSB) method, median-based estimators MEDL and MEDLA, and Theil-type (TT) weighted regression method), our methods reduce the variance of the estimators by not sacrificing the prediction precision in most cases.
Supplementary material can be found here.
A preliminary version of this paper was posted on arXiv https://arxiv.org/abs/1709.08775.
### ROBUST WAVELET-BASED ASSESSMENT OF SCALING WITH APPLICATIONS
Erin K. Hamilton, Minkyoung Kang, Seonghye Jeon, Pepa Ramírez Cobo, Kichun Sky Lee, and Brani Vidakovic
A number of approaches have dealt with statistical assessment of self-similary, and many of those are based on multiscale concepts. Most rely on certain distributional assumptions which are usually violated by real data traces, often characterized by large temporal or spatial mean level shifts, missing values or extreme observations. A novel, robust approach based on Theil-type weighted regression is proposed for estimating self-similarity in two-dimensional data (images). The method is compared to two traditional estimation techniques that use wavelet decompositions; {ordinary least squares} (OLS) and Abry-Veitch bias correcting estimator (AV). As an application, the suitability of the robust approach is illustrated in the classification of digitized mammogram images as cancerous or non-cancerous. The diagnostic employed here is based on the properties of image backgrounds, which is typically an unused modality in breast cancer screening. Classification results show nearly 68% of accuracy, varying slightly with the choice of wavelet basis, and the range of multirseolution levels used.
This paper is under revision for Communicatons in Statistics.
Because of the size of the paper the part of simulations results is deferred to this electronic Appendix B .
MATLAB codes for generating 1-D and 2-D fractional Brownian motions are: MakeFBM.m and MakeFBM2D.m , respectively. The 1-D fBm is generated by scaling the modulus and randomizing the phase of gaussians in FFT, while the 2-D fBm is authored by Olivier Barriere.
### WavmatND: A MATLAB Package for Non-Decimated Wavelet Transform and its Applications
Minkyoung Kang and Brani Vidakovic
A non-decimated wavelet transform (NDWT) is a popular version of wavelet transforms because of its many advantages in applications. The inherent redundancy of this transform proved beneficial in tasks of signal denoising and scaling assessment. To facilitate the use of NDWT, we built a MATLAB package, WavmatND, in which transforms are done by matrix multilication, and which has three novel features: First, for signals of moderate size the proposed method reduces computation time of the NDWT by replacing repetitive convolutions with matrix multiplications. Second, submatrices of an NDWT matrix can be rescaled, which enables a straightforward inverse transform. Finally, the method has no constraints on a size of the input signal in one or in two dimensions, so signals of non-dyadic length and rectangular two-dimensional signals with non-dyadic sides can be readily transformed. We provide illustrative examples and a tutorial to assist users in application of this stand-alone package.
The manuscript can be found HERE and the package with all MATLAB codes are zipped in WavmatND.zip.
### CHARACTERIZING EXONS AND INTRONS BY REGULARITY OF NUCLEOTIDE STRINGS
Tonya Woods, Thanawadee Preeprem, Kichun Lee, Woojin Chang, and Brani Vidakovic
Translation of nucleotides into a numeric form has been approached in many ways and has allowed researchers to investigate the properties of protein-coding sequences and noncoding sequences. Typically, more pronounced long-range correlations and increased regularity were found in intron-containing genes and in non-transcribed regulatory DNA sequences, compared to cDNA sequences or intron-less genes. The regularity is assessed by spectral tools defined on numerical translates. In most popular approaches of numerical translation the resulting spectra depend on the assignment of numerical values to nucleotides. Our contribution is to propose and illustrate a spectra which remains invariant to the translation rules used in traditional approaches.
We outline a methodology for representing sequences of DNA nucleotides as numeric matrices in order to analytically investigate important structural characteristics of DNA. This representation allows us to compute the 2-dimensional wavelet transformation and assess regularity characteristics of the sequence via the slope of the wavelet spectra. In addition to computing a global slope measure for a sequence, we can apply our methodology for overlapping sections of nucleotides to obtain an evolutionary slope." To illustrate our methodology, we analyzed 376 gene sequences from the first chromosome of the honeybee.
For the genes analyzed, we find that introns are significantly more regular (lead to more negative spectral slopes) than exons, which agrees with the results from the literature where regularity is measured on DNA walks." However, unlike DNA walks where the nucleotides are assigned numerical values depending on nucleotide characteristics (purine-pyrimidine, weak-strong hydrogen bonds, keto-amino, etc.) or other spatial assignments, the proposed spectral tool is invariant to the assignment of nucleotides. Thus, ambiguity in numerical translation of nucleotides is eliminated.
This paper is open access and can be found HERE .
MATLAB files supporting the paper are: evcdsplot.m and evdnaslopec.m. These two files are used for computing the cumulative evolutionary slope of a sequence and for creating the plots with exons, introns, and combination regions. The "evdnaslopec.m" program requires the use of WavMat.m function.
Reference:
Woods, T., Preeprem, T., Lee, K., Chang, W., and Vidakovic, B. (2016). Characterizing Exons and Introns by Regularity of Nucleotide Strings. Biology Direct, 11, 6, 1--17; DOI: 10.1186/s13062-016-0108-7
### A Constrained Wavelet Smoother for Pathway Identification Tasks in Systems Biology
Sepideh Dolatshahi, Brani Vidakovic, and Eberhard O. Voit
Metabolic time series data are being generated with increasing frequency, because they contain enormous information about the pathway from which the metabolites derive. This information is not directly evident, though, and must be extracted with advanced computational means. One typical step of this extraction is the estimation of slopes of the time courses from the data. Since the data are almost always noisy, and the noise is typically amplified in the slopes, this step can become a critical bottleneck. Several smoothers have been proposed in the literature for this purpose, but they all face the potential problem that smoothed time series data no longer correspond to a system that conserves mass throughout the measurement time period. To counteract this issue, we are proposing here a smoother that is based on wavelets and, through an iterative process, converges to a mass-conserving, smooth representation of the metabolic data. The degree of smoothness is user defined. We demonstrate the method with some didactic examples and with the analysis of actual measurements characterizing the glycolytic pathway in the dairy bacterium Lactococcus lactis. MATLAB code for the constrained smoother is available as a supplement.
Paper is here, and the reference is:
Dolatshahi, S., Vidakovic, B., and Voit, E. O. (2014). A constrained wavelet smoother for pathway identification tasks in systems biology. Computers and Chemical Engineering, 71, 728--733. doi: 10.1016/j.compchemeng.2014.07.019.
### DENOISING BY BAYESIAN MODELING IN THE DOMAIN OF DISCRETE SCALE MIXING 2D COMPLEX WAVELET TRANSFORMS
Norbert Remenyi, Orietta Nicolis, Guy Nason, and Brani Vidakovic
Wavelet shrinkage methods that use complex-valued wavelets provide additional insights to shrinkage process compared to standardly used real-valued wavelets. Typically, a location-type statistical model with an additive noise is posed on the observed wavelet coefficients and the true signal/image part is estimated as the location parameter. Under such approach the wavelet shrinkage becomes equivalent to a location estimation in the wavelet domain. The most popular type of models imposed on the wavelet coefficients are Bayesian. This popularity is well justified: Bayes rules are typically well behaved shrinkage rules, prior information about the signal can be incorporated in the shrinkage procedure, and adaptivity of Bayes rules can be achieved by data-driven selection of model hyperparameters. Several papers considering Bayesian wavelet shrinkage with complex wavelets are available. For example, Jean Marc Lina and coauthors focus on image denoising, in which the phase of the observed wavelet coefficients is preserved, but the modulus of the coefficients is shrunk by a Bayes rule. The procedure introduced in Barber and Nason in 2004 modifies both the phase and modulus of wavelet coefficients by a bivariate shrinkage rule.
We propose a Bayesian model in the domain of a complex scale-mixing discrete unitary, compactly supported wavelets that generalizes the method in Barber and Nason to 2-D signals. In estimating the signal part the model to allowed to modify both phase and modulus. The choice of wavelet transform is motivated by the symmetry / antisymmetry of decomposing wavelets, which is possible only in the complex domain under condition of orthogonality (unitarity) and compact support. Symmetry is considered a desirable property of wavelets, especially when dealing with images.
The 2-D discrete scale mixing wavelet transform is computed by left- and right-multiplying the image by a wavelet matrix W and its Hermitian transpose, respectively. Mallat's algorithm to perform this task is not used, but it is implicit in the construction of matrix $W.$
The resulting shrinkage procedures cSM-EB and cMOSM-EB are based on empirical Bayes approach and utilize non-zero covariances between real and imaginary parts of the wavelet coefficients. We discuss the possibility of phase-preserving shrinkage in this framework. Overall, the methods we propose are calculationally efficient and provide excellent denoising capabilities when contrasted with comparable and standardly used wavelet-based techniques.
A MATLAB toolbox developed by Norbert Remenyi cSM-EB2.zip illustrates cSM-EB and cMOSM-EB shrinkage.
Reference:
Remenyi, N., Nicolis, O., Nason, G., and Vidakovic, B. (2014). Image Denoising With 2D Scale-Mixing Complex Wavelet Transforms. IEEE Transactions on Image Processing, 23, 12, 5165--5174.
### LAMBDA NEIGHBORHOOD WAVELET SHRINKAGE
A wavelet-based denoising methodology based on total energy of a neighboring pair of coefficients plus their ‘‘parental’’ coefficient is proposed. The model is based on a Bayesian hierarchical model using a contaminated exponential prior on the total mean energy in a neighborhood of wavelet coefficients. The hyperparameters in the model are estimated by the empirical Bayes method, and the posterior mean, median and Bayes factor are obtained and used in the estimation of the total mean energy. Shrinkage of the neighboring coefficients are based on the ratio of the estimated and the observed energy. It is shown that the methodology is comparable and often superior to several existing and established wavelet denoising methods that utilize neighboring information, which is demonstrated by extensive simulations on a standard battery of test functions. An application to real-word data set from inductance plethysmography is also considered. A MATLAB toolbox developed by Norbert Remenyi LNWS.zip illustrates the methodology. The toolbox supports the manuscript: Lambda Neighborhood Wavelet Shrinkage, by Norbert Remenyi and Brani Vidakovic
Reference:
Reményi, N. and Vidakovic, B. (2013). Λ-neighborhood wavelet shrinkage. Computational Statistics \& Data Analysis, 57, 1, 404--416, doi:10.1016/j.csda.2012.07.008
### WAVELET SHRINKAGE WITH DOUBLE WEIBULL PRIORS
Bayesian wavelet shrinkage standardly employs point-mass at zero contamination priors for the signal part in nonparametric regression problems. In this paper a competitive methodology is achieved with a simple prior based on Double Weibull distribution without point mass at zero, but with a singularity at 0. A MATLAB toolbox developed by Norbert Remenyi DWWS.zip illustrates the methodology. The toolbox supports the manuscript: Wavelet Shrinkage with Double Weibull Prior, by Norbert Remenyi and Brani Vidakovic
Reference:
Reményi, N. and Vidakovic, B. (2015). Wavelet Shrinkage with Double Weibull Prior. Communications in Statistics - Simulation and Computation, 44, 1, 88--104.
### DENSITY ESTIMATION WHEN DATA ARE SIZE-BIASED: WAVELET-BASED MATLAB TOOLBOX
Often researchers need to estimate the density in the presence of size-biased data. The wavelet-based MATLAB toolbox biased.zip performs debiasing and estimattes density by smoothed linear projection wavelet esimator. The toolbox supports the manuscript: Wavelet Density Estimation for Stratified Size-Biased Sample, by Pepa Ramirez and Brani Vidakovic.
Reference:
Ramírez, P. and Vidakovic, B. (2010). Wavelet density estimation for stratified size-biased sample. Journal of Statistical Planning and Inference, 140, 2, 419 -- 432.
### LPM: Bayesian Wavelet Thresholding based on Larger Posterior Mode
This project explores the thresholding rules induced by a variation of the Bayesian MAP principle. The MAP rules are Bayes actions that maximize the posterior. Under the proposed model the posterior is neither unimodal or bimodal. The proposed rule is thresholding and always picks the mode of the posterior larger in absolute value, thus the name LPM. We demonstrate that the introduced shrinkage performs comparably to several popular shrinkage techniques. Exact risk properties of the thresholding rule are explored. We provide extensive simulational analysis and apply the proposed methodology to real-life experimental data coming from the field of Atomic Force Microscopy (AFM).
You could try the LPM thresholding if your MATLAB has access to WaveLab Module. MATLAB m-files, MATHEMATICA nb-files, data, and figures are zipped in the following archive file:
The manuscript (draft version in PDF) describing the introduced thresholding is here:
• Larger Posterior Mode Wavelet Thresholding and Applications. The authors are Luisa Cutillo, Yoon Young Jung, Fabrizio Ruggeri, and Brani Vidakovic. The provided files in LPM.zip are sufficient to fully REPRODUCE this manuscript.
Reference:
Cutillo, L., Jung, Y.-Y., Ruggeri, F., and Vidakovic, B. (2008). Larger Posterior Mode Wavelet Thresholding and Applications, Journal of Statistical Planning and Inference, 138, 3758--3773.
### Hunting for Dominant Straight-Line Features in Nanoscale Images Using Wavelets
Ilya Lavrik, PhD Graduate in Statistics at ISyE, has developed Matlab Toolbox which searches for significant straight line alignments of molecular structures in nano-scale images. The methodology is based on Hough and wavelet transforms. The software, excellent front end, and manual are available for download:
• nsia.zip NANOSCALE IMAGE ANALYSIS - zipped archive of NanoLab, a matlab suite with an excellent front.
• images.zip Several Genuine Nanoscale Images needed for the NSIA. The file is about 13.5MB in size.
• manual.pdf Manual for NSIA
Comments welcome! Partial support of this project by the Georgia Institute of Technology Molecular Design Institute, under prime contract N00014-95-1-1116 from the Office of Naval Research. Partial support for this work was also provided by National Security Agency Grant NSA E-24-60R at ISyE.
The manuscript (draft version in PDF) describing the methodology can be found here:
• LINEAR FEATURE IDENTIFICATION AND INFERENCE IN NANO-SCALE IMAGES
### THE WAVELETS PUZZLE
Fun with Wavelets! A rare R. J. Journet dexterity glass top puzzle*.
*Journet Dexterity Puzzles are glass top dexterity puzzles, made between 1890 and 1960 by an English manufacturer called R. J. Journet (affectionately known as RJ's). The diversity of subjects in these puzzles is humungous, ranging from an R.A.F trip to bomb Hitler in Berlin to Alice in Wonderland's Tea Party. To complete a puzzle you have to shake, rattle or roll it. They are called dexterity puzzles because you have to be dexterous (skillful in using hands) to complete them. Some are easy, some difficult, and some next to impossible.
### WAVELET MATRIX IN MATLAB
• Here a matlab routine to form a matrix performing discrete orthogional wavelet transformation. Once the matrix W is generated, the transformation d is obtained by multiplying the data vector y, d=W * y. Of course, y = W' * d. The m-file WavMat.m should be put in yourmatlabpath/toolbox/wavelab/Orthogonal/, although if you know the filter, no wavelab is needed -- the m-file is a stand alone.
> dat = [1 0 -3 2 1 0 1 2];
> filter = [sqrt(2)/2 sqrt(2)/2];
> W = WavMat(filter,2^3,3);
> wt = W * dat' %should be [sqrt(2) | -sqrt(2) | 1 -1 | ...
% 1/sqrt(2) -5/sqrt(2) 1/sqrt(2) - 1/sqrt(2) ]
> data = W' * wt % should return you to the 'dat'
If the matrix size exceeds 1024 x 1024, an average PC is getting slow. WavMat.m could be optimized. [In Ver. 1.2 built 12/1/04, functions 'modulo' and 'reverse' replaced by built-in functions. Thanks to Mr Deniz Sodiri for pointing out the original posting was not stand alone.] Some readers asked for the algorithm: Here are three pages describing it.
### AN OPEN PROBLEM OR EASY EXERCISE?
• Recently, working on Convex Rearrangements of wavelet filtered self-similar processes I looked at compactly supported orthogonal wavelet filters and for some "empirical evidence" could not find a proof. Let me know if you have an answer.
### A NICE 2D DATA TO NOISE/DENOISE
Here is a 2D data set free to use for tasks of image wavelet processing. This is a 3072 x 2048 (3.2MB, jpg) digital photo of von Klaus. Von Klaus is a two year old purebreed [AKC WR021286/04] Doberman Pinscher var. Warlock born in Marietta, Georgia. Although he looks quite intimidating, von Klaus is a gentle, playful, and devoted dog. The big (> 6 megapixel) JPG photo is imported to MATLAB using klaus.m m-file and 6 gray scale images of various dimensions are made. Here are all the 6 as an EPS file. (>11MB)
### BOOTSTRAPPING WAVELETS
• That is to say: WAVESTRAPPING. Also, a no-name article in Popular Mechanics [June 2004], but the two guys look very much like my graduate student Bin Shi and myself! The same from BRASIL!
### MINICOURSE IN MILANO WAVELETS AND SELF-SIMILARITY: THEORY AND APPLICATIONS
December, 14-17, 2004 at Consiglio Nazionale delle Ricerche Istituto di Matematica Applicata e Tecnologie Informatiche (Milano Department - formerly CNR-IAMI)
### BAMS-LP (Bayesian Adaptive Multiresolution Shrinker of Log Periodogram)
The matlab files that implement the BAMS-LP shrinker and a few examples of its use are zipped into archive BAMSP.zip . The software is tested with MATLAB6.5. The the theory behind the software and a paper describing Bayesianly induced wavelet shrinkage of Log-Periodogram, can be found HERE in the PDF format.
### Some ADDONS for WaveLab Module
#### Discrete Complex Orthogonal Wavelet Transformation
Here is some history. We started these m-files when Jean-Marc Lina from University of Montreal was visiting Duke University. At that time JML, being a complex wavelet guru, guided Gaby Katul and me how to do forward complex DISCRETE wavelet transformation. We originally used complex filters from Lina's papers and then discovered that Barry G. Sherlock now at UNC-Charlotte made an m-function a la Donoho's MakeONFilter.m for producing Daubechies complex filters. Getting into the complex wavelet domain was easy compared to returning back to the time'' domain. We made an m-file for inverse transformation but it was less than perfect...(a nice way to say it did not work as it was supposed to). Claudia Angelini, a bright graduate student visiting GaTech from Napoli's CNR, took a look at our pluses and minuses and fixed the inverse transform in a second. So here they are: FWTC_PO.m will mimic FWT_PO with complex filters and return the complex discrete wavelet transformation, two vectors [re, im] for the real and immaginary parts. IWTC_PO.m will get you back from the complex wavelet domain to the space of original discrete data. Finally, complex Daubechies wavelet filters are made by Sherlock's MakeCONFilter.m. To make Complex tools work just add FWTC_PO.m, IWTC_PO.m, and MakeCONFilter.m to ~/wavelab/Orthogonal/.
#### 2-D Continuous Wavelet Transformations
In the Spring of 2001 Xiaoming Huo and myself team-taught a graduate course on wavelets at GaTech. We had about 15 graduate students coming from various Tech's departments. Heejong Yoo, graduating PhD student from ECE, was an excellent programmer interested in implementing 2D Continuous Wavelet Transformation in his class-project. The idea came from commercial software Crit-tech Psilets 3.0; we decided to make a free clone! The theory behind the transformation is trivial: One (listably) multiplies the 2D object with the sampled fixed level 2D wavelet in the Fourier domain and then Fourier-inverts the product! Heejong's project is a standalone MATLAB program (no wavelab needed) with an excellent GUI. Zipped directory with all files needed to run the CWT2D is Project.zip and the PPT presentation of the project is: Cont2DWT.ppt . Only 2D Mexican hat is available right now. If you prefer the Wavelab environment, than you can add the function CWT2.m to ~/wavelab/Continuous/
#### 3-D Discrete Wavelet Transformation (Orthogonal, Tensor Product)
This pair of transformations naturally generalizes WaveLab's FWT2_PO.m and IWT2_PO.m. This is a part of wavelet-project of Vicki Yang, gifted graduate student at ISyE who took a course on wavelets with me. She was interested in wavelet processing of 3-D signals with applications.
The forward and inverse transformations are: FWT3_PO.m, for transforming the data to the wavelet domain, and IWT3_PO.m, for inverse-transforming the data back to the time domain. The function needed here is cubelength.m that is a 3-D counterpart of Donoho's quadlength.m utilized by the 2D pair. You will see that transforms are conceptually and algorithmically easy, and it would be quite starightforward to construct FWT4_PO, FWT5_PO, ... and their inverses. Now, both FWT3_PO and IWT3_PO transformations act on 3D data sets and such objects are difficult to visualize. We made several data related programs.
(i) Make3DData.m will make 3D ball with inscribed octahedron. Both bodies the ball and the octahedron are inscribed in a cube of (dyadic) side N. The noise can be added to both boundaries and interiors of objects.
(ii) DDD2Movie.m will make a movie from the 3D object taking frames along the dimension of choice. This is handy for viewing the 3D objects via their 2D cuts.
(iii) A small script test3d.m will take a 64 x 64 x 64 noisy object, view it, transfer it to the wavelet domain, view it again, threshold the object, view it, return the thresholded object to the original domain, and view it. For some misterious reasons, DDD2Movie.m will show the movie itself while recording, and built-in matlab function movie will show the movie twice! Be ready to watch the objects 4 x 3 = 12 times... To make 3D tools work just add FWT3_PO.m, IWT3_PO.m, and cubelength.m to ~/wavelab/Orthogonal/
#### A New Extended MakeONFilter, MakeONFilterExt.m.
This extension is final project in an undergraduate wavelet research course submitted by graduating ISyE student, Daphne Lai. Daphne added more Daubechies', Symmlets, and Coiflets, as well as some new filters. All added filters are numerically stable.
Standard WaveLab m-function dyad.m extracts particular level in the discrete wavelet transformation. If, for example, n=2^J, and the Discrete Wavelet Transfirmation is WT, the finest level is indexed by dyad(J-1), and extracted from WT as WT(dyad(J-1)). I needed dyad-like tools for 2-D wavelet transformations. A simple generalization is dyad2.m and it needs in addition to dyad.m the complement' function dyadc.m . The following matlab script shows use of dyad2:
>> pict = MakeImage('StickFigure',128);
>> wf = MakeONFilter('Haar',1);
>> wpict = FWT2_PO(pict, 5, wf);
>> diag_det = wpict(diagx, diagy);
>> imagesc(diag_det)
#### Some Shortcomings of WaveLab.
There is one problem with FWT_PO.m and its inverse in WaveLab that needs a fix! The problem propagates to other transformations, notably 2D, etc. It is well known that any scaling filter H=(h_0, ... ,h_N) can be matched with many quadrature mirror filters -- high pass counterparts G. Wavelet polygamy -- one father and many mothers.'' For example $g_n = (-1)^{n+x} h_{y - n},$ where $x=0,1$ and $y$ is arbitrary integer, is a valid QM wavelet filter. And not all the wavelet bases share the same proper'' translation and sign of G defined by $x$ and $y$ Not all H filters start with $h_0$! For example, proper start for Coiflet 1 (6 tap filter) is $h_{-2}.$ WaveLab does not allow for such flexibility. And although the reconstructions are perfect, the wavelet domain objects are circularly shifted. For example, if a period of a SINE function is sampled and transformed by wavelet transformation of depth 3 (log(n)-L=3), the resulting transformation should result in scaling coefficients that are degraded SINE function. This example shows that improper filter alignment causes smooth-part SINE to shift. To see this, please run the exercise under WaveLab. One may ask, why should we care when the reconstruction is perfect? The proper alignment is critical, because of simulational aspects of wavelets. Often one starts with the wavelet domain, feeds the empty levels with (simulated) coefficients and reconstructs. And if the alignment is not proper various problems and anomalies can occur. This could be an interesting project for a devoted grad student! Please take a look for an excellent solution of this alignment problem by UviWave software from University of Vigo. Unusual procedure is MirrorFilt.m. In it the high pass filter G is formed as
g = -( (-1).^(1:length(h)) ) .* h;
This leads to an orthogonal transformation, but more common filter g is obtained by
g = - reverse( (-1).^(1:length(h)) .* h );
In my version of Wavelab I modified MirrorFilt.m.
#### Daubechies-Lagarias Algorithm in Matlab
Calculate the value of \phi_{jk}(x_0) or \psi_{jk}(x_0) at ANY point x_0 for ANY orthonormal basis at ANY precision without going through Mallat's algorithm. The blurb DL.pdf describes the algorithm. The matlab programs used are: MakePollen1.m, MakePollen2.m, Phijk.m, Psijk.m, and m-script DLtest.m.
#### FWT2_POE and IWT2_POE for Rectangular Images of Dyadic Sides
In 2-D tensor product wavelet transformations, traditionally the inputs are square images of a dyadic side. Since performing the 2-D transformation amounts to subsequent application of 1-D transformations on rows and columns of an image, the restriction to square dimensions is inessential. Here are slight extensions of standard wavelab's FWT2_PO.m and IWT2_PO.m, the functions: FWT2_POE.m and IWT2_POE.m . The pair FWT2_POE, IWT2_POE will do the 2-D wavelet transformation and its inverse on rectangular images with dyadic sides. The m-file quadlength.m needed by FWT2_PO.m and IWT2_PO.m should be replaced by pow2length.m . All three files should reside in ...\Wavelab\Orthogonal\ directory. Now, take a look how the rectangular Lena ( lena21.eps or lena21.pdf) looks in the wavelet domain, ( lena21w.eps or lena21w.pdf). Data file is lena21.mat. The choice of size 256 x 512, rather than more interesting 512 x 256 or quite exciting 1024 x 256, was made by flipping a coin;).
Planned...
#### Practical Hints on Running WaveLab (when the names collide).
Many soon to be posted, stay tuned...
### Wavelet-history Curiosity
A wavelet-history curiosity I found interesting. Chapter 4 spanning 70 pages of the book Time Series Analysis and Applications'' by Enders A. Robinson is titled: Wavelet Composition of Time Series. The curios thing is that the book is published in 1981!!! Robinson's wavelets indeed have some of the wavelet spirit. A quote from page 84:
...The wavelets arrive in succession, and each wavelet eventually dies out. The wavelets all have the same basic form and shape, but the strength or impetus of each wavelet is random and uncorrelated with the strength of the other wavelets...
...Despite the foreordained death of any individual wavelet, the time-series does not die. The reason is that a new wavelet is born each day to take the place of the one that does die. On any given day, the time-series is composed of many living wavelets, all of a different age,-some young, others old.''
The chapter then formally describes the theory and practice of Robinson's atomic decompositions.
Reference: Robinson, Enders (1981). Time Series Analysis and Applications. Houston, Goose Pond Press 628p. Library of Congress Catalog 81-81825
A kind note from Laurent Duval: I am not esp. surprised since i consider Robinson (at least partly) as a geophysicist. One of the early mention i found in geophysics is: N. Ricker, A note on the determination of the viscocity of shale from the measurement of the wavelet breadth, Geophysics, Society of Exploration Geophysicists, vol. 06, pp. 254-258, 1941. See for instance: http://www.laurent-duval.eu/siva-wits-where-is-the-starlet.html The word wavelet has gone through a chain: wavelet (geophysics) -> ondelettes (geophysics) -> wavelet (as we know it). Even earliest spurs are in Huygens.
### Les Houches Center of Physics PHYSICS - SIGNAL - PHYSICS On the links between nonlinear physics and information sciences September 8-13, 2002
INFO On the links between nonlinear physics and information sciences
### CALL FOR PAPERS
Journal Applied Stochastic Models in Business and Industry [Wiley InterScience ISSN 1524-1904, http://www.interscience.wiley.com ] is considering a special issue on Wavelets and Other Multiscale Methods: Theory and Applications. Contributions for the Special Issue that are good balance of theory and applications of wavelets and other multiscale methods in industry, finance, and applied sciences are invited. Ascii Text
Poster
### BAMS Bayesian Adaptive Multiresolution Smoother; Matlab Demo Program; needs WaveLab Software
bams.m uses function bayesrule.m
Supporting Manuscript
• 00-06 Brani Vidakovic and Fabrizio Ruggeri
BAMS Method: Theory and Simulations
### BAMS: Matlab Front End (No Wavelab Necessary)
Dr Bin Shi, my former graduate student, made a simple front-end that demonstrates BAMS shrinkage in MATLAB. As of now, the only signal is doppler and, as tradditionally done, the standard normal noise is added to the rescaled signal to achieve desired SNR. The programs below should be on MATLAB's path and Wavelab is not needed.
### Wavelets in Statistics Week at CNR-IAMI, Milano
Eight Lectures!
### BOOK: STATISTICAL MODELING BY WAVELETS,
by Brani Vidakovic, Wiley Series in Probability and Statistics; ISBN: 0471293652, pp. 381.
• Supplemental WEB page [data sets, program codes, resources, reference updates, and more] is under preparation. Please check the site: wiley.html for the leatest updates.
### VOLUME: BAYESIAN INFERENCE IN WAVELET BASED MODELS, Springer-Verlag, Lecture Notes in Statistics 141.
(ISBN 0-387-98885-8)
• Peter Müller and Brani Vidakovic are editors a volume on Bayesian inference in the wavelet (multiscale) domain. The volume is just out of press [June 1999] and some of the contributors include: Abramovich, Aguilar, Albertson, Berliner, Bultheel, Chipman, Clyde, Corradi, Cressie, George, Huang, Jansen, Kalifa, Katul, Kohn, Kolaczyk, Krim, Leporini, Lynch, Mallat, Marron, Milliff, Müller, Nowak, Ogden, Pastor, Pensky, Pesquet, Rios Insua, Rodriguez, Ruggeri, Sapatinas, Simoncelli, Vannucci, Vidakovic, Wang, Wikle, Wolfson, and Yau.
The back-cover of the volume reads: This volume provides a thorough introduction and reference for any researcher who is interested in Bayesian inference for wavelet-based models. To achieve this goal, the book starts with an extensive introductory chapter providing a self contained introduction to the use of wavelet decompositions, and the relation to Bayesian inference. The remaining papers in this volume are divided into six parts: independent prior modeling; decision theoretic aspects; dependent prior modeling; spatial models using bivariate wavelet bases; empirical Bayes approaches; and case studies. Chapters are written by experts who published the original research papers establishing the use of wavelet based models in Bayesian inference.
### ('97, '98 ) Statistics 294 at ISDS, Duke
• The course STA 294 is a special topic'' course. In Spring 1997 (1998) the topic (one of the two topics) was WAVELETS in STATISTICS
• Archive
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rhine-0.1.0.0: Functional Reactive Programming with type-level clocks
Universal schedule for the ScheduleT monad transformer
Two clocks in the ScheduleT monad transformer can always be canonically scheduled. Indeed, this is the purpose for which ScheduleT was defined.
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## § Radical ideals, nilpotents, and reduced rings
A radical ideal of a ring $R$ is an ideal such that $\forall r \in R, r^n \in I \implies r \in I$. That is, if any power of $r$ is in $I$, then the element $r$ also gets "sucked into" $I$.
#### § Nilpotent elements
A nilpotent element of a ring $R$ is any element $r$ such that there exists some power $n$ such that $r^n = 0$. Note that every ideal of the ring contains $0$. Hence, if an ideal $I$ of a ring is known to be a radical ideal, then for any nilpotent $r$, since $\exists n, r^n = 0 \in I$, since $I$ is radical, $r \in I$. That is, a radical ideal with always contain all nilpotents! It will contain other elements as well, but it will contain nilpotents for sure.
#### § Radicalization of an ideal
Given a ideal $I$, it's radical idea $\sqrt I \equiv \{ r \in R, r^n \in I \}$. That is, we add all the elements $I$ needs to have for it to become a radical. Notice that the radicalization of the zero ideal $I$ will precisely contain all nilpotents. that is, $\sqrt{(0)} \equiv \{ r \in R, r^n = 0\}$.
#### § Reduced rings
A ring $R$ is a reduced ring if the only nilpotent in the ring is $0$.
#### § creating reduced rings (removing nilpotents) by quotienting radical ideals
Tto remove nilpotents of the ring $R$, we can create $R' \equiv R / \sqrt{(0}$. Since $\sqrt{(0)}$ is the ideal which contains all nilpotents, the quotient ring $R'$ will contain no nilpotents other than $0$. Similarly, quotienting by any larger radical ideal $I$ will remove all nilpotents (and then some), leaving a reduced ring.
A ring modulo a radical ideal is reduced
#### § Integral domains
a Ring $R$ is an integral domain if $ab = 0 \implies a = 0 \lor b = 0$. That is, the ring $R$ has no zero divisors.
#### § Prime ideals
An ideal $I$ of a ring $R$ is a prime ideal if $\forall xy \in R, xy \in I \implies x \in I \lor y \in I$. This generalizes the notion of a prime number diving a composite: $p | xy \implies p | x \lor p | y$.
#### § creating integral domains by quotenting prime ideals
Recall that every ideal contains a $0$. Now, if an ideal $I$ is prime, and if $ab = 0 \in I$, then either $a \in I$ or $b \in I$ (by the definition of prime). We create $R' = R / I$. We denote $\overline{r} \in R'$ as the image of $r \in R$ in the quotient ring $R'$. The intuition is that quotienting by a $I$, since if $ab = 0 \implies a \in I \lor b \in I$, we are "forcing" that in the quotient ring $R'$, if $\overline{a} \overline{b} = 0$, then either $\overline{a} = 0$ or $\overline{b} = 0$, since $(a \in I \implies \overline a = 0)$, and $b \in I \implies \overline b = 0)$.
A ring modulo a prime ideal is an integral domain.
I learnt of this explanation from this
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# zbMATH — the first resource for mathematics
## Alpern, Steve
Compute Distance To:
Author ID: alpern.steve Published as: Alperin, Steve; Alpern, S.; Alpern, Steve; Alpern, Steven Homepage: http://www.wbs.ac.uk/about/person/steve-alpern/ External Links: MGP · Wikidata · ORCID · GND
Documents Indexed: 108 Publications since 1974, including 4 Books
all top 5
#### Co-Authors
34 single-authored 18 Prasad, Vidhu S. 10 Fokkink, Robbert Johan 9 Gal, Shmuel 8 Lidbetter, Thomas F. 6 Baston, Vic J. 5 Beck, Anatole 4 Lim, Wei Shi 4 Papadaki, Katerina P. 3 Howard, John V. 3 Lindelauf, Roy 3 Morton, Alec 3 Reyniers, Diane J. 2 Ašić, Miroslav D. 2 Katrantzi, Ioanna 2 Olsder, Geert Jan 1 Auslander, Joseph 1 Chen, Bo 1 Chen, Bo 1 Choksi, J. R. 1 Edwards, Robert Duncan 1 Essegaier, Skander 1 Gąsieniec, Leszek Antoni 1 Kikuta, Ken 1 Leone, Pierre 1 Lidbetter, Tom 1 op den Kelder, Joram 1 Pelekis, Christos 1 Pikounis, Michael 1 Ramsey, David Mark 1 Silva, Cesar E. 1 Simanjuntak, Martin 1 Snower, Dennis J. 1 Solan, Eilon 1 Subrahmanian, V. S. 1 Timmer, Marco
all top 5
#### Serials
12 Operations Research 12 SIAM Journal on Control and Optimization 10 European Journal of Operational Research 4 Networks 3 Journal of Mathematical Analysis and Applications 3 The Annals of Probability 3 Bulletin of the London Mathematical Society 3 Canadian Journal of Mathematics 3 Naval Research Logistics 3 Proceedings of the American Mathematical Society 3 Ergodic Theory and Dynamical Systems 2 American Mathematical Monthly 2 Advances in Mathematics 2 Indiana University Mathematics Journal 2 International Journal of Game Theory 2 Inventiones Mathematicae 2 Journal of Applied Probability 2 Mathematics of Operations Research 2 Theory and Decision 2 Dynamics and Control 2 Cambridge Tracts in Mathematics 1 Mathematical Proceedings of the Cambridge Philosophical Society 1 Journal of Combinatorial Theory. Series B 1 Journal of the London Mathematical Society. Second Series 1 Journal of Optimization Theory and Applications 1 Mathematika 1 Theoretical Computer Science 1 Topology and its Applications 1 Stochastic Analysis and Applications 1 Annals of Operations Research 1 Games and Economic Behavior 1 International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 1 International Game Theory Review 1 International Series in Operations Research & Management Science 1 Dynamic Games and Applications 1 Journal of Theoretical Biology
all top 5
#### Fields
69 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 39 Operations research, mathematical programming (90-XX) 20 Measure and integration (28-XX) 15 Dynamical systems and ergodic theory (37-XX) 12 General topology (54-XX) 5 Probability theory and stochastic processes (60-XX) 4 Combinatorics (05-XX) 4 Calculus of variations and optimal control; optimization (49-XX) 3 Global analysis, analysis on manifolds (58-XX) 2 Linear and multilinear algebra; matrix theory (15-XX) 2 Systems theory; control (93-XX) 1 General and overarching topics; collections (00-XX) 1 History and biography (01-XX) 1 Manifolds and cell complexes (57-XX) 1 Computer science (68-XX) 1 Biology and other natural sciences (92-XX)
#### Citations contained in zbMATH
93 Publications have been cited 658 times in 278 Documents Cited by Year
The theory of search games and rendezvous. Zbl 1034.91017
Alpern, Steve; Gal, Shmuel
2003
The rendezvous search problem. Zbl 0837.90074
Alpern, Steve
1995
Typical dynamics of volume preserving homeomorphisms. Zbl 0970.37001
Alpern, Steve; Prasad, V. S.
2000
Minimax rendezvous on the line. Zbl 0857.90063
Lim, Wei Shi; Alpern, Steve
1996
Rendezvous search on labeled networks. Zbl 1008.91011
Alpern, Steve
2002
Rendezvous search on a graph. Zbl 0940.90038
Alpern, Steve; Baston, V. J.; Essegaier, Skander
1999
Patrolling games. Zbl 1233.91063
Alpern, Steve; Morton, Alec; Papadaki, Katerina
2011
Rendezvous search: A personal perspective. Zbl 1163.91324
Alpern, Steve
2002
Mining coal or finding terrorists: the expanding search paradigm. Zbl 1273.90221
Alpern, Steve; Lidbetter, Thomas
2013
Rendezvous search on the line with distinguishable players. Zbl 0837.90075
Alpern, Steve; Gal, Shmuel
1995
Search theory. A game theoretic perspective. Zbl 1263.91001
Alpern, Steve (ed.); Fokkink, Robbert (ed.); Gąsieniec, Leszek (ed.); Lindelauf, Roy (ed.); Subrahmanian, V. S. (ed.)
2013
Network search games with immobile hider, without a designated searcher starting point. Zbl 1151.91028
Alpern, Steve; Baston, Vic; Gal, Shmuel
2008
Search games on trees with asymmetric travel times. Zbl 1217.91021
Alpern, Steve
2010
The search value of a network. Zbl 0581.90040
Alpern, Steve; Asic, Miroslav
1985
Return times and conjugates of an antiperiodic transformation. Zbl 0496.28014
Alpern, Steve
1981
Optimal trade-off between speed and acuity when searching for a small object. Zbl 1377.91031
Alpern, Steve; Lidbetter, Thomas
2015
Searching a variable speed network. Zbl 1308.90072
Alpern, Steve; Lidbetter, Thomas
2014
Alternating search at two locations. Zbl 0996.90049
Alpern, Steve; Howard, J. V.
2000
Approximation to and by measure preserving homeomorphisms. Zbl 0405.28014
Alpern, Steve
1978
New proofs that weak mixing is generic. Zbl 0338.28012
Alpern, Steven
1976
Hide-and-seek games on a network, using combinatorial search paths. Zbl 1411.91108
Alpern, Steve
2017
Searching symmetric networks with Utilitarian-Postman paths. Zbl 1208.05143
Alpern, Steve; Baston, Vic; Gal, Shmuel
2009
Hide-and-seek games on a tree to which Eulerian networks are attached. Zbl 1180.91055
Alpern, Steve
2008
Find-and-fetch search on a tree. Zbl 1233.91046
Alpern, Steve
2011
Infiltration games on arbitrary graphs. Zbl 0784.90112
Alpern, Steve
1992
A mixed-strategy minimax theorem without compactness. Zbl 0662.90092
Alpern, Steve; Gal, Shmuel
1988
Ambush strategies in search games on graphs. Zbl 0596.90110
Alpern, Steve; Asic, Miroslav
1986
Search for point in interval, with high-low feedback. Zbl 0579.90056
Alpern, Steve
1985
Patrolling a border. Zbl 1354.91029
Papadaki, Katerina; Alpern, Steve; Lidbetter, Thomas; Morton, Alec
2016
A new approach to Gal’s theory of search games on weakly Eulerian networks. Zbl 1252.91023
Alpern, Steve
2011
Rendezvous on a planar lattice. Zbl 1165.91336
Alpern, Steve; Baston, Vic
2005
Maximally chaotic homeomorphisms of sigma-compact manifolds. Zbl 0977.54033
Alpern, Steve; Prasad, V. S.
2000
Rendezvous search on the line with more than two players. Zbl 0893.90094
Lim, Wei Shi; Alpern, Steve; Beck, Anatole
1997
On search games that include ambush. Zbl 1285.91004
Alpern, Steve; Fokkink, Robbert; Gal, Shmuel; Timmer, Marco
2013
On Ruckle’s conjecture on accumulation games. Zbl 1219.91008
Alpern, Steve; Fokkink, Robbert; Kikuta, Ken
2010
The “princess and monster” game on an interval. Zbl 1162.91008
Alpern, Steve; Fokkink, Robbert; Lindelauf, Roy; Olsder, Geert-Jan
2008
Asymmetric rendezvous on the line is a double linear search problem. Zbl 0944.90032
Alpern, Steve; Beck, Anatole
1999
Rotational representations of stochastic matrices. Zbl 0548.15018
Alpern, Steve
1983
The search game with mobile hider on the circle. Zbl 0312.90069
Alpern, Steve
1974
A sequential selection game with vetoes. Zbl 1197.91043
Alpern, Steve; Gal, Shmuel; Solan, Eilon
2010
Rendezvous in higher dimensions. Zbl 1151.90445
Alpern, Steve; Baston, Vic
2006
Asymmetric rendezvous search on the circle. Zbl 0987.90047
Alpern, Steve
2000
Typical recurrence for lifts of mean rotation zero annulus homeomorphisms. Zbl 0744.28015
Alpern, Steve; Prasad, V. S.
1991
Games with repeated decisions. Zbl 0661.90106
Alpern, Steve
1988
Approximate solutions for expanding search games on general networks. Zbl 1426.91046
Alpern, Steve; Lidbetter, Thomas
2019
Accumulation games on graphs. Zbl 1390.90364
Alpern, Steve; Fokkink, Robbert
2014
Disperse or unite? A mathematical model of coordinated attack. Zbl 1298.91054
Alpern, Steve; Fokkink, Robbert; op den Kelder, Joram; Lidbetter, Tom
2010
Equilibria of two-sided matching games with common preferences. Zbl 1176.90265
Alpern, Steve; Katrantzi, Ioanna
2009
Searching for an agent who May OR May not want to be found. Zbl 1163.90524
Alpern, Steve; Gal, Shmuel
2002
Spatial dispersion as a dynamic coordination problem. Zbl 1030.91045
Alpern, Steve; Reyniers, Diane J.
2002
Rendezvous search on the line with limited resources: Maximizing the probability of meeting. Zbl 1042.91516
Alperin, Steve; Beck, Anatole
1999
The symmetric rendezvous-evasion game. Zbl 0911.90374
Alpern, Steve; Lim, Wei Shi
1998
Typical transitivity for lifts of rotationless annulus or torus homeomorphisms. Zbl 0826.54031
Alpern, Steve; Prasad, V. S.
1995
End behaviour and ergodicity for homeomorphisms of manifolds with finitely many ends. Zbl 0628.58025
Alpern, S.; Prasad, V.
1987
Nonstable ergodic homeomorphisms of $$R^ 4$$. Zbl 0548.28005
Alpern, Steve
1983
Lusin’s theorem for measure preserving homeomorphisms. Zbl 0442.28024
Alpern, Steve; Edwards, Robert D.
1979
Search for an immobile hider in a known subset of a network. Zbl 1435.91033
Alpern, Steve
2019
The importance of voting order for jury decisions by sequential majority voting. Zbl 1394.91113
Alpern, Steve; Chen, Bo
2017
Patrolling a pipeline. Zbl 06667367
Alpern, Steve; Lidbetter, Thomas; Morton, Alec; Papadaki, Katerina
2016
Optimal search and ambush for a hider who can escape the search region. Zbl 1348.90345
Alpern, Steve; Fokkink, Robbert; Simanjuntak, Martin
2016
A proof of the Kikuta-Ruckle conjecture on cyclic caching of resources. Zbl 1248.91026
Alpern, Steve; Fokkink, Robbert; Pelekis, Christos
2012
Typical dynamics of volume preserving homeomorphisms. Reprint of the 2000 hardback original. Zbl 1208.37001
Alpern, Steve; Prasad, V. S.
2011
Analysis and design of selection committees: a game theoretic secretary problem. Zbl 1211.91055
Alpern, Steve; Gal, Shmuel
2009
Multitowers, conjugacies and codes: three theorems in ergodic theory, one variation on Rokhlin’s Lemma. Zbl 1151.37003
Alpern, S.; Prasad, V. S.
2008
A common notion of clockwise can help in planar rendezvous. Zbl 1142.90414
Alpern, Steve; Baston, Vic
2006
Properties generic for Lebesgue space automorphisms are generic for measure-preserving manifold homeomorphisms. Zbl 1025.37002
Alpern, Steve; Prasad, V. S.
2002
Combinatorial proofs of the Conley-Zehnder-Franks theorem on a fixed point for torus homeomorphisms. Zbl 0781.58013
Alpern, Steve; Prasad, V. S.
1993
Hex games and twist maps on the annulus. Zbl 0786.90108
Alpern, Steve; Beck, Anatole
1991
Cycles in extensive form perfect information games. Zbl 0745.90098
Alpern, Steve
1991
Coding a stationary process to one with prescribed marginals. Zbl 0691.60032
Alpern, S.; Prasad, V. S.
1989
Measure preserving homeomorphisms of $$\mathbb{R}^n$$. Zbl 0442.28025
Alpern, Steve
1979
Generic properties of measure preserving homeomorphisms. Zbl 0439.28012
Alpern, Steve
1979
Dyadic decompositions of the cube. Zbl 0358.05021
Alpern, Steven
1975
Optimizing periodic patrols against short attacks on the line and other networks. Zbl 1403.90392
Alpern, Steve; Lidbetter, Thomas; Papadaki, Katerina
2019
Rendezvous search with markers that can be dropped at chosen times. Zbl 1419.91007
Leone, Pierre; Alpern, Steve
2018
Winner-take-all games: the strategic optimisation of rank. Zbl 1411.91160
Alpern, Steve; Howard, J. V.
2017
Partnership formation with age-dependent preferences. Zbl 1292.91035
Alpern, S.; Katrantzi, I.; Ramsey, D. M.
2013
A numerical approach to the “princess and monster” game on an interval. Zbl 1179.91015
Alpern, Steve; Fokkink, Robbert; Lindelauf, Roy; Olsder, Geert Jan
2009
Line-of-sight rendezvous. Zbl 1144.90496
Alpern, Steve
2008
Rotational (and other) representations of stochastic matrices. Zbl 1152.60056
Alpern, Steve; Prasad, V. S.
2008
Rendezvous search with revealed information: applications to the line. Zbl 1171.90415
Alpern, Steve
2007
Strategic mating with common preferences. Zbl 1445.92313
Alpern, Steve; Reyniers, Diane
2005
Combinatorial approximation by Devaney-chaotic or periodic volume preserving homeomorphisms. Zbl 1089.37503
Alpern, Steve
1999
Chaotic homeomorphisms of $$R^n$$, lifted from torus homeomorphisms. Zbl 0923.54034
Alpern, Steve; Prasad, V. S.
1999
Rendezvous search on the line with bounded resources: Expected time minimization. Zbl 0930.90043
Alpern, Steve; Beck, Anatole
1997
Topological ergodic theory and mean rotation. Zbl 0792.54017
Alpern, Steve; Prasad, V. S.
1993
Return times for nonsingular measurable transformations. Zbl 0716.28010
Alpern, S.; Prasad, V. S.
1990
Area-preserving homeomorphisms of the open disk without fixed points. Zbl 0655.54031
Alpern, Steve
1988
Conjugates of infinite measure preserving transformations. Zbl 0653.28011
Alpern, S.; Choksi, J. R.; Prasad, V. S.
1988
Dynamics induced on the ends of a non-compact manifold. Zbl 0651.58016
Alpern, Steve; Prasad, V. S.
1988
Conjecture: In general a mixing transformation is not two-fold mixing. Zbl 0574.28012
Alpern, Steve
1985
Superhamiltonian graphs. Zbl 0392.05043
Alpern, Steven
1978
A topological analog of Halmos’ conjugacy lemma. Zbl 0381.28007
Alpern, Steve
1978
Approximate solutions for expanding search games on general networks. Zbl 1426.91046
Alpern, Steve; Lidbetter, Thomas
2019
Search for an immobile hider in a known subset of a network. Zbl 1435.91033
Alpern, Steve
2019
Optimizing periodic patrols against short attacks on the line and other networks. Zbl 1403.90392
Alpern, Steve; Lidbetter, Thomas; Papadaki, Katerina
2019
Rendezvous search with markers that can be dropped at chosen times. Zbl 1419.91007
Leone, Pierre; Alpern, Steve
2018
Hide-and-seek games on a network, using combinatorial search paths. Zbl 1411.91108
Alpern, Steve
2017
The importance of voting order for jury decisions by sequential majority voting. Zbl 1394.91113
Alpern, Steve; Chen, Bo
2017
Winner-take-all games: the strategic optimisation of rank. Zbl 1411.91160
Alpern, Steve; Howard, J. V.
2017
Patrolling a border. Zbl 1354.91029
Papadaki, Katerina; Alpern, Steve; Lidbetter, Thomas; Morton, Alec
2016
Patrolling a pipeline. Zbl 06667367
Alpern, Steve; Lidbetter, Thomas; Morton, Alec; Papadaki, Katerina
2016
Optimal search and ambush for a hider who can escape the search region. Zbl 1348.90345
Alpern, Steve; Fokkink, Robbert; Simanjuntak, Martin
2016
Optimal trade-off between speed and acuity when searching for a small object. Zbl 1377.91031
Alpern, Steve; Lidbetter, Thomas
2015
Searching a variable speed network. Zbl 1308.90072
Alpern, Steve; Lidbetter, Thomas
2014
Accumulation games on graphs. Zbl 1390.90364
Alpern, Steve; Fokkink, Robbert
2014
Mining coal or finding terrorists: the expanding search paradigm. Zbl 1273.90221
Alpern, Steve; Lidbetter, Thomas
2013
Search theory. A game theoretic perspective. Zbl 1263.91001
Alpern, Steve (ed.); Fokkink, Robbert (ed.); Gąsieniec, Leszek (ed.); Lindelauf, Roy (ed.); Subrahmanian, V. S. (ed.)
2013
On search games that include ambush. Zbl 1285.91004
Alpern, Steve; Fokkink, Robbert; Gal, Shmuel; Timmer, Marco
2013
Partnership formation with age-dependent preferences. Zbl 1292.91035
Alpern, S.; Katrantzi, I.; Ramsey, D. M.
2013
A proof of the Kikuta-Ruckle conjecture on cyclic caching of resources. Zbl 1248.91026
Alpern, Steve; Fokkink, Robbert; Pelekis, Christos
2012
Patrolling games. Zbl 1233.91063
Alpern, Steve; Morton, Alec; Papadaki, Katerina
2011
Find-and-fetch search on a tree. Zbl 1233.91046
Alpern, Steve
2011
A new approach to Gal’s theory of search games on weakly Eulerian networks. Zbl 1252.91023
Alpern, Steve
2011
Typical dynamics of volume preserving homeomorphisms. Reprint of the 2000 hardback original. Zbl 1208.37001
Alpern, Steve; Prasad, V. S.
2011
Search games on trees with asymmetric travel times. Zbl 1217.91021
Alpern, Steve
2010
On Ruckle’s conjecture on accumulation games. Zbl 1219.91008
Alpern, Steve; Fokkink, Robbert; Kikuta, Ken
2010
A sequential selection game with vetoes. Zbl 1197.91043
Alpern, Steve; Gal, Shmuel; Solan, Eilon
2010
Disperse or unite? A mathematical model of coordinated attack. Zbl 1298.91054
Alpern, Steve; Fokkink, Robbert; op den Kelder, Joram; Lidbetter, Tom
2010
Searching symmetric networks with Utilitarian-Postman paths. Zbl 1208.05143
Alpern, Steve; Baston, Vic; Gal, Shmuel
2009
Equilibria of two-sided matching games with common preferences. Zbl 1176.90265
Alpern, Steve; Katrantzi, Ioanna
2009
Analysis and design of selection committees: a game theoretic secretary problem. Zbl 1211.91055
Alpern, Steve; Gal, Shmuel
2009
A numerical approach to the “princess and monster” game on an interval. Zbl 1179.91015
Alpern, Steve; Fokkink, Robbert; Lindelauf, Roy; Olsder, Geert Jan
2009
Network search games with immobile hider, without a designated searcher starting point. Zbl 1151.91028
Alpern, Steve; Baston, Vic; Gal, Shmuel
2008
Hide-and-seek games on a tree to which Eulerian networks are attached. Zbl 1180.91055
Alpern, Steve
2008
The “princess and monster” game on an interval. Zbl 1162.91008
Alpern, Steve; Fokkink, Robbert; Lindelauf, Roy; Olsder, Geert-Jan
2008
Multitowers, conjugacies and codes: three theorems in ergodic theory, one variation on Rokhlin’s Lemma. Zbl 1151.37003
Alpern, S.; Prasad, V. S.
2008
Line-of-sight rendezvous. Zbl 1144.90496
Alpern, Steve
2008
Rotational (and other) representations of stochastic matrices. Zbl 1152.60056
Alpern, Steve; Prasad, V. S.
2008
Rendezvous search with revealed information: applications to the line. Zbl 1171.90415
Alpern, Steve
2007
Rendezvous in higher dimensions. Zbl 1151.90445
Alpern, Steve; Baston, Vic
2006
A common notion of clockwise can help in planar rendezvous. Zbl 1142.90414
Alpern, Steve; Baston, Vic
2006
Rendezvous on a planar lattice. Zbl 1165.91336
Alpern, Steve; Baston, Vic
2005
Strategic mating with common preferences. Zbl 1445.92313
Alpern, Steve; Reyniers, Diane
2005
The theory of search games and rendezvous. Zbl 1034.91017
Alpern, Steve; Gal, Shmuel
2003
Rendezvous search on labeled networks. Zbl 1008.91011
Alpern, Steve
2002
Rendezvous search: A personal perspective. Zbl 1163.91324
Alpern, Steve
2002
Searching for an agent who May OR May not want to be found. Zbl 1163.90524
Alpern, Steve; Gal, Shmuel
2002
Spatial dispersion as a dynamic coordination problem. Zbl 1030.91045
Alpern, Steve; Reyniers, Diane J.
2002
Properties generic for Lebesgue space automorphisms are generic for measure-preserving manifold homeomorphisms. Zbl 1025.37002
Alpern, Steve; Prasad, V. S.
2002
Typical dynamics of volume preserving homeomorphisms. Zbl 0970.37001
Alpern, Steve; Prasad, V. S.
2000
Alternating search at two locations. Zbl 0996.90049
Alpern, Steve; Howard, J. V.
2000
Maximally chaotic homeomorphisms of sigma-compact manifolds. Zbl 0977.54033
Alpern, Steve; Prasad, V. S.
2000
Asymmetric rendezvous search on the circle. Zbl 0987.90047
Alpern, Steve
2000
Rendezvous search on a graph. Zbl 0940.90038
Alpern, Steve; Baston, V. J.; Essegaier, Skander
1999
Asymmetric rendezvous on the line is a double linear search problem. Zbl 0944.90032
Alpern, Steve; Beck, Anatole
1999
Rendezvous search on the line with limited resources: Maximizing the probability of meeting. Zbl 1042.91516
Alperin, Steve; Beck, Anatole
1999
Combinatorial approximation by Devaney-chaotic or periodic volume preserving homeomorphisms. Zbl 1089.37503
Alpern, Steve
1999
Chaotic homeomorphisms of $$R^n$$, lifted from torus homeomorphisms. Zbl 0923.54034
Alpern, Steve; Prasad, V. S.
1999
The symmetric rendezvous-evasion game. Zbl 0911.90374
Alpern, Steve; Lim, Wei Shi
1998
Rendezvous search on the line with more than two players. Zbl 0893.90094
Lim, Wei Shi; Alpern, Steve; Beck, Anatole
1997
Rendezvous search on the line with bounded resources: Expected time minimization. Zbl 0930.90043
Alpern, Steve; Beck, Anatole
1997
Minimax rendezvous on the line. Zbl 0857.90063
Lim, Wei Shi; Alpern, Steve
1996
The rendezvous search problem. Zbl 0837.90074
Alpern, Steve
1995
Rendezvous search on the line with distinguishable players. Zbl 0837.90075
Alpern, Steve; Gal, Shmuel
1995
Typical transitivity for lifts of rotationless annulus or torus homeomorphisms. Zbl 0826.54031
Alpern, Steve; Prasad, V. S.
1995
Combinatorial proofs of the Conley-Zehnder-Franks theorem on a fixed point for torus homeomorphisms. Zbl 0781.58013
Alpern, Steve; Prasad, V. S.
1993
Topological ergodic theory and mean rotation. Zbl 0792.54017
Alpern, Steve; Prasad, V. S.
1993
Infiltration games on arbitrary graphs. Zbl 0784.90112
Alpern, Steve
1992
Typical recurrence for lifts of mean rotation zero annulus homeomorphisms. Zbl 0744.28015
Alpern, Steve; Prasad, V. S.
1991
Hex games and twist maps on the annulus. Zbl 0786.90108
Alpern, Steve; Beck, Anatole
1991
Cycles in extensive form perfect information games. Zbl 0745.90098
Alpern, Steve
1991
Return times for nonsingular measurable transformations. Zbl 0716.28010
Alpern, S.; Prasad, V. S.
1990
Coding a stationary process to one with prescribed marginals. Zbl 0691.60032
Alpern, S.; Prasad, V. S.
1989
A mixed-strategy minimax theorem without compactness. Zbl 0662.90092
Alpern, Steve; Gal, Shmuel
1988
Games with repeated decisions. Zbl 0661.90106
Alpern, Steve
1988
Area-preserving homeomorphisms of the open disk without fixed points. Zbl 0655.54031
Alpern, Steve
1988
Conjugates of infinite measure preserving transformations. Zbl 0653.28011
Alpern, S.; Choksi, J. R.; Prasad, V. S.
1988
Dynamics induced on the ends of a non-compact manifold. Zbl 0651.58016
Alpern, Steve; Prasad, V. S.
1988
End behaviour and ergodicity for homeomorphisms of manifolds with finitely many ends. Zbl 0628.58025
Alpern, S.; Prasad, V.
1987
Ambush strategies in search games on graphs. Zbl 0596.90110
Alpern, Steve; Asic, Miroslav
1986
The search value of a network. Zbl 0581.90040
Alpern, Steve; Asic, Miroslav
1985
Search for point in interval, with high-low feedback. Zbl 0579.90056
Alpern, Steve
1985
Conjecture: In general a mixing transformation is not two-fold mixing. Zbl 0574.28012
Alpern, Steve
1985
Rotational representations of stochastic matrices. Zbl 0548.15018
Alpern, Steve
1983
Nonstable ergodic homeomorphisms of $$R^ 4$$. Zbl 0548.28005
Alpern, Steve
1983
Return times and conjugates of an antiperiodic transformation. Zbl 0496.28014
Alpern, Steve
1981
Lusin’s theorem for measure preserving homeomorphisms. Zbl 0442.28024
Alpern, Steve; Edwards, Robert D.
1979
Measure preserving homeomorphisms of $$\mathbb{R}^n$$. Zbl 0442.28025
Alpern, Steve
1979
Generic properties of measure preserving homeomorphisms. Zbl 0439.28012
Alpern, Steve
1979
Approximation to and by measure preserving homeomorphisms. Zbl 0405.28014
Alpern, Steve
1978
Superhamiltonian graphs. Zbl 0392.05043
Alpern, Steven
1978
A topological analog of Halmos’ conjugacy lemma. Zbl 0381.28007
Alpern, Steve
1978
New proofs that weak mixing is generic. Zbl 0338.28012
Alpern, Steven
1976
Dyadic decompositions of the cube. Zbl 0358.05021
Alpern, Steven
1975
The search game with mobile hider on the circle. Zbl 0312.90069
Alpern, Steve
1974
all top 5
#### Cited by 364 Authors
41 Alpern, Steve 21 Czyzowicz, Jurek 20 Pelc, Andrzej 18 Lidbetter, Thomas F. 12 Kranakis, Evangelos Konstantinou 10 Dieudonné, Yoann 9 Baston, Vic J. 8 Gąsieniec, Leszek Antoni 8 Klasing, Ralf 8 Kosowski, Adrian 8 Krizanc, Danny 7 Fokkink, Robbert Johan 7 Gal, Shmuel 7 Kikuta, Kensaku 6 Prasad, Vidhu S. 6 Zoroa Alonso, Noemi 6 Zoroa Terol, Procopio 5 Angelopoulos, Spyros 5 Bampas, Evangelos 5 Chalopin, Jérémie 5 Das, Shantanu 5 Ilcinkas, David 5 Navarra, Alfredo 5 Tal, Fábio Armando 4 Bouchard, Sébastien 4 Di Stefano, Gabriele 4 Dobrev, Stefan 4 Flocchini, Paola 4 Georgiou, Konstantinos 4 Labourel, Arnaud 4 Lin, Kyle Y. 4 Markou, Euripides 4 Pajak, Dominik 4 Santoro, Nicola 3 Addas-Zanata, Salvador 3 Bose, Prosenjit K. 3 D’Angelo, Gianlorenzo 3 de Carufel, Jean-Lou 3 Dereniowski, Dariusz 3 Dürr, Christoph 3 Fernández-Sáez, M. José 3 Fernández-Sáez, María-José 3 Garnaev, Andrej Yu. 3 Guihéneuf, Pierre-Antoine 3 Kalpazidou, Sophia L. 3 López-Ortiz, Alejandro 3 MacQuarrie, Fraser 3 Narayanan, Lata 3 Opatrny, Jaroslav 3 Papadaki, Katerina P. 3 Pelekis, Christos 3 Ramsey, David Mark 2 Arsénio, Diogo 2 Basilico, Nicola 2 Bezuglyi, Sergey I. 2 Bingham, Nicholas Hugh 2 Blume, Andreas 2 Brandt, Sebastian F. 2 Csóka, Endre 2 Di Luna, Giuseppe Antonio 2 Ducourthial, Bertrand 2 Garnaev, Andrey Yu. 2 Garrec, Tristan 2 Gatti, Nicola 2 Gusev, Vasiliĭ Vasil’evich 2 Hassin, Refael 2 Heggernes, Pinar 2 Karpel, Olena 2 Kerr, David 2 Killick, Ryan 2 Kociumaka, Tomasz 2 Kuszner, Łukasz 2 Kutten, Shay 2 Martin, Russell A. 2 Mihai, Rodica 2 Mihalák, Matúš 2 Miller, Avery 2 Morton, Alec 2 Ostaszewski, Adam J. 2 Petit, Franck 2 Prencipe, Giuseppe 2 Prikhod’ko, Aleksandr A. 2 Ruckle, William Henry 2 Şahin, Ayşe Arzu 2 Shende, Sunil M. 2 Solan, Eilon 2 Thilikos, Dimitrios M. 2 Viglietta, Giovanni 2 Wattenhofer, Roger P. 2 Weiss, Benjamin 2 Widmayer, Peter 2 Yagasaki, Tatsuhiko 2 Yingst, Andrew Q. 1 Abdenur, Flavio 1 Agmon, Noa 1 Akin, Ethan 1 Amigoni, Francesco 1 Anaya, Julian 1 Andersson, Martin 1 Araujo, Jesús ...and 264 more Authors
all top 5
#### Cited in 83 Serials
36 Theoretical Computer Science 32 European Journal of Operational Research 13 Distributed Computing 9 Naval Research Logistics 8 Topology and its Applications 7 Proceedings of the American Mathematical Society 7 Ergodic Theory and Dynamical Systems 7 Algorithmica 6 Games and Economic Behavior 5 International Journal of Game Theory 5 Operations Research 4 Discrete Applied Mathematics 4 Networks 4 Transactions of the American Mathematical Society 3 Artificial Intelligence 3 Archive for Rational Mechanics and Analysis 3 Journal of Mathematical Analysis and Applications 3 Inventiones Mathematicae 3 Journal of Computer and System Sciences 3 Journal of Optimization Theory and Applications 3 Mathematics of Operations Research 3 Stochastic Analysis and Applications 3 Annals of Operations Research 3 Theory of Computing Systems 3 Qualitative Theory of Dynamical Systems 2 Mathematical Notes 2 Journal of Functional Analysis 2 SIAM Journal on Control and Optimization 2 Economics Letters 2 Indagationes Mathematicae. New Series 2 Mathematical Methods of Operations Research 2 Dynamical Systems 2 Dynamic Games and Applications 1 Communications in Mathematical Physics 1 Discrete Mathematics 1 Information Processing Letters 1 Israel Journal of Mathematics 1 Nonlinearity 1 Bulletin of the London Mathematical Society 1 Journal of Applied Probability 1 Journal of Combinatorial Theory. Series B 1 Journal of Economic Theory 1 Journal of the Mathematical Society of Japan 1 Journal of Pure and Applied Algebra 1 Manuscripta Mathematica 1 Mathematische Annalen 1 Mathematische Zeitschrift 1 Mathematika 1 SIAM Journal on Computing 1 Theory and Decision 1 Statistics & Probability Letters 1 Operations Research Letters 1 Annales de l’Institut Henri Poincaré. Analyse Non Linéaire 1 Computers & Operations Research 1 Random Structures & Algorithms 1 International Journal of Foundations of Computer Science 1 The Journal of Geometric Analysis 1 Discrete Mathematics and Applications 1 Geometric and Functional Analysis. GAFA 1 Automation and Remote Control 1 Proceedings of the Indian Academy of Sciences. Mathematical Sciences 1 Boletim da Sociedade Brasileira de Matemática. Nova Série 1 International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 1 Computational Optimization and Applications 1 NoDEA. Nonlinear Differential Equations and Applications 1 Annals of Mathematics and Artificial Intelligence 1 Journal of Difference Equations and Applications 1 Journal of Scheduling 1 Journal of the European Mathematical Society (JEMS) 1 Journal of Dynamical and Control Systems 1 Probability in the Engineering and Informational Sciences 1 Proceedings of the Steklov Institute of Mathematics 1 Journal of Modern Dynamics 1 Nonlinear Analysis. Hybrid Systems 1 Games 1 Communications in Mathematics and Statistics 1 Journal of the Operations Research Society of China 1 Computer Science Review 1 Journal of Mathematics 1 Mathematica Applicanda 1 Quantum Studies: Mathematics and Foundations 1 Nonlinear Analysis. Theory, Methods & Applications 1 Pacific Journal of Mathematics for Industry
all top 5
#### Cited in 32 Fields
107 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 90 Computer science (68-XX) 71 Operations research, mathematical programming (90-XX) 50 Dynamical systems and ergodic theory (37-XX) 24 Measure and integration (28-XX) 23 Combinatorics (05-XX) 12 Probability theory and stochastic processes (60-XX) 11 General topology (54-XX) 6 Functional analysis (46-XX) 6 Calculus of variations and optimal control; optimization (49-XX) 4 Real functions (26-XX) 4 Partial differential equations (35-XX) 4 Systems theory; control (93-XX) 3 Order, lattices, ordered algebraic structures (06-XX) 3 Manifolds and cell complexes (57-XX) 2 History and biography (01-XX) 2 Linear and multilinear algebra; matrix theory (15-XX) 2 Topological groups, Lie groups (22-XX) 2 Operator theory (47-XX) 2 Convex and discrete geometry (52-XX) 2 Global analysis, analysis on manifolds (58-XX) 2 Statistics (62-XX) 1 Mathematical logic and foundations (03-XX) 1 Category theory; homological algebra (18-XX) 1 Group theory and generalizations (20-XX) 1 Several complex variables and analytic spaces (32-XX) 1 Geometry (51-XX) 1 Fluid mechanics (76-XX) 1 Optics, electromagnetic theory (78-XX) 1 Quantum theory (81-XX) 1 Biology and other natural sciences (92-XX) 1 Information and communication theory, circuits (94-XX)
#### Wikidata Timeline
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# Secure Random Generators (CSPRNG)
Cryptography secure pseudo-random number generators (CSPRNG) are random generators, which guarantee that the random numbers coming from them are absolutely unpredictable. CSPRNG satisfy the next-bit test and withstand the state compromise extensions and are typically part of the operating system or come from secure external source. Depending on the level of security required, CSPRNG can be implemented as software components or as hardware devices or as combination of both.
For example, in the credit card printing centers the formal security regulations require certified hardware random generators to be used to generate credit card PIN codes, private keys and other data, designed to remain private.
Modern operating systems (OS) collect entropy (initial seed) from the environmental noise: keyboard clicks, mouse moves, network activity, system I/O interruptions, hard disk activity, etc. Sources of randomness from the environment in Linux, for example, include inter-keyboard timings, inter-interrupt timings from some interrupts, and other events which are both non-deterministic and hard to measure for an outside observer.
The collected in the OS randomness is usually accessible from /dev/random and /dev/urandom.
• Reading from the /dev/random file (the limited blocking random generator) returns entropy from the kernel's entropy pool (collected noise) and blocks when the entropy pool is empty until additional environmental noise is gathered.
• Reading the /dev/urandom file (the unlimited non-blocking random generator) returns entropy from the kernel's entropy pool or a pseudo-random data, generated from previously collected environmental noise, which is also unpredictable, but is based on secure entropy "stretching" algorithm.
Usually a CSPRNG should start from an unpredictable random seed from the operating system, from a specialized hardware or from external source. Random numbers after the seed initialization are typically produces by a pseudo-random computation, but this does not compromise the security. Most algorithms often "reseed" the CSPRNG random generator when a new entropy comes, to make their work even more unpredictable.
Typically modern OS CSPRNG APIs combine the constantly collected entropy from the environment with the internal state of their built-in pseudo-random algorithm with continuous reseeding to guarantee maximal unpredictability of the generated randomness with high speed and non-blocking behavior in the same time.
## Hardware Random Generators (TRNG)
Hardware random generators, known as true random number generators (TRNG), typically capture physical processes or phenomenа, such as the visible spectrum of the light, the thermal noise from the environment, the atmosphere noise, etc. The randomness from the physical environment is collected through specialized sensors, then amplified and processed by the device and finally transmitted to the computer through USB, PCI Express or other standard interface.
Modern microprocessors (CPU) provide a built-in hardware random generator, accessible through a special CPU instruction RdRand, which return a random integer into one of the CPU registers.
Most cryptographic applications today do not require a hardware random generator, because the entropy in the operating system is secure enough for general cryptographic purposes. Using a TRNG is needed for systems with higher security requirements, such as banking and finance applications, certification authorities and high volume payment processors.
## How as a Developer to Access the CSPRNG?
Typically developers access the cryptographically strong random number generators (CSPRNG) for their OS from a cryptography library for their language and platform.
• In Linux and macOS, it is considered that both /dev/random and /dev/urandom sources of randomness are secure enough for most cryptographic purposes and most cryptographic libraries access them internally.
• In Windows, random numbers for cryptographic purposes can be securely generated using the BCryptGenRandom function from the Cryptography API: Next Generation (CNG) or higher level crypto libraries.
• In C# use System.Security.Cryptography.RandomNumberGenerator.Create() from .NET Framework or .NET Core.
• In Python use os.urandom() or the secrets library.
• In Java use the java.security.SecureRandom system class.
• In JavaScript use window.crypto.getRandomValues(Uint8Array) for client side (in the Web browser) or crypto.randomBytes() or external module like node-sodium for server-side (in Node.js).
Never use Math.random() or similar insecure RNG functions for cryptographic purposes!
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# Etale descent of morphisms of schemes
Let $\pi: Y \to X$ be an etale morphism of finite-type schemes over a field $k$. Let $Z$ be a $k$-scheme. Suppose we have a morphism $f:Y \to Z$. When does $f$ descend to a morphism $g: X \to Z$? Is the answer simpler if any of $X,Y, Z$ are affine?
-
For a faithfully flat morphism $\pi\colon Y\to X$ of finite type, the morphism $f$ will descend if and only if the composites of it with the two projection morphisms $Y\times_X Y \to Y$ coincide. Etale maps are flat, and faithfully flat if surjective. If your etale morphism $\pi$ is Galois, then the second condition just says that $f$ is invariant under the action of the Galois group.
$\pi$ only has to be faithfully flat and quasi-compact, i.e. fpqc (in fact a weaker version of fpqc also works, see Angelo Vistoli's notes on descent theory). – Martin Brandenburg Nov 11 '13 at 16:31
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For other arrow notations, see down-arrow notation, mixed arrow notation, chained arrow notation, irrational arrow notation.
Arrow notation or up-arrow notation is a widely used notation for the hyper operators, devised by Donald Knuth in 1976 to represent large numbers.[1][2] Knuth roughly introduced $$a \uparrow b$$ as $$a^b$$ and $$a \underbrace{\uparrow \cdots \uparrow}_{n \textrm{ times}} b$$ as $$\underbrace{a \underbrace{\uparrow \cdots \uparrow}_{n-1 \textrm{ times}} \cdots a \underbrace{\uparrow \cdots \uparrow}_{n-1 \textrm{ times}} a}_{b \textrm{ times}}$$.
## Definition
We explain the precise intention of the original definition by Knuth. For any positive integers $$a$$, $$b$$, and $$n$$, $$a \uparrow^n b$$ is defined by the following recursive way:
\begin{eqnarray*} a \uparrow^n b = \left\{ \begin{array}{ll} a^b & (n = 1) \\ a & (n \geq 1, b = 1) \\ a \uparrow^{n-1} (a \uparrow^n (b-1)) & (n > 1, b > 1) \end{array} \right. \end{eqnarray*}
This notation gives a total computable function \begin{eqnarray*} \uparrow \colon \mathbb{N}_{>0}^3 & \to & \mathbb{N}_{>0} \\ (a,b,n) & \mapsto & a \uparrow^n b, \end{eqnarray*} where $$\mathbb{N}_{>0}$$ denotes the set of positive integers. Readers should be careful that some authors implicitly extend it so that $$a \uparrow^n 0 = 1$$ holds for any positive integers $$a$$ and $$n$$.
It satisfies the following relations, which also characterise the notation, for any positive integers $$a$$, $$b$$, and $$n$$:
\begin{eqnarray*} a \uparrow^1 b &=& a^b \\ a \uparrow^n 1 &=& a \\ a \uparrow^{n + 1} (b + 1) &=& a \uparrow^n (a \uparrow^{n + 1} b) \\ \end{eqnarray*}
Here, $$a \uparrow^{n} b$$ is regarded as a shorthand for $$a \uparrow\uparrow\cdots\uparrow\uparrow b$$ with $$n$$ arrows. So, for example, we have $$a \uparrow^2 b = a \uparrow\uparrow b$$. Arrow notation operators are right-associative; $$a \uparrow b \uparrow c$$ always means $$a \uparrow (b \uparrow c)$$.
Specifically, $$a \uparrow b$$ is exponentiation, $$a \uparrow\uparrow b$$ is tetration, $$a \uparrow\uparrow\uparrow b$$ is pentation, and in general $$a \uparrow^n b$$ is the (n+2)th hyper-operation. In ASCII, these are often written a^b, a^^b, a^^^b, ... or a**b, a***b, a****b as consistent with some programming languages which use a^b or a**b for exponentiation.
## Application to googology
The function $$f(n) = n \uparrow^n n$$ is a fast-growing function that eventually dominates all primitive recursive functions, and can be approximated using the fast-growing hierarchy as $$f_\omega(n)$$. This is the limit of the non-extended hyper operators, and by extension, arrow notation.
Arrow notation can relate to Hyper-E notation through the following rule: [3]
$$a \uparrow^c b = E[a]\underbrace{1\#1\#1\cdots\#b}_c$$
for positive integers a, b, c. For example,
• a↑b = E(a)b
• a↑↑b = E(a)1#b
• a↑↑↑b = E(a)1#1#b
Arrow notation has been generalized to other notations. A few notable ones are chained arrow notation, BEAF, and BAN. It has also been compared exactly with, among others, Notation Array Notation using the function (a{2, number of arrows}b).
Nathan Ho and Wojowu showed that arrow notation terminates — that is, $$a \uparrow^n b$$ exists for all $$a,b,n$$.[4][dead link]
## Examples
• $$2 \uparrow 3 = 2^3 = 8$$
• $$5 \uparrow 6 = 5^6 = 15,625$$
• $$10 \uparrow 100 = 10^{100} =$$ googol
• $$3 \uparrow\uparrow 4 = 3 \uparrow 3 \uparrow 3 \uparrow 3 = 3 \uparrow 3 \uparrow 27 = 3^{7,625,597,484,987} \approx 1.2580143*10^{3638334640024}$$
• $$5 \uparrow\uparrow 3 = 5 \uparrow 5 \uparrow 5 = 5^{5^5} \approx 1.9110126*10^{2184}$$
• $$2 \uparrow\uparrow\uparrow 2 = 2 \uparrow\uparrow 2 = 2 \uparrow 2 = 2^2 = 4$$
• $$2 \uparrow\uparrow\uparrow\uparrow 2 = 2 \uparrow\uparrow\uparrow 2 = 2 \uparrow\uparrow 2 = 2 \uparrow 2 = 2^2 = 4$$
• $$3 \uparrow\uparrow\uparrow 2 = 3 \uparrow\uparrow 3 = 3 \uparrow 3 \uparrow 3 = 3^{3^3} = 3^{27} = 7,625,597,484,987$$
• $$2 \uparrow\uparrow\uparrow 3 = 2 \uparrow\uparrow 2 \uparrow\uparrow 2 = 2 \uparrow\uparrow 4 = 2 \uparrow 2 \uparrow 2 \uparrow 2 = 2 \uparrow 2 \uparrow 4 = 2 \uparrow 16 = 65,536$$
• $$3 \uparrow\uparrow\uparrow 3 = 3 \uparrow\uparrow 3 \uparrow\uparrow 3 = 3 \uparrow\uparrow 7,625,597,484,987 =$$ Tritri
• $$a \uparrow^{n+1} 2 = a \uparrow^{n} a$$
## For non-integers
Fish defined[5] for $$x > 0, n \ge 1, n \in \mathbb{N}$$,
$$a \uparrow^n x = \begin{cases} a^x & \text{if } 0 < x \le 1 \text{ or } n=1 \\ a \uparrow^{n-1} (a \uparrow^n (x-1)) & \text{if } 1 < x, 1 < n \end{cases}$$
From this definition,
$$\begin{array}{rl} 10^{100} &= 10 \uparrow 10 \uparrow 2 = 10 \uparrow 10 \uparrow 10 \uparrow \log_{10}(2) \\ &\approx 10 \uparrow 10 \uparrow 10 \uparrow 0.301 = 10 \uparrow \uparrow 2.301 \\ 10^{10^{100}} &= 10 \uparrow 10 \uparrow 10 \uparrow 2 \approx 10 \uparrow 10 \uparrow 10 \uparrow 10 \uparrow 0.301 = 10 \uparrow \uparrow 3.301 \\ 3 \uparrow \uparrow \uparrow 3 &= 3 \uparrow \uparrow 7625597484987 \\ &\approx 10 \uparrow \uparrow 7625597484986.041 \\ &\approx 10 \uparrow \uparrow 10 \uparrow 12.88227 \\ &\approx 10 \uparrow \uparrow 10 \uparrow 10 \uparrow 10 \uparrow 0.04532 \\ &= 10 \uparrow \uparrow 10 \uparrow \uparrow 2.04532 \\ &\approx 10 \uparrow \uparrow 10 \uparrow \uparrow 10 \uparrow \uparrow 0.31076 \\ &= 10 \uparrow \uparrow \uparrow 2.31076 \\ \end{array}$$
## Turing machine code
Input form: to represent $$a \uparrow^{c} b$$, place a 1's, then c+2 ^'s, then b 1's. For example, 111^^^^^111 computes tritri.
Turing machine code
0 * * r 0
0 _ _ l 1
1 1 _ l 2
2 ^ _ l 3
2 1 1 l 4
3 ^ _ l 3
3 1 _ l 2
4 1 1 l 4
4 ^ 1 l 4'
4 _ 1 l halt
4' ^ ^ l 5
4' 1 1 r 0
5 ^ ^ l 5
5 1 1 r 6
6 ^ x r 7
6 1 y r 9
6 | ^ l 12
7 * * r 7
7 _ | r 8
7 | | r 8
8 * * r 8
8 _ ^ l 11
9 * * r 9
9 | | r 10
10 * * r 10
10 _ 1 l 11
11 * * l 11
11 x ^ r 6
11 y 1 r 6
12 * * l 12
12 ^ ^ l 12'
12' ^ ^ l 12'
12' * * l 13
12' 1 x r 14
13 * * l 13
13 ^ ^ r 20
13 _ _ r 20
13 1 x r 14
14 * * r 14
14 ^ ^ r 15
15 ^ ^ r 15
15 x x r 16
15 1 x l 12
16 x x r 16
16 1 x l 12
16 ^ x r 17
17 ^ ^ r 17
17 1 ^ r 18
18 ^ 1 r 17
18 1 1 r 18
18 _ 1 l 19
19 * * l 19
19 x x l 12
20 x 1 r 20
20 ^ ^ r 21
21 ^ ^ r 21
21 x x r 22
22 x x r 22
22 1 _ r 23
22 ^ ^ l 30
23 1 _ r 23
23 ^ ^ l 24
24 _ ^ r 25
25 ^ ^ r 25
25 1 1 l 26
26 ^ 1 r 27
27 1 1 r 27
27 _ _ l 28
28 1 _ l 29
29 * * l 29
29 _ ^ r 25
29 x 1 l 30
30 x 1 l 30
30 ^ ^ r 31
31 * * r 31
31 _ _ l 32
32 1 _ l 1
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### Home > CCAA > Chapter 2 Unit 2 > Lesson CC2: 2.2.3 > Problem2-62
2-62.
Sketch each expression using $+$ and $-$ tiles or draw it on a number line. Then find the simplified value of each expression.
1. $-5+6+4$
Arrange the following symbols to make zero pairs and then determine the value of the remaining symbols.
1. $7+(-3)+(-4)$
This problem is very similar to part (a). Try drawing the zeros on your paper.
1. $8+|(-2)+(-3)|$
This problem is very similar to part (a). The vertical lines indicate absolute value. Try drawing the zeros on your paper.
1. $|-5|+3+6$
This problem is very similar to part (a). Try drawing the zeros on your paper.
Explore the problem with the eTool below.
Click the link to the right to view full version.
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# Ticket #13808: trac_13808-v3.patch
File trac_13808-v3.patch, 43.8 KB (added by dcoudert, 8 years ago)
new combined version
• ## doc/en/reference/graphs.rst
# HG changeset patch
# User dcoudert <david.coudert@inria.fr>
# Date 1356046910 -3600
# Parent 653a76423407a118217595cc3e5acea9370bd531
trac #13808 -- Gromov hyperbolicity of a graph
diff --git a/doc/en/reference/graphs.rst b/doc/en/reference/graphs.rst
a sage/graphs/distances_all_pairs sage/graphs/graph_latex sage/graphs/graph_list sage/graphs/hyperbolicity
• ## module_list.py
diff --git a/module_list.py b/module_list.py
a Extension('sage.graphs.genus', sources = ['sage/graphs/genus.pyx']), Extension('sage.graphs.hyperbolicity', sources = ['sage/graphs/hyperbolicity.pyx']), ################################ ## ## sage.graphs.base
• ## new file sage/graphs/hyperbolicity.pyx
diff --git a/sage/graphs/hyperbolicity.pyx b/sage/graphs/hyperbolicity.pyx
new file mode 100644
- r""" Hyperbolicity of a graph **Definition** : The hyperbolicity \delta of a graph G has been defined by Gromov [Gromov87]_ as follows (we give here the so-called 4-points condition): Let a, b, c, d be vertices of the graph, let S_1, S_2 and S_3 be defined by .. MATH:: S_1 = dist(a, b) + dist(b, c)\\ S_2 = dist(a, c) + dist(b, d)\\ S_3 = dist(a, d) + dist(b, c)\\ and let M_1 and M_2 be the two largest values among S_1, S_2, and S_3. We define hyp(a, b, c, d) = M_1 - M_2, and the hyperbolicity \delta(G) of the graph is the maximum of hyp over all possible 4-tuples (a, b, c, d) divided by 2. That is, the graph is said \delta-hyperbolic when .. MATH:: \delta(G) = \frac{1}{2}\max_{a,b,c,d\in V(G)}hyp(a, b, c, d) (note that hyp(a, b, c, d)=0 whenever two elements among a,b,c,d are equal) **Some known results** : - Trees and cliques are 0-hyperbolic - n\times n grids are n-1-hyperbolic - Cycles are approximately n/4-hyperbolic - Chordal graphs are \leq 1-hyperbolic Besides, the hyperbolicity of a graph is the maximum over all its biconnected components. **Algorithms and complexity** : The time complexity of the naive implementation (i.e. testing all 4-tuples) is O( n^4 ), and an algorithm with time complexity O(n^{3.69}) has been proposed in [FIV12]_. This remains very long for large-scale graphs, and much harder to implement. An improvement over the naive algorithm has been proposed in [CCL12]_, and is implemented in the current module. Like the naive algorithm, it has complexity O(n^4) but behaves much better in practice. It uses the following fact : Assume that S_1 = dist(a, b) + dist(c, d) is the largest among S_1,S_2,S_3. We have .. MATH:: S_2 + S_3 =& dist(a, c) + dist(b, d) + dist(a, d) + dist(b, c)\\ =& [ dist(a, c) + dist(b, c) ] + [ dist(a, d) + dist(b, d)]\\ \geq &dist(a,b) + dist(a,b)\\ \geq &2dist(a,b)\\ Now, since S_1 is the largest sum, we have .. MATH:: hyp(a, b, c, d) =& S_1 - \max\{S_2, S_3\}\\ \leq& S_1 - \frac{S_2+ S_3}{2}\\ =& S_1 - dist(a, b)\\ =& dist(c, d)\\ We obtain similarly that hyp(a, b, c, d) \leq dist(a, b). Consequently, in the implementation, we ensure that S_1 is larger than S_2 and S_3 using an ordering of the pairs by decreasing lengths. Furthermore, we use the best value h found so far to cut exploration. The worst case time complexity of this algorithm is O(n^4), but it performs very well in practice since it cuts the search space. This algorithm can be turned into an approximation algorithm since at any step of its execution we maintain an upper and a lower bound. We can thus stop execution as soon as a multiplicative approximation factor or an additive one is proven. TODO: - Add exact methods for the hyperbolicity of chordal graphs - Add method for partitioning the graph with clique separators **This module contains the following functions** At Python level : .. csv-table:: :class: contentstable :widths: 30, 70 :delim: | :meth:~hyperbolicity | Return the hyperbolicity of the graph or an approximation of this value. :meth:~hyperbolicity_distribution | Return the hyperbolicity distribution of the graph or a sampling of it. REFERENCES: .. [CCL12] N. Cohen, D. Coudert, and A. Lancin. Exact and approximate algorithms for computing the hyperbolicity of large-scale graphs. Research Report RR-8074, Sep. 2012. [http://hal.inria.fr/hal-00735481]. .. [FIV12] H. Fournier, A. Ismail, and A. Vigneron. Computing the Gromov hyperbolicity of a discrete metric space. ArXiv, Tech. Rep. arXiv:1210.3323, Oct. 2012. [http://arxiv.org/abs/1210.3323]. .. [Gromov87] M. Gromov. Hyperbolic groups. Essays in Group Theory, 8:75--263, 1987. AUTHORS: - David Coudert (2012): initial version, exact and approximate algorithm, distribution, sampling Methods ------- """ ############################################################################### # Copyright (C) 2012 David Coudert # # Distributed under the terms of the GNU General Public License (GPL) # http://www.gnu.org/licenses/ ############################################################################### # imports from sage.graphs.graph import Graph from sage.graphs.distances_all_pairs cimport c_distances_all_pairs from sage.rings.arith import binomial from sage.rings.integer_ring import ZZ from sage.functions.other import floor from sage.misc.bitset import Bitset from libc.stdint cimport uint32_t, uint64_t include "../ext/stdsage.pxi" # Defining a pair of vertices as a C struct ctypedef struct pair: int s int t ###################################################################### # Speedup functions ###################################################################### def _my_subgraph(G, vertices, relabel=False, return_map=False): r""" Return the subgraph containing the given vertices This method considers only the connectivity. Therefore, edge labels are ignored as well as any other decoration of the graph (vertex position, etc.). If relabel is True, the vertices of the new graph are relabeled with integers in the range '0\cdots |vertices|-1'. The relabeling map is returned if return_map is also True. TESTS: Giving anything else than a Graph:: sage: from sage.graphs.hyperbolicity import _my_subgraph as mysub sage: mysub([],[]) Traceback (most recent call last): ... ValueError: The input parameter must be a Graph. Subgraph of a PetersenGraph:: sage: from sage.graphs.hyperbolicity import _my_subgraph as mysub sage: H = mysub(graphs.PetersenGraph(), [0,2,4,6]) sage: H.edges(labels=None) [(0, 4)] sage: H.vertices() [0, 2, 4, 6] """ if not isinstance(G,Graph): raise ValueError("The input parameter must be a Graph.") H = Graph() if not vertices: return (H,{}) if (relabel and return_map) else H if relabel: map = dict(zip(iter(vertices),xrange(len(vertices)))) else: map = dict(zip(iter(vertices),iter(vertices))) B = {} for v in G.vertex_iterator(): B[v] = False for v in vertices: B[v] = True H.add_vertex(map[v]) for u in vertices: for v in G.neighbor_iterator(u): if B[v]: H.add_edge(map[u],map[v]) return (H,map) if (relabel and return_map) else H ###################################################################### # Building blocks ###################################################################### cdef inline int __hyp__(unsigned short ** distances, int a, int b, int c, int d): """ Return the hyperbolicity of the given 4-tuple. """ cdef int S1, S2, S3, h S1 = distances[a][b] + distances[c][d] S2 = distances[a][c] + distances[b][d] S3 = distances[a][d] + distances[b][c] if S1 >= S2: if S2 > S3: h = S1-S2 else: h = abs(S1-S3) else: if S1 > S3: h = S2-S1 else: h = abs(S2-S3) return h ###################################################################### # Basic algorithm for the hyperbolicity ###################################################################### cdef tuple __hyperbolicity_basic_algorithm__(int N, unsigned short ** distances, verbose = False): """ Returns **twice** the hyperbolicity of a graph, and a certificate. This method implements the basic algorithm for computing the hyperbolicity of a graph, that is iterating over all 4-tuples. See the module's documentation for more details. INPUTS: - N -- number of vertices of the graph. - distances -- path distance matrix (see the distance_all_pairs module). - verbose -- (default: False) is boolean. Set to True to display some information during execution. OUTPUTS: This function returns a tuple ( h, certificate ), where: - h -- the maximum computed value over all 4-tuples, and so is twice the hyperbolicity of the graph. If no such 4-tuple is found, -1 is returned. - certificate -- 4-tuple of vertices maximizing the value h. If no such 4-tuple is found, the empty list [] is returned. """ cdef int a, b, c, d, S1, S2, S3, hh, h_LB cdef list certificate h_LB = -1 for 0 <= a < N-3: for a < b < N-2: for b < c < N-1: for c < d < N: # We compute the hyperbolicity of the 4-tuple hh = __hyp__(distances, a, b, c, d) # We compare the value with previously known bound if hh > h_LB: h_LB = hh certificate = [a, b, c, d] if verbose: print 'New lower bound:',ZZ(hh)/2 # Last, we return the computed value and the certificate if h_LB != -1: return ( h_LB, certificate ) else: return ( -1, [] ) ###################################################################### # Decomposition methods ###################################################################### def elimination_ordering_of_simplicial_vertices(G, max_degree=4, verbose=False): r""" Return an elimination ordering of simplicial vertices. An elimination ordering of simplicial vertices is an elimination ordering of the vertices of the graphs such that the induced subgraph of their neighbors is a clique. More precisely, as long as the graph as a vertex u such that the induced subgraph of its neighbors is a clique, we remove u from the graph, add it to the elimination ordering (list of vertices), and repeat. This method is inspired from the decomposition of a graph by clique-separators. INPUTS: - G -- a Graph - max_degree -- (default: 4) maximum degree of the vertices to consider. The running time of this method depends on the value of this parameter. - verbose -- (default: False) is boolean set to True to display some information during execution. OUTPUT: - elim -- A ordered list of vertices such that vertex elim[i] is removed before vertex elim[i+1]. TESTS: Giving anything else than a Graph:: sage: from sage.graphs.hyperbolicity import elimination_ordering_of_simplicial_vertices sage: elimination_ordering_of_simplicial_vertices([]) Traceback (most recent call last): ... ValueError: The input parameter must be a Graph. Giving two small bounds on degree:: sage: from sage.graphs.hyperbolicity import elimination_ordering_of_simplicial_vertices sage: elimination_ordering_of_simplicial_vertices(Graph(), max_degree=0) Traceback (most recent call last): ... ValueError: The parameter max_degree must be > 0. Giving a graph build from a bipartite graph plus an edge:: sage: G = graphs.CompleteBipartiteGraph(2,10) sage: G.add_edge(0,1) sage: from sage.graphs.hyperbolicity import elimination_ordering_of_simplicial_vertices sage: elimination_ordering_of_simplicial_vertices(G) [2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 1, 11] sage: elimination_ordering_of_simplicial_vertices(G,max_degree=1) [] """ if verbose: print 'Entering elimination_ordering_of_simplicial_vertices' if not isinstance(G,Graph): raise ValueError("The input parameter must be a Graph.") elif max_degree < 1: raise ValueError("The parameter max_degree must be > 0.") # We make a copy of the graph. We use a NetworkX graph since modifications # are a bit faster this way. import networkx ggnx = networkx.empty_graph() for u,v in G.edge_iterator(labels=None): ggnx.add_edge(u,v) from sage.combinat.combinat import combinations_iterator cdef list elim = [] cdef set L = set() # We identify vertices of degree at most max_degree for u,d in ggnx.degree_iter(): if d<=max_degree: L.add(u) while L: # We pick up a vertex and check if the induced subgraph of its neighbors # is a clique. If True, we record it, remove it from the graph, and # update the list of vertices of degree at most max_degree. u = L.pop() X = ggnx.neighbors(u) if all(ggnx.has_edge(v,w) for v,w in combinations_iterator(X,2)): elim.append(u) ggnx.remove_node(u) for v,d in ggnx.degree_iter(X): if d<=max_degree: L.add(v) if verbose: print 'Propose to eliminate',len(elim),'of the',G.num_verts(),'vertices' print 'End elimination_ordering_of_simplicial_vertices' return elim ###################################################################### # Compute the hyperbolicity using a path decreasing length ordering ###################################################################### cdef inline _invert_cells(pair * tab, uint32_t idxa, uint32_t idxb): cdef pair tmp = tab[idxa] tab[idxa] = tab[idxb] tab[idxb] = tmp cdef _order_pairs_according_elimination_ordering(elim, int N, pair *pairs, uint32_t nb_pairs, uint32_t *last_pair): r""" Re-order the pairs of vertices according the elimination of simplicial vertices. We put pairs of vertices with an extremity in elim at the end of the array of pairs. We record the positions of the first pair of vertices in elim. If elim is empty, the ordering is unchanged and we set last_pair=nb_pairs. """ cdef uint32_t j, jmax jmax = nb_pairs-1 if nb_pairs<=1 or not elim: last_pair[0] = nb_pairs else: B = Bitset(iter(elim)) j = 0 while j0: jmax -= 1 else: # This pair is at a correct position. j += 1 # We record the position of the first pair of vertices in elim last_pair[0] = jmax+1 cdef tuple __hyperbolicity__(int N, unsigned short ** distances, int D, int h_LB, float approximation_factor, float additive_gap, elim, verbose = False): """ Return the hyperbolicity of a graph. This method implements the exact and the approximate algorithms proposed in [CCL12]_. See the module's documentation for more details. INPUTS: - N -- number of vertices of the graph - distances -- path distance matrix - D -- diameter of the graph - h_LB -- lower bound on the hyperbolicity - approximation_factor -- When the approximation factor is set to some value larger than 1.0, the function stop computations as soon as the ratio between the upper bound and the best found solution is less than the approximation factor. When the approximation factor is 1.0, the problem is solved optimaly. - additive_gap -- When sets to a positive number, the function stop computations as soon as the difference between the upper bound and the best found solution is less than additive gap. When the gap is 0.0, the problem is solved optimaly. - elim -- elimination ordering of simplicial vertices (list of vertices to eliminate when the lower bound is larger or equal to 1). - verbose -- (default: False) is boolean set to True to display some information during execution OUTPUTS: This function returns a tuple ( h, certificate, h_UB ), where: - h -- is an integer. When 4-tuples with hyperbolicity larger or equal to h_LB are found, h is the maximum computed value and so twice the hyperbolicity of the graph. If no such 4-tuple is found, it returns -1. - certificate -- is a list of vertices. When 4-tuples with hyperbolicity larger that h_LB are found, certificate is the list of the 4 vertices for which the maximum value (and so the hyperbolicity of the graph) has been computed. If no such 4-tuple is found, it returns the empty list []. - h_UB -- is an integer equal to the proven upper bound for h. When h == h_UB, the returned solution is optimal. """ cdef int i, j, l, l1, l2, x, y, h, hh, h_UB, a, b, c, d, S1, S2, S3 cdef dict distr = {} cdef list certificate = [] # We count the number of pairs of vertices at distance l for every l cdef uint32_t * nb_pairs_of_length = sage_calloc(D+1,sizeof(uint32_t)) for 0 <= i < N: for i+1 <= j < N: nb_pairs_of_length[ distances[i][j] ] += 1 # We organize the pairs by length in an array of pairs cdef pair ** pairs_of_length = sage_malloc(sizeof(pair *)*(D+1)) cdef uint32_t * cpt_pairs = sage_calloc(D+1,sizeof(uint32_t)) for 1 <= i <= D: pairs_of_length[i] = sage_malloc(sizeof(pair)*nb_pairs_of_length[i]) # cpt_pairs[i] = 0 for 0 <= i < N: for i+1 <= j < N: l = distances[i][j] pairs_of_length[ l ][ cpt_pairs[ l ] ].s = i pairs_of_length[ l ][ cpt_pairs[ l ] ].t = j cpt_pairs[ l ] += 1 sage_free(cpt_pairs) if verbose: print "Current 2 connected component has %d vertices and diameter %d" %(N,D) print "Paths length distribution:", [ (l, nb_pairs_of_length[l]) for l in range(1, D+1) ] # We improve the ordering of the pairs according the elimination # ordering. We store in last_pair[l] the index of the last pair of length l # to consider when the lower bound on the hyperbolicity >=1. cdef uint32_t * last_pair = sage_malloc(sizeof(uint32_t)*(D+1)) for 1 <= l <= D: _order_pairs_according_elimination_ordering(elim, N, pairs_of_length[l], nb_pairs_of_length[l], last_pair+l) approximation_factor = min(approximation_factor, D) additive_gap = min(additive_gap, D) # We create the list of triples (sum,length1,length2) sorted in # co-lexicographic order: decreasing by sum, decreasing by length2, # decreasing length1. This is to ensure a valid ordering for S1, to avoid # some tests, and to ease computation of bounds. cdef list triples = [] for i in range(D,0,-1): for j in range(D,i-1,-1): triples.append((i+j,j,i)) # We use some short-cut variables for efficiency cdef pair * pairs_of_length_l1 cdef pair * pairs_of_length_l2 cdef uint32_t nb_pairs_of_length_l1, nb_pairs_of_length_l2 h = h_LB h_UB = D for S1, l1, l2 in triples: # If we cannot improve further, we stop if l2 <= h: h_UB = h break if h_UB > l2: h_UB = l2 if verbose: print "New upper bound:",ZZ(l2)/2 # Termination if required approximation is found if certificate and ( (h_UB <= h*approximation_factor) or (h_UB-h <= additive_gap) ): break pairs_of_length_l1 = pairs_of_length[l1] nb_pairs_of_length_l1 = last_pair[l1] if h>=1 else nb_pairs_of_length[l1] x = 0 while x < nb_pairs_of_length_l1: a = pairs_of_length_l1[x].s b = pairs_of_length_l1[x].t if l2 <= h: # We cut current exploration if lower bound cannot be improved break elif l1 == l2: y = x+1 else: y = 0 pairs_of_length_l2 = pairs_of_length[l2] nb_pairs_of_length_l2 = last_pair[l2] if h>=1 else nb_pairs_of_length[l2] while y < nb_pairs_of_length_l2: c = pairs_of_length_l2[y].s d = pairs_of_length_l2[y].t # If two points are equal, the value will be 0, so we skip the # test. if a == c or a == d or b == c or b == d: y += 1 continue # We compute the hyperbolicity of the 4-tuple. We have S1 = l1 + # l2, and the order in which pairs are visited allow us to claim # that S1 = max( S1, S2, S3 ). If at some point S1 is not the # maximum value, the order ensures that the maximum value has # previously been checked. S2 = distances[a][c] + distances[b][d] S3 = distances[a][d] + distances[b][c] if S2 > S3: hh = S1 - S2 else: hh = S1 - S3 if h < hh: # We update current bound on the hyperbolicity and the # search space h = hh certificate = [a, b, c, d] if h>=1: nb_pairs_of_length_l2 = last_pair[l2] nb_pairs_of_length_l1 = last_pair[l1] if x >= nb_pairs_of_length_l1: break if verbose: print "New lower bound:",ZZ(hh)/2 # We go for the next pair c-d y += 1 # We cut current exploration if we know we can not improve lower bound if l2 <= h: h_UB = h break # We go for the next pair a-b x += 1 # We now release the memory sage_free(nb_pairs_of_length) for 1 <= i <= D: sage_free(pairs_of_length[i]) sage_free(pairs_of_length) sage_free(last_pair) # Last, we return the computed value and the certificate if len(certificate) == 0: return ( -1, [], h_UB ) else: return (h, certificate, h_UB) def hyperbolicity(G, algorithm='cuts', approximation_factor=1.0, additive_gap=0, verbose = False): r""" Return the hyperbolicity of the graph or an approximation of this value. The hyperbolicity of a graph has been defined by Gromov [Gromov87]_ as follows: Let a, b, c, d be vertices of the graph, let S_1 = dist(a, b) + dist(b, c), S_2 = dist(a, c) + dist(b, d), and S_3 = dist(a, d) + dist(b, c), and let M_1 and M_2 be the two largest values among S_1, S_2, and S_3. We have hyp(a, b, c, d) = |M_1 - M_2|, and the hyperbolicity of the graph is the maximum over all possible 4-tuples (a, b, c, d) divided by 2. The worst case time complexity is in O( n^4 ). INPUT: - G -- a Graph - algorithm -- (default: 'cuts') specifies the algorithm to use among: - 'basic' is an exhaustive algorithm considering all possible 4-tuples and so have time complexity in O(n^4). - 'cuts' is an exact algorithm proposed in [CCL12_]. It considers the 4-tuples in an ordering allowing to cut the search space as soon as a new lower bound is found (see the module's documentation). This algorithm can be turned into a approximation algorithm. - 'cuts+' is an extension of the cuts algorithm that uses elimination ordering of the simplicial vertices as documented in [CCL12_]. For some classes of graphs it allows a significant speed-up. Try it to know if it is intersting for you. - approximation_factor -- (default: 1.0) When the approximation factor is set to some value larger than 1.0, the function stop computations as soon as the ratio between the upper bound and the best found solution is less than the approximation factor. When the approximation factor is 1.0, the problem is solved optimaly. This parameter is used only when the chosen algorithm is 'cuts'. - additive_gap -- (default: 0.0) When sets to a positive number, the function stop computations as soon as the difference between the upper bound and the best found solution is less than additive gap. When the gap is 0.0, the problem is solved optimaly. This parameter is used only when the chosen algorithm is 'cuts'. - verbose -- (default: False) is a boolean set to True to display some information during execution: new upper and lower bounds, etc. OUTPUT: This function returns the tuple ( delta, certificate, delta_UB ), where: - delta -- the hyperbolicity of the graph (half-integer value). - certificate -- is the list of the 4 vertices for which the maximum value has been computed, and so the hyperbolicity of the graph. - delta_UB -- is an upper bound for delta. When delta == delta_UB, the returned solution is optimal. Otherwise, the approximation factor if delta_UB/delta. EXAMPLES: Hyperbolicity of a 3\times 3 grid:: sage: from sage.graphs.hyperbolicity import hyperbolicity sage: G = graphs.GridGraph([3,3]) sage: hyperbolicity(G,algorithm='cuts') (2, [(0, 0), (0, 2), (2, 0), (2, 2)], 2) sage: hyperbolicity(G,algorithm='basic') (2, [(0, 0), (0, 2), (2, 0), (2, 2)], 2) Hyperbolicity of a PetersenGraph:: sage: from sage.graphs.hyperbolicity import hyperbolicity sage: G = graphs.PetersenGraph() sage: hyperbolicity(G,algorithm='cuts') (1/2, [0, 1, 2, 3], 1/2) sage: hyperbolicity(G,algorithm='basic') (1/2, [0, 1, 2, 3], 1/2) Asking for an approximation:: sage: from sage.graphs.hyperbolicity import hyperbolicity sage: G = graphs.GridGraph([2,10]) sage: hyperbolicity(G,algorithm='cuts', approximation_factor=1.5) (1, [(0, 0), (0, 9), (1, 0), (1, 9)], 3/2) sage: hyperbolicity(G,algorithm='cuts', approximation_factor=4) (1, [(0, 0), (0, 9), (1, 0), (1, 9)], 4) sage: hyperbolicity(G,algorithm='cuts', additive_gap=2) (1, [(0, 0), (0, 9), (1, 0), (1, 9)], 3) Comparison of results:: sage: from sage.graphs.hyperbolicity import hyperbolicity sage: for i in xrange(10): # long test ... G = graphs.RandomBarabasiAlbert(100,2) ... d1,_,_ = hyperbolicity(G,algorithm='basic') ... d2,_,_ = hyperbolicity(G,algorithm='cuts') ... d3,_,_ = hyperbolicity(G,algorithm='cuts+') ... l3,_,u3 = hyperbolicity(G,approximation_factor=2) ... if d1!=d2 or d1!=d3 or l3>d1 or u3= 1.0. Giving negative additive gap:: sage: from sage.graphs.hyperbolicity import hyperbolicity sage: G = Graph() sage: hyperbolicity(G,algorithm='cuts', additive_gap=-1) Traceback (most recent call last): ... ValueError: The additive gap must be >= 0. Asking for an unknown algorithm:: sage: from sage.graphs.hyperbolicity import hyperbolicity sage: G = Graph() sage: hyperbolicity(G,algorithm='tip top') Traceback (most recent call last): ... ValueError: Algorithm 'tip top' not yet implemented. Please contribute. """ if not isinstance(G,Graph): raise ValueError("The input parameter must be a Graph.") if algorithm=='cuts': if approximation_factor < 1.0: raise ValueError("The approximation factor must be >= 1.0.") if additive_gap < 0.0: raise ValueError("The additive gap must be >= 0.") elif not algorithm in ['basic', 'cuts+']: raise ValueError("Algorithm '%s' not yet implemented. Please contribute." %(algorithm)) # The hyperbolicity is defined on connected graphs if not G.is_connected(): raise ValueError("The input Graph must be connected.") # The hyperbolicity of some classes of graphs is known. If it is easy and # fast to test that a graph belongs to one of these classes, we do it. if G.num_verts() <= 3: # The hyperbolicity of a graph with 3 vertices is 0. # The certificate is the set of vertices. return 0, G.vertices(), 0 elif G.num_verts() == G.num_edges() + 1: # G is a tree # Any set of 4 vertices is a valid certificate return 0, [G.vertices()[x] for x in range(4)], 0 elif G.is_clique(): # Any set of 4 vertices is a valid certificate return 0, [G.vertices()[z] for z in xrange(4)], 0 cdef unsigned short * _distances_ cdef unsigned short ** distances cdef int i, j, k, iN, N, hyp, hyp_UB, hh, hh_UB, D cdef dict distr = {} cdef list certificate = [] cdef list certif cdef dict mymap, myinvmap # We search for the largest 2-connected component. Indeed, the hyperbolicity # of a graph is the maximum over its 2-connected components. B,C = G.blocks_and_cut_vertices() if verbose: # we compute the distribution of size of the blocks for V in B: i = len(V) if i in distr: distr[ i ] += 1 else: distr[ i ] = 1 print "Graph with %d blocks" %(len(B)) print "Blocks size distribution:", distr hyp = 0 for V in B: # The hyperbolicity of a graph with 3 vertices is 0, and a graph cannot # have hyperbolicity larger than N/2. So we consider only larger # 2-connected subgraphs. if len(V) > max( 3, 2*hyp) : # We build the subgraph H = _my_subgraph(G,V) N = H.num_verts() # We test if the block belongs to a graph class with known # hyperbolicity and a fast test. if H.is_clique(): continue # We relabel the vertices to ensure integer vertex names in the # range [0..N-1] since the c_distances_all_pairs uses integer labels # in the range [0..N-1]. mymap = H.relabel( return_map=True ) # We compute the distances and store the results in a 2D array # Although it seems irrelevant to use a 2D array instead of a 1D # array, it seems to be faster in practice. We also compute the # diameter. _distances_ = c_distances_all_pairs(H) distances = sage_malloc(sizeof(unsigned short *)*N) D = 0 for 0 <= i < N: distances[i] = _distances_+i*N for 0 <= j < N: if distances[i][j] > D: D = distances[i][j] # We call the cython function for computing the hyperbolicity with # the required parameters. if algorithm == 'cuts' or algorithm == 'cuts+': if algorithm == 'cuts+': # We compute the elimination ordering of simplicial vertices of H elim = elimination_ordering_of_simplicial_vertices(H, max(2,floor(N**(1/2.0))), verbose) else: elim = [] hh, certif, hh_UB = __hyperbolicity__(N, distances, D, hyp, approximation_factor, 2*additive_gap, elim, verbose) elif algorithm == 'basic': hh, certif = __hyperbolicity_basic_algorithm__(N, distances, verbose) hh_UB = hh # We test if the new computed value improves upon previous value. if hh > hyp: hyp = hh hyp_UB = hh_UB # We construct the inverse mapping of the relabeling of the # vertices of the subgraph myinvmap = dict([(mymap[x],x) for x in mymap]) # We then construct the correct certificate certificate = [myinvmap[u] for u in certif] # We now release the memory sage_free(distances) sage_free(_distances_) # Last, we return the computed value and the certificate return ZZ(hyp)/2, sorted(certificate), ZZ(hyp_UB)/2 ###################################################################### # Distribution of the hyperbolicity of 4-tuples ###################################################################### cdef dict __hyperbolicity_distribution__(int N, unsigned short ** distances): """ Return the distribution of the hyperbolicity of the 4-tuples of the graph. The hyperbolicity of a graph has been defined by Gromov [Gromov87]_ as follows: Let a, b, c, d be vertices of the graph, let S_1 = dist(a, b) + dist(b, c), S_2 = dist(a, c) + dist(b, d), and S_3 = dist(a, d) + dist(b, c), and let M_1 and M_2 be the two largest values among S_1, S_2, and S_3. We have hyp(a, b, c, d) = |M_1 - M_2|, and the hyperbolicity of the graph is the maximum over all possible 4-tuples (a, b, c, d) divided by 2. The computation of the hyperbolicity of each 4-tuple, and so the hyperbolicity distribution, takes time in O( n^4 ). We use unsigned long int on 64 bits, so uint64_t, to count the number of 4-tuples of given hyperbolicity. So we cannot exceed 2^64-1. This value should be sufficient for most users. INPUT: - N -- number of vertices of the graph (and side of the matrix) - distances -- matrix of distances in the graph OUTPUT: - hdict -- A dictionnary such that hdict[i] is the number of 4-tuples of hyperbolicity i among the considered 4-tuples. """ # We initialize the table of hyperbolicity. We use an array of unsigned long # int instead of a dictionnary since it is much faster. cdef int i cdef uint64_t * hdistr = sage_calloc(N+1,sizeof(uint64_t)) # We now compute the hyperbolicity of each 4-tuple cdef int a, b, c, d for 0 <= a < N-3: for a < b < N-2: for b < c < N-1: for c < d < N: hdistr[ __hyp__(distances, a, b, c, d) ] += 1 # We prepare the dictionnary of hyperbolicity distribution to return Nchoose4 = binomial(N,4) cdef dict hdict = dict( [ (ZZ(i)/2, ZZ(hdistr[i])/Nchoose4) for 0 <= i <= N if hdistr[i] > 0 ] ) sage_free(hdistr) return hdict # We use this trick since it is way faster than using the sage randint function. cdef extern from "stdlib.h": long c_libc_random "random"() void c_libc_srandom "srandom"(unsigned int seed) cdef dict __hyperbolicity_sampling__(int N, unsigned short ** distances, uint64_t sampling_size): """ Return a sampling of the hyperbolicity distribution of the graph. The hyperbolicity of a graph has been defined by Gromov [Gromov87]_ as follows: Let a, b, c, d be vertices of the graph, let S_1 = dist(a, b) + dist(b, c), S_2 = dist(a, c) + dist(b, d), and S_3 = dist(a, d) + dist(b, c), and let M_1 and M_2 be the two largest values among S_1, S_2, and S_3. We have hyp(a, b, c, d) = |M_1 - M_2|, and the hyperbolicity of the graph is the maximum over all possible 4-tuples (a, b, c, d) divided by 2. We use unsigned long int on 64 bits, so uint64_t, to count the number of 4-tuples of given hyperbolicity. So we cannot exceed 2^64-1. This value should be sufficient for most users. INPUT: - N -- number of vertices of the graph (and side of the matrix) - distances -- matrix of distances in the graph - sampling_size -- number of 4-tuples considered. Default value is 1000. OUTPUT: - hdict -- A dictionnary such that hdict[i] is the number of 4-tuples of hyperbolicity i among the considered 4-tuples. """ cdef int i, a, b, c, d cdef uint64_t j # We initialize the table of hyperbolicity. We use an array of unsigned long # int instead of a dictionnary since it is much faster. cdef uint64_t * hdistr = sage_calloc(N+1,sizeof(uint64_t)) # We now compute the hyperbolicity of each quadruple for 0 <= j < sampling_size: a = c_libc_random() % N b = c_libc_random() % N c = c_libc_random() % N d = c_libc_random() % N while a == b: b = c_libc_random() % N while a == c or b == c: c = c_libc_random() % N while a == d or b == d or c == d: d = c_libc_random() % N hdistr[ __hyp__(distances, a, b, c, d) ] += 1 # We prepare the dictionnary of hyperbolicity distribution from sampling cdef dict hdict = dict( [ (ZZ(i)/2, ZZ(hdistr[i])/ZZ(sampling_size)) for 0 <= i <= N if hdistr[i] > 0 ] ) sage_free(hdistr) return hdict def hyperbolicity_distribution(G, algorithm='sampling', sampling_size=10**6): r""" Return the hyperbolicity distribution of the graph or a sampling of it. The hyperbolicity of a graph has been defined by Gromov [Gromov87]_ as follows: Let a, b, c, d be vertices of the graph, let S_1 = dist(a, b) + dist(b, c), S_2 = dist(a, c) + dist(b, d), and S_3 = dist(a, d) + dist(b, c), and let M_1 and M_2 be the two largest values among S_1, S_2, and S_3. We have hyp(a, b, c, d) = |M_1 - M_2|, and the hyperbolicity of the graph is the maximum over all possible 4-tuples (a, b, c, d) divided by 2. The computation of the hyperbolicity of each 4-tuple, and so the hyperbolicity distribution, takes time in O( n^4 ). INPUT: - G -- a Graph. - algorithm -- (default: 'sampling') When algorithm is 'sampling', it returns the distribution of the hyperbolicity over a sample of sampling_size 4-tuples. When algorithm is 'exact', it computes the distribution of the hyperbolicity over all 4-tuples. Be aware that the computation time can be HUGE. - sampling_size -- (default: 10^6) number of 4-tuples considered in the sampling. Used only when algorithm == 'sampling'. OUTPUT: - hdict -- A dictionnary such that hdict[i] is the number of 4-tuples of hyperbolicity i. EXAMPLES: Exact hyperbolicity distribution of the Petersen Graph:: sage: from sage.graphs.hyperbolicity import hyperbolicity_distribution sage: G = graphs.PetersenGraph() sage: hyperbolicity_distribution(G,algorithm='exact') {0: 3/7, 1/2: 4/7} Exact hyperbolicity distribution of a 3\times 3` grid:: sage: from sage.graphs.hyperbolicity import hyperbolicity_distribution sage: G = graphs.GridGraph([3,3]) sage: hyperbolicity_distribution(G,algorithm='exact') {0: 11/18, 1: 8/21, 2: 1/126} TESTS: Giving anythin else than a Graph:: sage: from sage.graphs.hyperbolicity import hyperbolicity_distribution sage: hyperbolicity_distribution([]) Traceback (most recent call last): ... ValueError: The input parameter must be a Graph. Giving a non connected graph:: sage: from sage.graphs.hyperbolicity import hyperbolicity_distribution sage: G = Graph([(0,1),(2,3)]) sage: hyperbolicity_distribution(G) Traceback (most recent call last): ... ValueError: The input Graph must be connected. """ if not isinstance(G,Graph): raise ValueError("The input parameter must be a Graph.") # The hyperbolicity is defined on connected graphs if not G.is_connected(): raise ValueError("The input Graph must be connected.") # The hyperbolicity distribution of some classes of graphs is known. If it # is easy and fast to test that a graph belongs to one of these classes, we # do it. if (G.num_verts()==G.num_edges()+1) or G.is_clique(): return {0: sampling_size if algorithm=='sampling' else binomial(G.num_verts(),4)} cdef int N = G.num_verts() cdef int i, j cdef unsigned short ** distances cdef unsigned short * _distances_ cdef dict hdict # We compute the all pairs shortest path and store the result in a 2D array # for faster access. H = G.relabel( inplace = False ) _distances_ = c_distances_all_pairs(H) distances = sage_malloc(sizeof(unsigned short *)*N) for 0 <= i < N: distances[i] = _distances_+i*N if algorithm == 'exact': hdict = __hyperbolicity_distribution__(N, distances) elif algorithm == 'sampling': hdict = __hyperbolicity_sampling__(N, distances, sampling_size) else: raise ValueError("Algorithm '%s' not yet implemented. Please contribute." %(algorithm)) # We release memory sage_free(distances) sage_free(_distances_) return hdict
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# Twofish key length
I'm using twofish. Say you have a 120 bit key, is there any difference in using a 128 bit bit keylength or a 256 bit one?
Since the most significant digits of the key will be allo zero in both cases, does the keylen parameter really matter?
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Well, it turns out that, both from a security and a performance standpoint, it doesn't really matter.
From a security standpoint, the goal (which, as far as we know, Twofish achieves) is that if you know all but N bits of the key, it still takes about $2^N$ trial decrypts to recover the remaining bits. So, it doesn't matter if you have a 128 bit Twofish key (and give the attacker 8), or if you have a 256 bit Twofish key (and give the attacker 136); he still needs to iterate through the remaining 120 bits.
From a performance standpoint, the only difference between 128 bit Twofish keys and 256 bit Twofish keys is during the key scheduling operation; after that, the encryption and decryption operations do not depend on the key size.
On the other hand, if you go through the Twofish specification, we find that they do specify how to use odd-sized keys; you append zero bytes up to the next "standard" size (which would be 128 bits in this case); this is section 4.3.1 of the book.
Given that there is a standard way to handle it, well, you might as well go with it.
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{}
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## Learn from others! (1)
Question from raxacoricofallapatorius: Why orbits don’t eventually decay?
Response from anna v:
You are right, the planetary model of the atom does not make sense when one considers the electromagnetic forces involved. The electron in an orbit is accelerating continuously and would thus radiate away its energy and fall into the nucleus.
One of the reasons for “inventing” quantum mechanics was exactly this conundrum.
The Bohr model was proposed to solve this, by stipulating that the orbits were closed and quantized and no energy could be lost while the electron was in orbit, thus creating the stability of the atom necessary to form solids and liquids. It also explained the lines observed in the spectra from excited atoms as transitions between orbits.
Question from anna v: How does uranium from supernovae explosions end up in mineral veins in a planet?
Response from Martin Beckett:
Mostly because they are heavy.
Rocks erode putting their constituents into solution, the heavy stuff settles out in river/sea beds, and metals are heavy.
For many metals hydrothermal process are more important. Super hot water deep in the earth dissolves the rock containing the minerals, it moves along cracks in the rock and cools depositing the salt and metals as lines in the rock.
In an asteroid with no geological process the metals are found in their raw state having cooled directly from the original ball of primeval gas
Question from jcw: Why don’t metals bond when touched together?
Response from Hasan:
I think that mere touching does not bring the surfaces close enough. The surface of a metal is not perfect usually. Maybe it has an oxide layer that resists any kind of reaction. If the metal is extremely pure and if you bring two pieces of it extremely close together, then they will join together. It’s also called cold welding.
Question from Nogwater: How does gravity escape a black hole?
Response from Vagelford:
Well, the information doesn’t have to escape from inside the horizon, because it is not inside. The information is on the horizon.
One way to see that is from the fact that from the perspective of an observer outside the horizon of a black hole, nothing ever crosses the horizon. It asymptotically gets to the horizon in infinite time (as it is measured from the perspective of an observer at infinity).
On how big a bubble would have to be for us to live inside? http://physics.stackexchange.com/questions/67970/surviving-under-water-in-air-bubble
Why do airplanes fly? (sersiously) http://home.comcast.net/~clipper-108/lift.pdf
Sun’s light “to be” in phase, or not in phase? http://physics.stackexchange.com/questions/69929/stupid-yet-tricky-question-why-do-we-actually-see-the-sun
Event horizon vs black hole: http://physics.stackexchange.com/questions/95366/why-does-stephen-hawking-say-black-holes-dont-exist
## My questions to others! (1)
My problem:
Let $G(n,k)$ be the n-th k-almost prime. Prove that for every for every $n \in N$ there exists infinitely many $k \in N$ satisfying $2*G(n,k) = G(n,k+1)$.
And a solution from Ross Millikan:
$G(n,1)=p_n$, the $n^{\text{th}}$ prime.
$G(n,k) \le 2^{k-1}p_n$ because we can display $n-1$ numbers that must be smaller; $2^{k-1}$ times all the smaller primes.
Given $n$, we can find $m$ such that $3^m > 2^{m-1}p_n \ge G(n,m)$
Then for all $k \ge m, 2G(n,k)=G(n,k+1)$
My question:
Every program P which built of function sequence (order counts): $F_1,..,F_n$, where $F_i$ returns $R_i$ and $F_{i+1}$ takes $R_i$ as an argument, can be shown as $F_1(F_2(F_3(...(F_n))))$, i.e. we do not need to store intermediary program states. Can every program be transformed to such without intermediary states?
Yes, as long as the semantics of the underlying programming language are effective and we allow higher-order functions.
Assume for the sake of the argument that we are given big-step semantics for the underlying programming language, using the notation $s_1 \to s_2$, where $s_1$ and $s_2$ are program states. Suppose furthermore that the underlying programming language is “roughly imperative”: We have several base commands (e.g., assignment) that are joined into a sequence. Then the execution of a sequence $c_1; c_2; \ldots c_n$ of commands corresponds to executing the big-step reductions for $c_1, c_2, \ldots, c_n$ in sequence. Since we assumed that the semantics are effective, in the sense that there are recursive functions implementing the reduction rules, this boils down to applying the corresponding functions implementing the semantics in sequence.
If we add control-flow constructors like if or while, things get a bit more interesting. Consider for example the following simple if command: $\text{if } e \text{ then } c$, where $e$ is an expression and $c$ is a command.
The semantics could be given by the two big-step reduction rules
$$\frac{e(s_1)=1\quad s_1 \to_c s_2}{s_1 \to_{\text{if } e \text{ then } c} s_2} \qquad \frac{e(s_1)=0}{s_1 \to_{\text{if } e \text{ then } c} s_1}$$
This gives rise to an evaluation function of the form $E_\text{if}(E_e,E_c,s)$ which takes two functions as arguments, namely the evaluation functions for $e$ and $c$, and the execution state $s$. $E_\text{if}$ is clearly recursive. Since the evaluation functions for $E_e$ and $E_c$ can be derived from the program being considered, we can treat them as parameters and therefore get an
implementation function $E_{\text{if } e \text{ then } c}$ that maps execution states to execution states, as above.
My question: What is the rigorous definition of the Aufbau principle and the mathematical model used for its description?
The Aufbau principle isn’t rigorous because it’s based upon the approximation that the electron-electron interaction can be averaged into a mean field. This is called the [Hartree-Foch][1] or self consistent field method. The centrally symmetric mean field results in a set of atomic orbitals that you can populate 2 electrons at a time.
The trouble is that the electron correlations mix up the atomic orbitals so that distinct atomic orbitals no longer exist. Instead you have a single wavefunction that describes all the electrons and does not factor into parts for each electron. For example this is explicitly done in the [configuration interaction][2] technique for improving the accuracy of Hartree-Foch calculations.
[1]: http://en.wikipedia.org/wiki/Hartree%E2%80%93Fock_method
[2]: http://en.wikipedia.org/wiki/Configuration_interaction
My question: Unstable atomic nuclei will spontaneously decompose to form nuclei with a higher stability. What is the algorithm for deciding what sort of it is? (alpha, beta, gamma, etc. Also, given that alpha and beta emission are often accompanied by gamma emission, what is an algorithm for deciding about the distribution of the radiation?
Gamma emission is emission of a photon upon a nucleus transitioning from an excited state to a lower or ground state **of the same nucleus**. The number of neutrons and protons in the nucleus is exactly the same before and after the gamma photon is emitted.
Beta decay results from a nucleus having too few or too many neutrons relative to the number of protons to be stable. If there are too many neutrons, a neutron becomes a proton, an electron and an anti electron-neutrino. If there are too few neutrons, a proton may become a nuetron by positron emission or electron capture. Whether beta decay is favorable can be calculated based upon the energies of the parent and daughter nuclei, as well as the energies of other particles.
Alpha decay is only observed in heavy nuclei, with at least 52 protons. Iron (26 protons, 30 neutrons) is the most stable nucleus. The [semi-empirical mass formula][1] may be used to determine if alpha decay is energetically favorably, but even if it is, the rate of decay may be extremely slow. There is a potential energy barrier to the particle’s escape from the nucleus. See [this reference][2] for further information.
[1]: http://en.wikipedia.org/wiki/Semi-empirical_mass_formula
[2]: http://www.astro.uwo.ca/~jlandstr/p467/lec8-alpha_reactions/
## Learn from Bertrand Russell!
- “I have lived in the pursuit of a vision, both personal and social. Personal: to care for what is noble, for what is beautiful, for what is gentle; to allow moments of insight to give wisdom at more mundane times. Social: to see in imagination the society that is to be created, where individuals grow freely, and where hate and greed and envy die because there is nothing to nourish them. These things I believe, and the world, for all its horrors, has left me unshaken.”, Me: physical body life-term goal lives as long as physical body works, a goal beyond is just beyond; to “see in imagination” means to imagine “what is should be” (at his current level of understanding); the big goal: “to care for what is noble” is cursory- still, it highlights seeking for explicit features
- “The trouble with the world is that the stupid are cocksure and the intelligent are full of doubt.”, Me: “stupid” have limited observation, hence must be more “sure” of many claims; given the Gaussian distribution of people (given that we assume naive genetics behind the birth of new beings), optimizing for quality requires optimizing for a tight range; Still, lets remember that we still understand barely anything, so it is also a good idea to seek in others different talents than intelligence (understood as the ability to effectively solve problems)
- “Love is something far more than desire for sexual intercourse; it is the principal means of escape from the loneliness which afflicts most men and women throughout the greater part of their lives.”, Me: but why do people want to escape from loneliness? cannot they talk to arbitrary people in the street? why is love understood as tight (selective) feature so often? what really is loneliness?
- “I would never die for my beliefs because I might be wrong.”, Me: we just see and model the observable (our observable)
- “The good life is one inspired by love and guided by knowledge.”, Me: love would be just “love for life”, i.e. appreciation of the given, and knowledge would be “learning about the black box” through the (arbitrarily effective) walk in the parameter space; here Russell adds: “Three passions, simple but overwhelmingly strong, have governed my life: the longing for love, the search for knowledge, and unbearable pity for the suffering of mankind.”, so he names three features, i.e. includes suffering, but I would remove suffering from here, understanding it as one of things we experience when learning
- “Conventional people are roused to fury by departure from convention, largely because they regard such departure as a criticism of themselves.”, Me: a fixed set of unchallenged rules, or more generically, ceasing to challenge own rules, refrains many from effective learning; Russell adds more here: The world is full of magical things patiently waiting for our wits to grow sharper.”, i.e. by looking at learning with more happiness, we learn to live better; why do we die instead of living forever learning?
- “One should respect public opinion insofar as is necessary to avoid starvation and keep out of prison, but anything that goes beyond this is voluntary submission to an unnecessary tyranny.”, Me: as noted before, when learning, use tight range for choosing, and constantly challenge claims
- “Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.”, Me: we remember Russell’s paradox, we understand how many axioms and assumptions are introduced to mathematics; we need to think more and more as how to revise assumptions using knowledge from experiments
- “Life is nothing but a competition to be the criminal rather than the victim.”, Me: important words, but I need time to investigate them deeply
- “We are faced with the paradoxical fact that education has become one of the chief obstacles to intelligence and freedom of thought.”, Me: remember than to learn you need to perceive (hear, see) and then understand at your level of understanding, you often learn without understanding; secondly, it is misleading to many than “school” ends, in fact- “school” is all the time, for more – also refer to Warren Buffet’s words quotes in one of my former posts
- “Science is what you know, philosophy is what you don’t know.”, Me: science describes the perceived using our models, philosophy or thinking in general challenges the models, and dreams of a larger picture
- “It is preoccupation with possessions, more than anything else, that prevents us from living freely and nobly.”, Me: I hope that it is not even the case, but the fact that people are afraid to go for what they like because they need to survive, and they need food before self-development; an intelligent being will never focus on implicit and misleading information, such as money, but that does not imply one will not have money; an intelligent being will focus on important features, going beyond physical life, and will crave for and do things that are worth doing
- “Do not fear to be eccentric in opinion, for every opinion now accepted was once eccentric.”, Me: you have to define what “eccentric” means; if an opinion different than the opinion of majority, then by definition of Gauss distribution and the naive assumption about the birth-rate, we will have the answer
- “The only thing that will redeem mankind is cooperation.”, Me: about taking the picture of group rather than one being; in such a case, we do not optimize what is good for one being; however, what is not “good” for being will not be “good” for the mankind; still, being good for 2-year perspective is different than 20-year perspective, and different than 1M-year pearspective
More to come
## Widening the range of the perceived
Increasing the perceived (see: Goedel) allows us to iteratively improve our model of the universe. We can also use it for negative, e.g. non-existence, proofs. The range of the perceived are clustered to: sight, smell, taste, hearing, touch, vestibular, proprioception. Sight, smell, taste, hearing, touch – all those boil down to sensing certain features of data. Vestibular and proprioception: the former is about sensing positioning, the latter is about knowing about the structure of our body.
So, all in all, senses are about knowing certain features of the given at the perceived level. This means we could have more perspectives as how to observe our data set and therefore more senses. In this post I aim at learning more about the features determining the senses. I want to investigate and cluster current methods for enhancing the known senses. And finally, I also want to touch the problem of finding new features.
Sight. Recognized and observed by Leonardo da Vinci: “The function of the human eye … was described by a large number of authors in a certain way. But I found it to be completely different.”. From Gestalt theory (regarding eyes) we should distinguish between: proximity, similarity, closure, symmetry, common Fate (i.e. common motion), and continuity. But from our perspective, in the 22nd century, these are just different, but correlated, perspectives on the visual data. The question is why visual data is visualized and not heard. Why cannot we hear the visual data, and see the speech? We model it with waves. So theoretically, we’d be able to extract information about arbitrary observable data in way we want. The key is that we should aim at having a larger picture of things rather than focusing on correlated perspectives.
Hearing. Famous 20-20k Hz. Also modeled with frequencies. Still, we non-rigorously allow for “many” frequencies. Not as if we used integers. Might change due to on-going quantum research. Visual spectrum’s (sight) 430-790 THz is much faster peaking than our hearing frequencies. Taste, olfaction are defined with cations and ion channels.
There are very many more senses mentioned in related articles. The question is to find correlations between them as well as simplify the model of description. Another question is whether our model would enable us to build a generic sensor for learning about data and this data would be rendered and further analyzed. That could enable us to widen the range of the perceived as well as face the real issues regarding building such a tool.
So, the initial question would be. Would would we “see” the sound and different other “frequencies”? Which of the “frequencies” would be correlated? How could automate learning from newly acquired data? How could we employ computers to widen the range of perception for us? We have microscopes for transforming nano to larger-scale, we have telescope for transforming macro to lower-scale, and for now we still learn from this apparently different pictures and only occasionally draw big conclusions. But, at the very same time, we should always try to learn what is there rather than what is visible.
A more general question- how would we build the general sensor for learning from the entire spectrum? And finally, what is this entire spectrum? How does this spectrum look like? Is our model with waves satisfactory for the game at scale?
## Elaborating on programming paradigms in the context of the observable
We perceive the objects in the universe as if they had intrinsic properties. Those intrinsic properties last a moment (“how long is now?” on one of Berlin’s houses). For this moment they have a value. If we “omit” now, then we say “flow”. “Flow” indicates that value of property changes in every moment and therefore, as such, can be modeled the way we do we it in hydrodynamics, with connected pipes. Constant flow. So, we either have objects with properties with values that characterize the “now”, or there is only the connection of pipes, and logic behind it. If we have the latter, then in every moment we have objects with properties in current states, unless moment in given sense does not exist (but is still a good approximation for many of our use cases). But, even if we have the object with states, we might be interested about the flow rather than objects and their states, and then we way omit storing information of states. The big thing behind is that when we forget about states, we focus on the flow. If we focus on the flow, we focus on the bigger mechanism of interaction. We can then focus on more complex flows.
Focusing on the flow requires from us understanding the structure of the problem we deal with. A bigger picture is required before we start to deal with a problem. But also solutions are more like sculptures. Now, we will go both directions. But finally better solutions will be chosen. As long as we are not interested in current object properties and more in the logic behind the problem, we will focus on the flow.
A couple of examples. Focusing on processes is secondary since it is based on implicit feedback, i.e. they are derived from the fact that some problems have their data structure such that can be handled with parallel processing. Procedure is a general idea that an action can be encapsulated. But, as I mentioned before, either we care about the result or not. If not then we care about what comes next. Is it possible to turn every program that uses states into a program with no intermediary states?
If yes, can we automatedly learn about the flow of the problem based on its data structure and then heuristically model the solution, and finally iteratively arrive at final solution without states? If this is possible, then finding the formula for primes numbers would first involve learning about prime data set, learning about its informative features. Then re-learning about these informative features until we arrive at truly informative ones that enable us to see the final true picture. And then would we need states in between?
And now, which is the direction for augmented reality featured brains exploiting automated learning about data sets? And how to develop better learning methods and will arrive faster at what really informative is? One thing is that we can lie with statistics a lot, since most of people just see pictures “going up” or “going down”, or “clusters”. Iterative and automated learning for decreasing the amount of states used in between could be an interesting idea.
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# Given the complex number 5 – 3i how do you graph the complex number in the complex plane?
Draw two perpendicular axes, like you would for a $y , x$ graph, but instead of $y \mathmr{and} x$ use $i \mathmr{and} r$.
A plot of $\left(r , i\right)$ will be so the $r$ is the real number, and $i$ is the imaginary number.
So, plot a point on $\left(5 , - 3\right)$ on the $r , i$ graph.
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Equation with factorials
by Alesak
Tags: equation, factorials
P: 134 hello everyone, I usually post my questions on one small czech mathematical forum, but here is an equation noone knows how to "solve". Ive came to it by accident, when I made an mistake in one combinatorics equation. $$x! + (x-3)! = 16x - 24$$ its fairly simple to solve in one way(x has to be greater than 2, and from some point left side is greater than right side, because (x-3)! is always greater then 0 and we dont have to care about -24 on the right side, so we can check for which x is x! > 16x. this leaves us only very few possibilities for x to check). this is nice, but Id like to know if its possible to get it in form x = something. I cant think of any way how to do it. also, this equation doesn`t have any solution, its rather theoretical question.
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# Axelrod Tournament
Thought this would be a fun thing to code, basically it is a simulation of the prisoners dilemma. I'm trying to improve my coding so any critiques about how to get better would be appreciated. Also if you have any suggestions for other strategies for the game, those would be appreciated as well. The goal is to have the lowest score. I think the part of the codes I am most worried about at the moment is the weird way I take the output from the strategies and put them into the play function, and everyone online says the eval function is dangerous.
from random import randint
class Player(object):
def __init__(self):
pass
class TFT(Player):
def strat():
global lg #last game
global round #round number
global TFTv #TFT output variable
if round==0:
x=0
elif round !=0 and Player1=='TFT' and (lg=="out1" or lg=="out2"):
x=0
elif round !=0 and Player2=='TFT' and (lg=="out1" or lg=="out3"):
x=0
else:
x=1
TFTv=[x]
class Rand1(Player):
def strat():
global Rand1v #Rand1 output variable
x=randint(0,1)
Rand1v=[x]
class Rand2(Player):
def strat():
global Rand2v #Rand2 output variable
x=randint(0,1)
Rand2v=[x]
class Doomsday(Player):
def strat():
global Doomsdayv #doomsday output variable
if round==0:
x=0
elif lg=="out1":
x=0
elif Doomsdayv[0]==1:
x=1
else:
x=1
Doomsdayv=[x]
def open():
global MaxRounds
global round
global p1 #player 1 points
global p2 #player 2 points
MaxRounds=(int(input("How many rounds would you like?\n> "))+1)
round=0
p1=0
p2=0
global Player1
global Player2
players={'Description':'Player Name','Tit for Tat':'TFT','Random50/50':'Rand1',"Doomsday":'Doomsday'}
print("Possible Players:", players)
Player1=input("Player 1?\n> ")
Player2=input("Player 2?\n> ")
return Arena.play()
class Counter(object):
def count():
global p1
global p2
global lg
global MaxRounds
if round==MaxRounds:
print(f"Player 1 score={p1}, and Player 2 score={p2}")
elif lg=="out1":
p1+=0
p2+=0
elif lg=="out2":
p1+=0
p2+=60
elif lg=="out3":
p1+=60
p2+=0
elif lg=="out4":
p1+=30
p2+=30
else:
pass
class Arena(object):
def play():
global lg
global round
global Player1
global Player2
eval(Player1).strat()
eval(Player2).strat()
R1=eval(Player1+'v')[0]
R2=eval(Player2+'v')[0]
round+=1
if round==MaxRounds:
Counter.count()
elif R1==R2==0:
lg="out1"
print("Neither Snitched")
Counter.count()
return Arena.play()
elif R1==1 and R2==0:
print("Player 1 snitched on Player 2")
lg="out2"
Counter.count()
return Arena.play()
elif R1==0 and R2==1:
print("Player 2 snitched on Player 1")
lg="out3"
Counter.count()
return Arena.play()
else:
print("Both Snitched")
lg="out4"
Counter.count()
return Arena.play()
### design
1. You seem to have designed your system arround changing a whole mess of global variables. please don't do this. It is impossible to understand what variables trigger what effects.
2. Instead of outputing to a variable, and then getting that variable with eval, why not just return the value from strat()?
3. Why use classes when all they have is one static method? In this exact case functions will do just fine, however, I assume you want to make an object oriented design, so my next points will be on that.
4. Seperate your input/output from your game logic, in order to make the game logic reusable. What if you wanted to make a grpahical interface?
### suggested design
1. If a global variable is used only by the class it is in, make it an instance variable. (i.e. Player1 and Player2). Other global variables can be passed as arguments or return values.
2. have the opining menu create an Arena which contains players. leave it up the main program to call Arena.play().
3. Get rid of counter entirly.
### other
1. to avoid using eval() you can have a dictionary that maps the player entered strings to stretegies
{"TFT": TFT, "Rand1": Rand1, ...}
2. I don't understand why the counter class exist since it just checks a variable then sets other variables, why not just set the numbers in the first place.
3. def __init__(self): pass and class Something(object): are implicit and do not need to be defined.
4. Why does Arena.play recurse instead of looping over the rounds. Python has no tail call optimization so the all variables from the previous round are still kept in memory. also the Players are re-evald each time.
5. Having two Rand classes is a side effect of your design not instantiating classes.
### summary
you might have noticed that most of these critiques are fairly simmilar. you should be able to vastly improve the quality of this code by following these guidelines:
1. Don't use global variables in your classes.
2. Don't use eval().
3. Seperate your game logic and output.
4. use return.
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# Lesson goal: Plotting functions
Previous: Graph a parabola | Home | Next: The Lorenz Attractor
Using computers to plot mathematical functions is a very common task, and is also a great way to strengthen your understanding of how functions appear and behave, without the hassle of using pencil and paper (or the small buttons on your calculator).
new_chart(x-label,y-label,chart-type)
graph_point(x,y)
# Now you try. Try fixing the y= statement to be the function you wish to plot. Also, try testing the ideas of shrinking, expanding, and translating functions, as in $x^2$ to $2*(x-5)^2$.
This code will not run. You have to fix the y= statement first to define your parabola (hint: try y=x^2). Next, you have to fix the pset statement to plot $x$ on the x-axis, and $y$ on the y-axis. Can you do it? Dismiss.
Show a friend, family member, or teacher what you've done!
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Home > 20 (4), 5
# Cooperation Via Intimidation: An Emergent System of Mutual Threats can Maintain Social Order
and
aUniversity of Lausanne, Switzerland; bComenius University in Bratislava, Slovakia
Journal of Artificial Societies and Social Simulation 20 (4) 5
<http://jasss.soc.surrey.ac.uk/20/4/5.html>
DOI: 10.18564/jasss.3336
Received: 10-Aug-2016 Accepted: 18-Apr-2017 Published: 31-Oct-2017
### Abstract
Can human aggressiveness promote peaceful cooperation? Despite the seeming contradiction of these phenomena, our study suggests the answer is yes. We develop two agent-based models of cooperative interactions among aggressive agents threatening each other. In Model 1, we show that aggressive displays performed by dominance-seeking individuals create a system of mutual threats that effectively enforces cooperation and inhibits agents from escalating conflicts. This happens because agents observe each other fighting, which deters them from attacking each other due to aggressive reputations. In Model 2 we extend this effect to third-party interventions showing that forming alliances makes attacks more efficient and promotes the emergence of common rules determining whom to fight against. In such a state, social order is maintained by the existence of moral alliances – groups of agents willing to fight against norm violators. In summary, we argue that reputation for toughness and the aggressive predisposition of humans could have played an important role in the evolution of cooperation and moral systems.
Keywords: Cooperation, Punishment, Revenge, Conflict, Aggression, Morality
The beginning of the slaves’ revolt in morality occurs when ressentiment itself turns creative
and gives birth to values: the ressentiment of those beings who, denied the proper response of action,
compensate for it only with imaginary revenge.
- Nietzsche (2006 [1887], p. 20)
### Introduction
Cooperation among genetically unrelated organisms is a puzzling phenomenon because an individual has to bear a cost to perform it while it provides a benefit in a short run that is smaller than selfishly exploiting others (Clutton-Brock 2009). It is rare among rational individuals, because even if cooperation that strives for a common goal is in everyone's interest, there is no guarantee that it will be achieved unless it is also in every individual's personal interest (Olson 1971).
In this study, we focus on the role of aggressive displays in the emergence of social cooperation. Previous research has shown that humans invest their own resources to harm other individuals without any conscious self-interested motive (Fehr & Gächter 2002). We are interested in the question how such sentiments evolved and whether they foster cooperative order.
While there is agreement on what are the proximate mechanisms controlling cooperation and punishment, their ultimate function remains debated (Alexander 1993; Burnham & Johnson 2005; Hagen & Hammerstein 2006; Krasnow & Delton 2016; Raihani & Bshary 2015; West, Griffin & Gardner 2007; West, El Mouden & Gardner 2011). In other words, while mechanistic explanations of (non-)cooperative decisions seem uncontroversial, the question why and how these mechanisms evolved remains unanswered. Agent-based models are well suited to contribute to such an investigation because they can illustrate how given strategies perform in different environments and what macro-level patterns emerge from local interactions (Bandini, Manzoni & Vizzari 2009).
In general, agent-based models allow rigorous specification of and direct control over agent-internal mechanisms hypothesized to cause a certain behavioral pattern. They also allow large-scale simulations (both in terms of population size and number of interactions) unfeasible in empirical experiments. They can model behavioral patterns emerging within a fixed population (e.g., Shoham & Tennenholtz 1997) or be combined with evolutionary replication dynamics to model multi-generational populations with combinations of genetic and cultural transmission (e.g., Gilbert et al. 2006). Last but not least, the models allow simulating interventions that would be unethical in real experiments.
The main purpose of our research is to explore adaptiveness of different aggression strategies and their impact on rate of cooperativeness in the population. We are also interested in whether effectively spread information about an aggressive reputation can lead to systems of mutual threats and dominance hierarchies that would keep the number of actual fights low. In the next section, we review some interesting findings from experimental studies on cooperation and punishment. We distinguish two types of aggressive behavior: (1) individual aggression among partners directly involved in cooperative interactions, and (2) moralized aggression performed by third parties observing interactions of others. We argue that the uses of these behaviors constitute manifestations of dominance-seeking behavior of increasing complexity, and we investigate emergent properties of societies built on them.
We then formalize verbal theories proposed to explain these findings into two evolutionary agent-based models with the goal of evaluating their feasibility. Both of them model social interactions based on the use of aggressive behavior:
• In the first model, we investigate the effectiveness of different strategies regulating whom agents attack upon encounter. Agents are paired and play the Prisoner’s Dilemma (PD) Game. Depending on their strategy, they can decide to fight against their partners afterwards. We introduce 4 types of agents: those who are never aggressive (pacifists), those who are willing to attack partners who defected against them (avengers), those who are willing to attack those who cooperated with them (bullies) and those who always want to aggress against their partner regardless of his decision (harassers). When agents are observed defeating their opponents, they gain a reputation for toughness which discourages others from attacking them in the future.
• The second model is an extension of the first one, where we investigate how patterns of interactions change when neutral observers become engaged in conflicts. Therein we introduce (morally sensitive) agents who care about the interactions of others and engage in fights after observing an interaction. We consider two additional strategies: agents who fight against defectors (rangers) and agents who fight against cooperators (thugs) also when they merely observe an interaction. That leads to group fighting and analogically to the first model, agents participating in the victorious group gain a reputation for being tough.
We then analyze emergent patterns of interaction within generations and also observe success of different strategies in the long evolutionary run. Based on obtained results, we argue that agents with aggressive and exploitative predispositions in their decision-making faculties can give rise to a system where violence is suppressed due to the existence of credible mutual threats. We conclude that the evolution of moral systems may in part be based on adaptations to secure agents’ own interests. The agent-based models were developed using the NetLogo software (version 5.2) and all data were analyzed using R software (version 3.2.2).
### The Logic of Aggression
#### Individual aggression
The supposed evolutionary reason for the existence of mechanisms motivating aggressive behavior is the need for negotiating status and forcing desirable treatment from others – in a situation of conflict of interests individuals need to fight for their position and the feeling of anger is a proximate mechanism that motivates them to achieve that (Sell, Tooby & Cosmides 2009).
It was observed that such aggressive behaviors are commonly used by dominant individuals in order to enforce cooperation; this stabilizes social functioning through the system of threats (Clutton-Brock & Parker 1995). However, when analyzing patterns of aggression, it is crucial to emphasize that there are no a priori rules regarding who is an acceptable target of violence. Experimental results show an interesting variability in how humans choose targets of aggression. It has been shown that in the context of Public Goods Games, some people are willing to attack those who do not contribute to the common pool, whereas others are inclined to attack prosocial contributors (Herrmann, Thöni & Gächter 2008). While punitive attitudes are usually directed at violators of prosocial norms, it has also been observed that people spitefully attack others only in order to ensure their well-being is relatively worse (Abbink & Herrmann 2011; Abbink & Sadrieh 2009; Houser & Xiao 2010; Sheskin, Bloom, & Wynn 2014). The relative prevalence of these behaviors is supposed to be related to local competitiveness (Barker & Barclay 2016; Lahti & Weinstein 2005; Sylwester, Herrmann, & Bryson 2013; see also Charness, Masclet & Villeval 2013). Due to this variability, it has been argued that researchers need to investigate all possibilities of the direction in which aggression is inflicted (Dreber & Rand 2012).
A relatively common type of aggressive behavior is revenge, targeted at individuals believed to have caused some harm to the aggressor in the past. It seems to be the most hard-wired and inflexible mechanism in determining decisions to inflict harm on others, since it is automatic and can be carried out by an individual without any regard to other information (Carlsmith & Darley 2008; Carlsmith, Darley & Robinson 2002; Crockett, Özdemir & Fehr 2014).
Three main functions of revenge can be distinguished: (1) reducing the relative fitness advantage of the target, (2) changing the target’s expectations about the aggressor who signals his unwillingness to accept similar treatment in the future, (3) changing audience expectations about the aggressor’s status (Giardini & Conte 2015; Griskevicius et al. 2009; Kurzban & DeScioli 2013; McCullough, Kurzban & Tabak 2013; Nakao & Machery 2012; Price, Cosmides & Tooby 2002). Note that these functions do not have to be consciously realized – often the only pursued goal of an individual inflicting such a harm is to make the target suffer (Giardini, Andrighetto & Conte 2010; Andrighetto, Giardini & Conte 2013; Giardini & Conte 2015; see also Marlowe et al. 2011) without understanding of complex social interdependencies by avengers, their victims, and observers. All that matters is the showcase of aggressiveness through which agents prove their toughness.
In such a way, harming another individual serves a self-presentational function, that is, its purpose is to restore challenged self-esteem of the aggressor (Miller 2001). By achieving this, deterrence emerges as another function of such behavior (McCullough et al. 2013). Its adaptiveness depends on a correlation between inflicting harm and benefits received in subsequent encounters (Gardner & West 2004; Jensen 2010; Johnstone & Bshary 2004) and it will emerge only when deterrence is effective and if future interactions with the same agent are likely (Krasnow et al. 2012).
Because of these adaptations, willingness to harm others and fear of others being hostile is an important motivator of human action (Abbink & de Haan 2014; see also Schelling 1960). Aggression serves thereby a self-protective function – it has been shown that reputation for performing aggressive acts decreases the likelihood of being attacked by others in subsequent interactions (Benard 2013, 2015).
It seems that aggression is the default behavioral predisposition of humans that manifests itself in the absence of complex social organizations, as famously observed by Hobbes (1651/2011).
In Model 1, we investigate which of the aggression strategies becomes prevalent in the population and what social order emerges from interactions between aggressive agents who instigate fights after social exchanges they are involved in.
#### Moralized aggression
What is perplexing about the logic of human intervention is that people are willing to pay a cost to harm others in cases when they are not directly affected by the interaction; this is known as a third-party punishment (Fehr & Fischbacher 2004). People usually perform such behavior with a conviction that they uphold some moral norms. What benefit can humans get from becoming involved in conflicts of others?
Becoming involved in conflicts first and foremost provides an opportunity to signal a dominant social position by performing an aggressive act (Krasnow et al. 2016). It can also appear in case of helping one’s group member or ally, or in the reciprocity case when the third parties can expect a repayment of their investment in the future (Asao & Buss 2016; Marczyk 2015; Rabellino et al. 2016). Additional benefit can be reaped by increasing one’s own trustworthiness in the eyes of norm-followers. In line with this understanding, it was shown that engaging in third-party punishment increases the observer-rated likeability of an individual (Gordon, Madden & Lea 2014; Jordan et al. 2016; but see Przepiórka & Liebe 2016).
This type of aggression is closely related to moralization of actions – targets of aggression are those who are believed to break some social or moral norms. It is typically accompanied by an aggressive use of moral language (accusing, enforcing behavior by claiming one should do it) which emerges if using it is effective in manipulating others (Cronk 1994). In that perspective, moralization should be understood as a process of attributing objective moral properties to actions in order to manipulate others with implicit threats of aggression – if one is able to present some behavior not as serving a particular interest but as serving moral principles, then the manipulation is successful (Maze 1973). An individual benefits from this by engaging third-parties in fight for his case (Petersen 2013).
In conjunction with coalitional psychology, this mechanism paves the way for the emergence of large-scale moral systems by enabling humans to form and assess strength of alliances (Tooby & Cosmides 2010). In such a setting, there is a pressure for developing a common set of rules allowing for dynamic coordination among observers of conflicts (DeScioli & Kurzban 2009, 2013; DeScioli 2016).
Moving from a system based on individual conflicts to a system based on moralized actions evoking group conflicts requires the existence of agents that are more cognitively complex. While in the case of simple revenge it was sufficient to detect harm done to oneself and the agent who caused it, third-party punishment requires some understanding of social relations. Specifically, observers need to negotiate the status of an action – something that is absent in a world ruled by personal revenge only. This created an evolutionary pressure for developing common code and might have aided the evolution of language (Boyd & Matthew 2015).
The evolutionary emergence of such moral alliances is not well understood and has not been investigated in simulation work before. In order to fill this gap, in Model 2 we investigate third-party involvement in conflicts, which presents an attempt to formally examine circumstances under which these strategies can be adaptive and moral order can emerge in the community.
### Model 1: Individual Aggression
#### Model description
The first model (Figure 1) investigates the role of aggressive displays in enforcing cooperation in a multi-generation evolutionary setting. Agents in each generation engage in interactions, where they can lose or gain fitness. In each interaction round, one fifth of the agents are paired randomly for interaction, with the rest of the agents evenly distributed to be observers of the interactions (8 observers assigned for each interacting pair). The interacting agents first play the Prisoner’s Dilemma Game and afterwards they can decide to fight their opponent based on their fighting strategy (see below) and perceived relative strength. The fitness of the interacting agents is influenced by both the outcome of the game and a potential subsequent fight (see Appendix A for a detailed algorithm).
We are interested in the question of what fighting strategies are most adaptive from an evolutionary point of view. These are a hereditary trait of agents: (1) pacifists are agents who never fight after an interaction, (2) avengers fight against defectors, (3) bullies fight against cooperators, and (4) harassers fight against everyone. In the first generation, the population is composed of pacifists only. See Table 1 for comparison of agent strategies.
Fight against Cooperators Defectors Observed cooperators Observed defectors Pacifists - - - - Avengers - + - - Bullies + - - - Harassers + + - - Rangers - + - + Thugs + - + -
The result of a fight depends on relative strengths of the agents involved[1]. The existence of power asymmetry among agents is the crucial element of the model: stronger agents are more aggressive, decide to fight more frequently, and are more likely to win if a fight takes place.
A behavioral rule for decision whether to cooperate or defect in the PD game is also related to strength, but this time to perceived relative strength of one’s opponent: the agents cooperate when they feel weaker than their opponent and defect when they feel stronger. Thus, decisions to cooperate and decisions not to fight are closely connected in our model, as both of them are manifestations of weak sense of entitlement, a characteristic of people who are physically feeble (Sell et al. 2009; Sell, Eisner & Ribeaud 2016). This belief is established probabilistically according to strength differences among contestants – see Appendix A for details.
The choice of such a decision rule is deliberate since we want to exclude any other factors contributing to these decisions and focus on aggressive displays only. Each time an agent loses a fight against an opponent, the loser and all observers of this fight will add the winner to their fear_list – a privately maintained list of ‘tough’ agents[2]. When assessing the strength of an opponent, the fear_list biases perception, i.e. the opponents in the fear_list are perceived to be stronger than they are (again, for details see Appendix A). This assumption is motivated by the observation that the formation of animal dominance hierarchies depends on organisms observing each other what leads to development of expectations about their status and strength (Chase et al. 2002). In our model this emergent effect coming from observation and experience is conceptualized as deterrence – overestimation of strength of successful agents.
To investigate the influence of information spreading in the population besides the direct observers, the information from the fear_list is further passed on (gossiped) to other randomly chosen agents. After 200 interaction rounds, all the agents die and are replaced with a new generation according to the replicator equation (see Equation 2 in Appendix A), i.e. fighting strategies are spread proportionally to the fitness of their bearers in the previous generation.
The strength parameter that the agents possess can be thought of as the upper-body muscle mass, a feature argued to be the most important for estimating fighting ability (Sell, Hone & Pound 2012) and it is normally distributed within population. It has been shown that the fighting ability influences beliefs of individuals about their entitlements – stronger individuals are more likely to bargain for outcomes advantageous to them and have lower thresholds for aggression (Petersen et al. 2013; Sell et al. 2016).
Although strength is not the only feature determining the assessment of fighting ability (see e.g., Arnott & Elwood 2009; Pietraszewski & Shaw 2015), for the sake of simplicity we assume the agents make decisions based on this value only. We further assume that each agent knows accurately their own fighting ability (but see Fawcett & Johnstone 2010; Garcia et al. 2014) and can observe the fighting ability of others, although these observations are systematically biased, as already noted (Fawcett & Mowles 2013; Prenter, Elwood & Taylor 2006). We also assume that both contestants are equally motivated to compete for resources.
In this way, our model does not suffer from unrealistic assumptions found in previous models – that everybody is, in principle, able to punish everybody else[3]. In real life, it is not true, because individuals differ in their fighting ability. At the same time, the outcome of a fight is never certain (see Gordon et al. 2014). Agents make assessments that are probabilistic in their nature and mistakes are very common in real life (see Figures 10, 11 in the Appendix A). Perfectly rational and omniscient individuals always decide to fight when they are stronger and retreat from the conflict when they are weaker. In a population of all infallible individuals, fights would never occur. The fact they do occur suggests that individuals can make errors in their assessments, some of them resulting from adaptive biases (Johnson & Fowler 2011).
Our primary research question is which fighting strategies become dominant and which conditions promote cooperation in the PD game and reduce number of fights. We were especially interested in the impact of population size (we ran simulations for sizes equal to: 50, 100, or 500), the impact of deterrence effect strength (conceptualized as the number of standard deviations added to the opponent’s strength in case he was observed to have been aggressive in the past), the impact of differing cost of punishing, and the importance of gossiping (to how many agents information about conflict outcome is passed). The number of different agents one can remember (size limit on the fear_list) was equal to 50 and held constant across all simulations. The independent parameters are listed in Table 2 and used in the algorithm described in Appendix A. Particular values were chosen to sample from ranges that make sense relative to one other.[4]
Parameter Used values Description population {50, 100, 500} regulates the size of the population punishment 9 regulates the magnitude of a punishment inflicted on the agent losing a fight pun_cost {3, 6, 9} regulates the cost of inflicting punishment borne by the winner of a fight submit_cost 6 regulates the cost of retreating from conflict by submitting to one's opponent deterrence {1, 3, 6} regulates the magnitude of overestimation of the opponent's strength if he is known to be aggressive sensitivity 0.8 regulates how sensitive the agents are to differences in strength gossip {5, 20} regulates to how many individuals one can pass information about the outcome of the conflict memory 50 regulates the number of other agents one can remember (stored in fear_list)
The dependent variable was the rate of Cooperate-Cooperate (CC) outcomes. We put forward the following hypotheses:
• H1: Harassers will be an evolutionary stable strategy under all parameter combinations.
• H2: Cooperation rates will be higher in smaller groups than in bigger groups.
• H3: Cooperation rates will be higher under strong deterrence effect.
• H4: Cooperation rates will be higher when gossiping is more frequent.
• H5: Cooperation rates will be higher when the cost of punishment is low.
#### Results
The results of Model 1 clearly show that the most adaptive strategy across all various environmental conditions is to fight against everyone (harasser strategy, see Table B1 in Appendix B). This is the case because these agents establish a status of being tough most effectively due to a deterrence mechanism present in our model; this causes more of their opponents to cooperate with them. They have a particular advantage over avenger agents because they attack agents who cooperated with them, effectively showing their aggressiveness, since cooperators are mostly those who are weaker and are easy to damage (this relationship follows directly from the decision rule agents possess).
In our model it is the aggression inflicted upon weaker agents that drives the success of harasser agents – a very cost-effective strategy of increasing one’s own status in a group, as every bully understands. Avenger agents lose this opportunity and cannot establish their status. Thus, we find a support for H1. Nevertheless, an unexpected finding needs to be noted – when the group size is large (n = 500), evolution leads to coexistence of harasser and bully agents (see Figure 3). Bully agents perform well too since they also fight against cooperators after defecting them, but they forego opportunities to fight after Defect-Defect (DD) games. It seems that the extreme aggressiveness expressed by harasser agents pays off in small groups where valuable information about their toughness spreads effectively. In larger groups there exists a significant group of bully agents because harassers’ costly fighting after DD games where both contestants are of similar strength is not so strongly compensated by gaining a tough reputation as it is in smaller populations.
Hypothesis Result Comment H1: Harassers will be an evolutionary stable strategy under all parameter combinations. Confirmed The harasser strategy is most adaptive since these agents engage in fights most frequently and build the reputation for being tough H2: Cooperation rates will be higher in smaller groups than in bigger groups. Confirmed Cooperation is stronger in smaller groups because agents observe each other fighting frequently what leads them to think others are tough; this is not the case in big groups with anonymous interactions H3: Cooperation rates will be higher under strong deterrence effect. Partially confirmed Deterrence promotes cooperation in small groups where agents observe each other frequently; in large groups where repeated interactions are not likely its effect becomes negligible (see also Figure 4) H4: Cooperation rates will be higher when gossiping is more frequent. Not Confirmed This parameter did not significantly affect results of our simulations H5: Cooperation rates will be higher when the cost of punishment is low. Not Confirmed This parameter did not significantly affect results of our simulations
Figures 24 illustrate simulation dynamics in two different runs. As can be seen in Figure 2, in small groups the CC rate increases as the proportion of aggressive individuals increases. However, this is not the case for large groups (Figure 3). This difference is due to the availability of information about other agents in small, but not large, populations.
In large populations the lack of information about other agents’ past behavior in most encounters leads to a strict dominance hierarchy – most interactions are CD outcomes (Figure 3). This happens because agents assess each other relatively accurately and “correctly” play Defect (the stronger player) and Cooperate (the weaker player). Such interactions are often followed by the weaker agent submitting (Dominations, Figure 4). Interestingly, due to dominance hierarchies, the rate of post-game fights remain low.
In smaller populations agents have information about others’ past behavior in most cases and adjust their decisions accordingly. Because more of the assigned partners are remembered to be aggressive, agents overestimate their fighting ability and “incorrectly” play Cooperate. When both players fear their opponent, it leads to a CC outcome. This situation corresponds to the Hobbesian state of war in which agents live in a system of mutual threats. The rate of fighting is similarly low as in large populations, the rate of dominations is also lower (because agents fear each other, conflicts are instigated less frequently).
Figure 5 summarizes the rates of CC at the end of different simulations. A baseline to which these rates are compared is a situation when deterrence parameter is set to 0. In this case, agents do not develop reputation for being tough so the mechanism proposed by us is not at work. Under the assumption that agents can classify themselves as either weaker or stronger without making errors, this baseline would be equal to 0, as the only possible outcome would be CD. Because we made more realistic assumption about messy classifier (see Appendix A), random errors lead also to CC and DD outcomes. We verified this by running our model with deterrence parameter set to 0 and the average CC rate achieved was equal to 19.79%[5].
As can be seen in Figure 5, our results indicate an interaction between the group size and deterrence. Effective threats can deter defections and sustain a relatively high ratio of CC outcomes in a population when its size is small. We thus find support for H2 and a limited support for H3. When the population size is large, the deterrence effect becomes irrelevant, because most interactions will be anonymous. In that case the prevailing outcome of PD Game will be Cooperate-Defect (CD, see Figure 3). For the effectiveness of aggressive displays, it needs to be the case that the information is passed to other agents. The group size and the effectiveness of deterrence are contributing to the maintenance of social order. However, we do not find support for hypotheses H4 and H5, as the effects of gossiping and the cost of punishment parameters were negligible (see Appendix B).
#### Discussion
The results show that the existence of aggressive agents does not lead to a higher number of fights. This is the case because agents who have information about each other’s toughness develop dominance hierarchies that allow them to resolve conflicts without the need to fight. Such dominance hierarchies are frequently observed in various animal societies (Franz et al. 2015; Hsu, Earley & Wolf 2006) and our model illustrates their possible evolutionary emergence.
Our results are also consistent with findings showing that sociality promotes suppression of fights and that conflicts are resolved through submissive displays (Hick et al. 2014).
How is it possible that cooperation is promoted based on a decision rule that promotes defection against everyone who is thought to be weaker? The only way to achieve it is in a situation of mutual fear – when two agents meet each other and both of them think the opponent is stronger. This follows from the formula for estimating the opponent’s strength (see Appendix A). If the two agents have each other in their fear_lists, their perception of the other agent’s strength will be biased by the deterrence value, which causes the stronger agent to choose cooperation and decide not to fight because of its false belief. It follows then that the only way to enforce cooperation is to make oneself stronger in the eyes of others. We assume that it can be achieved through aggressive displays which deter future partners from defecting. Deterrence is conceptualized as a tendency to overestimate the strength of individuals that have been observed to be aggressive. We show that this effect is especially pronounced in small groups (see Figure 5).
In this model, we provide a novel way of conceptualizing deterrence in agent interactions. Previous approaches modeled deterrence in the context of the Public Goods Game by having agents develop a reputation for being a punisher, which in turn enforced contribution when agents observed these punishers in their groups (e.g., dos Santos & Wedekind 2015; Krasnow et al. 2015). In another related model, Nowak et al. (2016) investigated the adaptiveness of different strategies regulating behavior in response to being challenged to fight. A novel feature, missing in their approaches, is the linking of these predispositions to cooperative interactions which, in our model, also affect agents’ fitness. Through this linking, our model focuses on emergent mechanisms curbing agents’ exploitative and aggressive predispositions, which were assumed to be universal.
### Model 2: Moralized Aggression
#### Model description
Model 2 (see Figure 6) is an extension of Model 1 in that some of the observers (so called morally sensitive) will not remain passive bystanders but will actively take sides and join the fight. The outcome of the fight is then determined by relative total strengths of the two fighting groups. The model investigates the adaptiveness of different strategies regulating side-taking rules in conflicts. We introduce two new types of (morally sensitive) agents: (1) Rangers who engage in fights as a third-party supporting cooperators, and (2) Thugs who support defectors.
The latter strategy can be understood as a mafia-like alliance in which antisocial individuals will join their forces to inflict harm and suffering even more effectively. These agents will also possess a different decision rule whether to cooperate or defect in the PD game: in order to defect, they need not be stronger themselves; it suffices if the number of antisocial fighters in the society is high enough to support them: another feature that, this time, a group of bullies perfectly understands[6].
The ranger strategy can be understood as a moral alliance, which demands moral integrity from its members. Thus, the decision rule for these agents is also different. We investigate three possibilities regarding how these agents can behave: (1) they cooperate with everyone (Always Cooperate); (2) they defect against the agents who have been observed defecting in the past (Defect Defectors[7]); (3) they defect against defectors and against the agents who remain neutral in conflicts (Defect Defectors and Neutrals), thereby creating an additional pressure against the agents who do not care about interactions they are not involved in. The algorithmic details of Model 2 are described in Appendix C (see also Table 4 for newly introduced parameters).
Parameter Used values Description coal_sensitivity[8] 0.3 regulates how sensitive the agents are to differences in the coalitional strength. Since the variance of strength differences is higher in case of fighting coalitions, this number was set to be smaller than sensitivity Rule {alwaysC, DefectD, DefectD&N} regulates the decision rule for rangers: 1) Always Cooperate; 2) Defect against agents who have been observed defecting in the past; 3) Defect against agents who have been observed defecting in the past and against agents who have been observed remaining neutral at conflicts
Our main research question for Model 2 is which conditions make an intervention performed by a morally sensitive third-party adaptive. To the best of our knowledge, there are no models that would explicitly investigate this effect[9].
The logic behind a potential success of the intervention strategies relies on several factors. The first is that group fighting can decrease the cost borne by the participating individuals (Jaffe & Zaballa 2010; Okada & Bingham 2008). The second factor is that these strategies allow establishment of toughness more easily, especially for weaker agents who can display aggressiveness in coalitions supporting them. Third, fighting coalitions can more easily eliminate lone agents – while this can have a destructive effect on the society, it can very often increase the relative fitness of attackers. Finally, the more individuals are involved in punishment, the stronger the deterrence effect appears to be, even if material losses are the same (Villatoro et al. 2014).
We put forward the following hypotheses:
• H6: Thug will be an evolutionary stable strategy under Always Cooperate decision rule, or when the population size is large.
• H7: Under Defect Defectors and Defect Defectors and Neutrals decision rules, Ranger will be an evolutionary stable strategy, but only when the population size is small (n = 50 and n = 100).
#### Results
In accordance with hypothesis H6, results of Model 2 show that, under ranger decision rule Always Cooperate, the only possible outcome is that thugs invade the population and the rate of DD outcomes approaches 100%, leading to costly private fights among all members of the population (see Appendix D). Rangers maintain their integrity by having a cooperative reputation; they inflict punishment on defectors, but this is not enough to outweigh benefits acquired by the defectors.
Hypothesis Result Comment H6: Thug will be an evolutionary stable strategy under rangers’ Always Cooperate decision rule, or when the population size is large. Confirmed Unconditional cooperation makes agents an easy target of exploitation that leads to the success of thugs This happens both in the case of cooperating with everyone by default (Always Cooperate rule) and when interactions are anonymous (as in large groups) H7: Under Defect Defectors and Defect Defectors and Neutrals decision rules, Ranger will be an evolutionary stable strategy, but only when the population size is small (n = 50 and n = 100). Partially confirmed This can happen, but success is not always guaranteed (see Figure 8 and Sections 4.11-4.14). Successful order maintenance happens due to the existence of a moral alliance between rangers
In the case of the other two rangers’ decision rules, when the group size is large (n = 500), thug agents also invade the population regardless of the values of remaining parameters (see Figure 7). This happens because rangers are not able to keep all individuals in memory, and under these circumstances of anonymity they play Cooperate in cases when they should not, also leading to their demise. Thus, we find full support for H6.
In the case of smaller group sizes, results are sensitive to random differences at the beginning of each simulation run. Figure 8 depicts a simulation run where ranger invaded a population promoting very high rate of CC outcomes. Such a result, however, was not obtained in all parameter combinations (see Table D1 in Appendix D). In order to verify this seemingly random pattern of results, we ran Model 2 again for small and moderate group sizes, as well as Defect Defectors and Defect Defectors and Neutrals decision rules and now each parameter combination was repeated 10 times (with different random generator seeds).
Figure 9 depicts the number of simulations in which these strategies were successful. There appears to be a threshold above which one of the coalitions gets strong enough to intimidate other players in all interactions and spreads very quickly. Once one of the strategies becomes prevailing, it is evolutionary stable. Thus, while it seems that a small group size and efficient information spread help rangers enforce cooperative order, we do not find full support for H7 – under favorable conditions, this strategy invades the population only 65 % of time. Further research needs to identify different factors contributing to the success of emerging coalitions.
#### Discussion
The results of our simulation suggest there are two factors necessary for the emergence of moral alliances: (1) small group size where agents are not anonymous, and (2) common rules allowing for justified defection (defecting those who have defected, see also Axelrod & Hamilton 1981). The maintenance of social order in runs where rangers prevailed suggests a possible mechanism in which threat of aggression serves a prosocial role. Still, even if these conditions hold, the success of prosocial moral strategies (rangers) is not guaranteed.
In this model, enforcement is based on the presence of individuals willing to fight for a given norm. The success of moral alliance composed of rangers shows the paradox inherent in the use of morality – that the weapon originally invented to fight against others by constraining their behavior turns against its inventors who need to constrain their own behavior as well. We believe that the decision rule possessed by those agents exemplifies the central mechanism of morality – they refrain from exploiting others not because they expect to be beaten (as did the agents present in Model 1), but because they act according to norms (e.g., defecting only against defectors). Adherence to this alliance enforces cooperation on other non-moral agents when they believe they are in a minority, what leads to even higher CC rates than those observed in Model 1.
In this model we formalized the verbal argument regarding the emergence of impartial morality put forward by DeScioli and Kurzban (2009, 2013; DeScioli 2016). We thus provide some support for their proposal by showing that agents committing themselves to a moral alliance and upholding norms agreed upon by them can perform well despite their refraining from exploitation of others. This happens because they effectively bring order into society and promote cooperation.
However, it is not clear what other factors contribute to the success of rangers as opposed to thugs. In our model the success of rangers is not guaranteed. It suggests that other factors such as social exclusion of non-cooperative individuals and forming closed communities with a limited number of partners might also be important in the emergence of cooperation (see e.g., Baumard, André & Sperber 2013; Helbing et al. 2010).
### General Discussion
In the current study, we developed two agent-based models of social conflict, each leading to a system of social aggression with characteristic complexities. The models illustrate a way in which social order can be maintained among aggressive agents with minimal assumptions regarding their cognitive capacities.
In the first model, we assumed that agents seek dominance and use aggression in order to fight for resources. Their social life was limited to observing fights performed by others, which helped in making their own decisions regarding escalating conflicts. The results of this model showed that this system would lead to vengefulness and social exploitation of weaker agents by those who had power to do so. Despite such a grim world, we showed that the system of mutual threats established by everyone’s aggressiveness could constitute an effective deterrent discouraging the agents from seeking dominance violently. Interactions of such intimidated agents produced a relatively high level of cooperative outcomes.
We believe that such understanding of evolutionary roots of norm-related behavior sheds light on the existence of moral hypocrisy in social interactions (see Batson 2016). Rather than conceptualizing morality as a device for controlling one’s own behavior, we conceptualized it as a device for controlling others’ behavior. As we argued, this presents a more reasonable approach to the problem from the evolutionary perspective. We have shown that in a group of agents possessing primarily aggressive and other-directed morality, adherence to moral rules can nevertheless be promoted. Adherence happens not because agents intrinsically value adherence to moral rules, as there are no evolutionary reasons for such a sentiment to be adaptive. Instead, adherence happens because agents fear others who are willing to defend their (other-directed) moral rules.
Such a state makes moral hypocrisy an adaptive strategy (when unobserved, agents would try to break the rules themselves) and creates a pressure for the development of behavioral strategies that control one’s own behavior depending on whether others are observing the agent (see also Trivers 1971). We think that such framing illuminates the roots of defensive morality which is concerned with securing a cooperative reputation within a group (Alexander 1987). An emergent property of a system based on aggression is that in the presence of individuals willing to fight for a given norm it is adaptive to follow the norm, because it diminishes the probability of being attacked (or, to be more precise, it is adaptive to appear to be a norm follower in the eyes of others).
In the second model, we made an additional assumption regarding members of the society – we endowed agents with the ability to form alliances and wage group wars. We showed that such a system established a selection pressure for developing common rules for forming alliances and conflict resolution. Our results illustrate that in the absence of moral alliances, societies are invaded by antisocial groups (thugs), whose interactions lead to mutual defections and costly fights. They also show that under favorable circumstances, reliable coalitions can enforce cooperative outcomes. For this to happen, it is necessary for agents promoting moral order to allow for justified defections according to widely accepted norms.
In such a way, our study builds on the philosophical argument on the emergence of morality developed by Nietzsche (1887/2006). Cooperative behavior in the Prisoner’s Dilemma Game is usually thought to be a moral or socially desirable response, while decision to defect is thought to be a selfish or at least distrustful choice. However, on a more basic level, these decisions can be analyzed not on a cooperative versus non-cooperative continuum, but on an inferiority versus assertiveness continuum. It is primarily not a moral choice, but a choice between asserting one’s own superiority and aiming for higher payoffs (defecting) or relinquishing potential payoff to the partner (cooperating). Cooperative behavior in our first model was not chosen by agents who were the fittest, it was chosen by those who were weak and intimidated. Such a conceptualization sheds light on findings suggesting that strictness in moral judgments about one’s own transgressions is typical for individuals who have low power, as they do not feel entitled to fight for their interests (Lammers, Stapel & Galinsky 2010).
In Nietzsche’s terms, moral prescription to cooperate has its roots in the slave morality. It is not based on striving to be superior; it is based on being inferior. The evolutionary emergence of moralization (the slave’s revolt) happens at the moment when agents following cooperative strategies (rangers) form an alliance in which they enforce cooperative norms on others. This corresponds to the moment of transition from the system where cooperation was a necessary manifestation of inferiority when one feared his opponent (Model 1) to the system where, due to the newly-formed alliance, the concept of behavioral consistency enters the picture and rangers need to practice what they preach by following cooperative norm themselves (Model 2).
We would like to conclude our paper with emphasizing some important limitations of our results. While we found support for our hypotheses regarding evolutionary pressures created by modeled mechanism, in both models these failed to promote cooperative outcomes for larger populations. This suggests that there are likely also other mechanisms at play in promoting cooperative outcomes. Specifically, the reason why we obtained such results is that we assumed interactions in well-mixed populations, where each agent is equally likely to be paired with every other member of the group. Modeling variations on this grouping scheme was beyond scope of this study but it could be expected that taking into consideration local interactions within larger population would make information spread more efficient and promote more cooperative outcomes. These effects are not inconsistent with the framework outlined here and future work needs to focus on them in investigating the emergence of moral alliances.
### Acknowledgements
We would like to thank the anonymous reviewers for their helpful comments. Figures 1 and 6 contain graphics designed by Freepik.
### Notes
1. The strength of agents is distributed normally within the population (mean = 10, SD = 1).
2. At the beginning of every generation this list is empty – agents only start to learn through their experience and observation.
3. Bone et al. (2015) criticize previous models for exactly the same reason and explicitly introduce power asymmetries. Nevertheless, they still assume that weak players are able to punish strong ones, albeit inflicting smaller relative damage.
4. For example, punishment is roughly the same as benefit from the defection, submit_cost is smaller, because this is the case of accepting defeat to avoid fighting, etc.
5. Note that when deterrence parameter is set to 0, values of other parameters should become irrelevant – that was indeed the case – in these simulations, we did not observe any variability in CC rates across combinations of other parameters.
6. In the first generation, thugs are born with a belief that they are in majority; this gives them a slight advantage as they defect in the first round. This is motivated by our assumption that they should feel entitled to exploit others by default – they will only stop doing so if they learn they are in minority. In all subsequent generations they are born with the average belief present in the previous generation (see Appendix C for details).
7. This is a generalization of the Grim Trigger strategy (Friedman 1971) that is even more grim, since justified defections will follow after merely observing the other player playing defect.
8. In case of group conflicts, sensitivity used in decision to fight (Equation 1, Appendix A) needs to be lower as strength differences taken as input increase (see Appendix C for explanation).
9. Halevy and Halali (2015) investigate third-party interventions, but in their framework benefits are introduced directly to the payoff structure, which allows strategic rather than morally-motivated interventions.
### Appendix
#### Appendix A: Description of Model 1
In this appendix, we provide an algorithmic description of the first model developed in the study.
$$sigmoid = \frac{1}{1+\exp{(-diff\cdot sensitivity)}}$$ (1)
$$Pr_{t+1}(i)=\frac{Pr_t(i)\cdot F_t(i)}{\sum_{j=1}^N \, Pr_t(j)\cdot F_t(j)}$$ (2)
Equation 1 depicts the sigmoid function that returns the probability of feeling stronger. Equation 2 depicts the Replicator Dynamics used in the model. Prt(i) denotes the proportion and Ft(i) denotes the average fitness of agents following strategy i at time t.
#### Appendix B: Results of Model 1
Table 6 presents the full results of Model 1 runs:
population pun_cost gossip deterrence pacifists avengers harassers bullies DDs CDs CCs 50 3 5 1 10.00 8.00 74.00 8.00 10.200 57.770 32.030 50 3 5 3 2.00 4.00 88.00 6.00 2.950 38.490 58.560 50 3 5 6 4.20 2.00 84.20 9.60 1.830 26.790 71.380 50 3 20 1 10.00 12.60 64.60 12.80 9.850 55.460 34.690 50 3 20 3 4.40 6.00 76.00 13.60 2.560 34.810 62.630 50 3 20 6 5.60 7.00 77.80 9.60 1.560 24.030 74.410 50 6 5 1 6.00 3.40 73.00 17.60 9.360 56.860 33.780 50 6 5 3 4.00 12.60 67.40 16.00 3.110 38.580 58.310 50 6 5 6 4.60 5.40 78.00 12.00 1.950 28.000 70.050 50 6 20 1 9.20 14.60 63.00 13.20 9.630 56.640 33.730 50 6 20 3 3.00 7.00 72.80 17.20 2.330 35.850 61.820 50 6 20 6 2.00 2.80 89.20 6.00 1.280 23.660 75.060 50 9 5 1 16.00 8.80 51.20 24.00 10.440 57.960 31.600 50 9 5 3 5.80 4.00 74.60 15.60 3.360 38.420 58.220 50 9 5 6 3.80 4.00 84.20 8.00 2.120 27.150 70.730 50 9 20 1 11.80 14.00 18.00 56.20 9.900 57.130 32.970 50 9 20 3 3.80 7.20 77.60 11.40 2.450 35.100 62.450 50 9 20 6 0.00 6.00 87.40 6.60 1.360 23.600 75.040 100 3 5 1 7.90 9.00 54.70 28.40 13.710 58.335 27.955 100 3 5 3 5.20 6.30 75.90 12.60 7.435 51.465 41.100 100 3 5 6 5.60 5.20 81.90 7.30 5.955 45.800 48.245 100 3 20 1 9.00 7.90 65.50 17.60 13.520 58.495 27.985 100 3 20 3 3.90 5.30 84.20 6.60 7.455 51.325 41.220 100 3 20 6 2.00 3.00 90.30 4.70 5.685 45.855 48.460 100 6 5 1 10.00 8.40 49.20 32.40 13.590 58.180 28.230 100 6 5 3 8.10 4.70 76.00 11.20 7.405 51.815 40.780 100 6 5 6 4.70 4.20 77.50 13.60 6.055 45.970 47.975 100 6 20 1 10.10 6.80 65.50 17.60 13.135 59.495 27.370 100 6 20 3 4.80 5.10 79.80 10.30 7.415 50.865 41.720 100 6 20 6 3.60 4.60 82.70 9.10 5.775 45.280 48.945 100 9 5 1 9.50 11.20 42.70 36.60 13.470 58.955 27.575 100 9 5 3 2.30 4.20 80.90 12.60 7.530 51.715 40.755 100 9 5 6 3.00 3.50 84.70 8.80 5.895 45.155 48.950 100 9 20 1 4.70 9.20 46.00 40.10 13.235 59.115 27.650 100 9 20 3 3.40 6.20 73.60 16.80 7.235 50.945 41.820 100 9 20 6 3.50 4.70 79.70 12.10 6.080 45.970 47.950 500 3 5 1 13.50 17.54 47.42 21.54 18.368 60.754 20.878 500 3 5 3 8.24 10.40 63.56 17.80 16.704 60.263 23.033 500 3 5 6 6.60 7.02 70.00 16.38 16.634 58.965 24.401 500 3 20 1 12.76 16.96 45.04 25.24 18.506 60.219 21.275 500 3 20 3 8.04 8.58 60.34 23.04 17.065 59.551 23.384 500 3 20 6 9.80 8.36 67.72 14.12 16.450 59.134 24.416 500 6 5 1 17.38 11.78 30.16 40.68 18.366 60.251 21.383 500 6 5 3 11.12 7.44 38.66 42.78 17.034 59.735 23.231 500 6 5 6 7.40 8.00 52.90 31.70 16.316 59.205 24.479 500 6 20 1 17.72 10.46 27.46 44.36 18.304 60.512 21.184 500 6 20 3 12.04 8.46 42.12 37.38 16.971 59.448 23.581 500 6 20 6 7.68 9.34 50.22 32.76 16.720 58.872 24.408 500 9 5 1 21.42 13.18 20.94 44.46 18.411 60.330 21.259 500 9 5 3 9.38 9.88 30.38 50.36 17.098 59.740 23.162 500 9 5 6 8.28 7.20 38.42 46.10 16.081 59.770 24.149 500 9 20 1 19.36 9.76 24.24 46.64 18.271 60.617 21.112 500 9 20 3 12.46 9.28 30.90 47.36 17.086 59.573 23.341 500 9 20 6 9.82 6.86 35.02 48.30 16.736 58.975 24.289
#### Appendix C: Description of Model 2
In this appendix, we provide a detailed description of the second model developed in the study. The structure of interactions is related to those present in Model 1. Agents are assigned to roles of players and observers just like in Model 1. The difference is in the way types of agents introduced here make decisions regarding playing the PD Game, see lines 9-18 of the algorithm in the Appendix A.
Ranger agents maintain in memory decisions made by other players by updating defector_list and neutral_list (players following strategies present in model 1 who do not engage in group fights).
When rule parameter is set to alwaysC, rangers always cooperate with the partner.
When it is set to DefectD, rangers make the decision in a following way:
When it is set to DefectD&N, rangers make the decision in a following way:
Thug agents maintain an estimate of the frequency of their strategy relative to rangers (rangers / thugs). In the first generation, they are born with an estimate equal to 0.9, which is then subsequently updated as agents observe new contests. This value was chosen such that thugs receive a slight advantage at the beginning of simulation – they decide to defect in their first interactions and stop doing that only if they learn they are in minority (see below).
Within the generation, this estimate is updated in each interaction round 3.
$$new\_ratio=(0.8\cdot old\_ratio)+(0.2\cdot\frac{ranger}{thug})$$ (3)
ranger denotes the number of altruistic fighters (supporting the cooperator) perceived in this round, whereas thug denotes the number of antisocial fighters (supporting the defector) perceived in this round.
In subsequent generations, thug agents are born with the estimate equal to the average estimate present in the previous generation.
Thug agents make a decision how to play the PD Game, according to the following rule:
For the punishment stage of the game (lines 21-29 of the algorithm in Appendix A), there are two possibilities: either there are no morally sensitive agents among observers, in which case the conflict resolution will be based on a private fight just like in Model 1, or there are morally sensitive observers, and that will lead to a coalitional fight.
In the latter case, the procedure is the following:
$$cost = \cfrac{pun\_cost}{\#members^2}$$ (4)
Equation 4 shows how the cost borne by one group member is calculated. Agents replicate and mutate in the same way that in Model 1, see Appendix A, the algorithm lines 32-33.
Note that the equation determining decisions to fight (Equation 1, Appendix A) in case of group conflicts uses different sensitivity value than in the case of individual conflicts. This is because strength differences between groups of individuals can be higher than differences between single individuals – in other words, the range of inputs to the sigmoid function increases (see Figure 10). For that reason, in order to keep the baseline of random cooperation similar to the level found in individual contests (see Figure 5), we needed to change the value sensitivity parameter. Assigning higher value would very slightly decrease number of CC outcomes, but would not qualitatively change our results.
#### Appendix D: Results of Model 2
Table 7 presents the full results of Model 2 runs:
rule population deterrence pacifists avengers harassers bullies rangers thugs DDs CDs CCs AC 50 1 2.20 0.00 3.00 0.00 2.00 92.80 89.520 10.290 0.190 DD 50 1 7.00 2.00 3.60 6.60 74.80 6.00 1.360 14.990 83.650 DD&N 50 1 3.00 2.00 2.00 2.00 6.20 84.80 88.400 11.140 0.460 AC 50 3 2.40 0.00 2.60 0.40 2.00 92.60 88.140 11.650 0.210 DD 50 3 6.00 0.00 6.00 11.40 67.20 9.40 0.640 11.410 87.950 DD&N 50 3 2.20 2.00 3.00 4.00 82.40 6.40 2.180 21.800 76.020 AC 50 6 0.00 0.00 0.00 2.20 0.40 97.40 95.550 4.450 0.000 DD 50 6 1.20 1.80 1.40 2.00 2.80 90.80 87.330 12.240 0.430 DD&N 50 6 4.00 0.00 3.40 2.80 88.60 1.20 0.880 13.800 85.320 AC 100 1 1.10 1.00 1.30 0.00 1.00 95.60 93.925 6.045 0.030 DD 100 1 4.90 4.40 1.60 3.10 78.10 7.90 1.355 12.210 86.435 DD&N 100 1 1.80 1.50 1.00 1.00 92.20 2.50 2.595 7.460 89.945 AC 100 3 1.30 1.00 1.00 1.20 1.20 94.30 91.595 8.310 0.095 DD 100 3 7.00 3.00 3.20 6.00 74.10 6.70 1.410 12.345 86.245 DD&N 100 3 1.30 1.40 1.00 0.00 1.90 94.40 94.010 5.880 0.110 AC 100 6 1.50 0.50 1.00 1.20 0.90 94.90 92.160 7.775 0.065 DD 100 6 1.20 1.00 1.00 1.10 1.00 94.70 93.080 6.830 0.090 DD&N 100 6 1.20 2.70 2.50 2.30 87.10 4.20 2.745 14.060 83.195 AC 500 1 0.56 0.60 0.78 0.36 0.26 97.44 96.992 2.996 0.012 DD 500 1 0.40 0.60 0.54 0.74 0.24 97.48 97.137 2.844 0.019 DD&N 500 1 0.64 0.40 0.50 0.36 0.30 97.80 97.604 2.387 0.009 AC 500 3 0.58 0.50 0.28 0.40 0.34 97.90 97.578 2.407 0.015 DD 500 3 0.36 0.48 0.54 0.38 0.26 97.98 97.600 2.389 0.011 DD&N 500 3 0.52 0.84 0.36 0.34 0.30 97.64 97.349 2.629 0.022 AC 500 6 0.70 0.52 0.56 0.46 0.36 97.40 96.833 3.144 0.023 DD 500 6 0.56 0.54 0.48 0.56 0.28 97.58 97.459 2.520 0.021 DD&N 500 6 0.74 0.44 0.46 0.42 0.34 97.60 97.388 2.591 0.021
As can be seen in Table 6, when the population size was large (n = 500) and when the AlwaysCooperate rule was used, the only possible outcome was that thugs invaded the population. When the population size was smaller, the results appeared to be random, so we ran the simulation for the second time, varying the following parameters (Table 7):
Parameter Used values population {50, 100} deterrence {1, 3, 6} rule {DefectD, DefectD&N}
We then counted the number of simulations for each combination of parameters invaded by ranger and thug agents. These results are presented in Table 8.
Population Deterrence Rule Rangers Thugs 50 1 DD 6 4 50 1 DD&N 6 4 50 3 DD 3 7 50 3 DD&N 6 4 50 6 DD 6 4 50 6 DD&N 8 2 100 1 DD 9 1 100 1 DD&N 9 1 100 3 DD 5 5 100 3 DD&N 5 5 100 6 DD 7 3 100 6 DD&N 8 2
We also decided to change the assumption that thugs are born with the estimate equal to 0.9 (see Appendix C) and observe if giving them even more advantage at the beginning will lead to more invasions. We ran our simulation once again varying the parameters as indicated in Table 7, but this time setting initial estimate of thugs to 0.1. It turned out that this does not significantly affect results and leads to similar pattern as depicted in Table 8 (63.33% invaded by rangers if initial estimate set to 0.1 vs. 65% with initial estimate set to 0.9).
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# Derivative of Dirac Delta function
Hello I'm trying to figure out how to evaluate(in the distribution sense)
$\delta'(g(x))$. Where $\delta(x)$ is the dirac delta function. Please notice that what I want to evaluate is not $\frac{d}{dx}(\delta(g(x)))$ but the derivative of the delta function calculated in g(x).
If anyone could post a proof, an idea to find the proof or a link it would be greatly appreciated!
## Answers and Replies
$\delta'$ is a linear operator on functions. Are you referring to its value when you pair it with g(x), as in
$$\langle \delta',\, g\rangle = \int_{\mathbb{R}} \delta'(x)g(x)\, dx \, ?$$
I'll assume you are. In that case, you use the integral notation above and then symbolically do integration by parts. Don't worry if it is not a well-defined operation because the answer you get is literally the definition of what you want.
Look here under distributional derivatives for more info:
http://en.wikipedia.org/wiki/Dirac_delta_function
No I'm sorry if I wasn't clear. I understand the value of
$\int \delta'(x)g(x)dx$
I'm asking the value of
$\int \delta'(g(x))\phi(x)dx$
Where $\phi(x)$ is the test function. Here it is not immediately obvious to me how to integrate by parts. I thought about this(but I am unsure of whether it is correct):
Assume that g(x) is an invertible function with as many derivatives as necessary(to keep things simple for now), so we substitute $y = g(x)$ and get
$\int \delta'(y)\frac{\phi(g^{-1}(y))}{g'(g^{-1}(y))} dy$ Now I could integrate by parts and get
$-\int \delta(y)\frac{d}{dy}(\frac{\phi(g^{-1}(y))}{g'(g^{-1}(y))}) dy$
Do you think my reasoning is correct up to here?
The dirac delta is just a normal distribution who's standard deviation approaches 0. Take the derivative of the normal dist. then take the limit as stdev =>0. I'm not sure if that's a valid way to do the problem, but its what I would try.
Ok thanks, I'll try that
Your calculation looks right to me, and what Aero said makes sense too. As far as proof goes, I can't remember exactly how general the rules are for changing variables like that. A book like Friedlander would probably have it...
Check out the very last post here for a similar problem:
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# Using same AES-GCM setup for multiple packets?
Most cryptographic libraries I've encountered uses some variant of processPlaintext(...) and then doFinal() to produce the ciphertext and resets the state.
In the context of packet communications, it feels like it would be optimal to setup key and IV once and then use that single initialisation in some way for all packets on the same channel.
In the case of CBC + padding, one could re-initialise the cipher with the last cipher block as IV.
(If I use a stream cipher, or manually implement CTR, it's straightforward of course.)
It would be nice to be able to use GCM to avoid manually adding authentication. However, it would appear that the usual implementations are geared towards encrypting a single packet - and resets all internal state on doFinal.
I would like to provide only a single IV for GCM, and then in some safe way either keeping the original state, or derive consequent IVs from the initial IV in a secure manner.
What I wonder is consequently:
Are there implementations that allow you not to reset the state when finalising a packet (i.e. making encryption and authentication dependent on the previous packet)? Failing that, what would be a safe way to derive consequent IVs from an initial IV value?
(I've not read about GCM sufficiently in depth - so I have no idea if there is a significant cost to setting up GCM with an IV)
-
In the context of packet communications, it feels like it would be optimal to setup key and IV once and then use that single initialisation in some way for all packets on the same channel.
It might be optimal for efficiency standpoint, but we also care about security; it isn't necessary very good for security. You mentioned that, with CBC mode, you can use the last cipherblock as the next IV; well, you could, but there are known weaknesses with that involving chosen plaintext. Other modes don't share this specific weakness.
As for GCM, well, it was designed to be used in a very specific manner (and proven to be safe in that manner). However, if you use it in a different manner (such as reusing the same GCM state to both output an authenticate tag for one message, and continue to encrypt with another) would be, at best, playing with fire.
Are there implementations that allow you not to reset the state when finalising a packet (i.e. making encryption and authentication dependent on the previous packet)?
I should hope not. Using the same nonce with two different messages is known to be VERY BAD, and the obvious way to do it with continuing to use the current GCM state would appear to fall into that weakness.
Failing that, what would be a safe way to derive consequent IVs from an initial IV value?
This one is easy: the only requirement that GCM has for IVs (nonces in GCM terminology) is that you can't use the same one twice; you can use any method of generating them that abides with this requirement.
Hence, one easy way to generate IVs is to generate them sequentially, as in: $$IV_{next} := IV_{previous} + 1$$
Oh, and you did mention that:
I have no idea if there is a significant cost to setting up GCM with an IV
Assuming you use a 96 bit IV (which is the cheap size for GCM), the total cost is "one AES block encryption" operation.
Hence, selecting a new IV for each packet is both moderately cheap, and safe.
And, on a side topic: you mention that you were intending this for use with "packet communication". Well, normally with this level of communication, you need to be able to deal with packet drops; that is, if the sender transmits a packet, but the receiver doesn't receive it correctly (or at all). Because of this, I suspect you'll want to design you system so that the receiver can process packet 5, even if has never received packet 4. Among other things, this implies that you might not be able to use implicit IVs; you may need to send them explicitly in the encrypted packet.
-
Sorry, I meant packet as in a "packet constructed from a [TCP/IP] stream using some delimiting scheme", so there are no "packet drops". – Nuoji Jul 20 '13 at 18:24
Under the requirement that each key is randomly generated during handshake, it should then be quite ok to start the IV at 0 and increase it from there? (Not that it is necessary) – Nuoji Jul 20 '13 at 18:25
A quick benchmark of Bouncy Castle's implementation shows that setting up a new IV takes more than ten times the cost of encrypting a short message. :( – Nuoji Jul 20 '13 at 18:45
@Nuoji: starting the IV at 0, and incrementing it for each encrypted message is fine; it meets the criteria that "no IV is used more than once" – poncho Jul 20 '13 at 19:08
@Nuoji - Bouncy Castle allows you to re-init the IV/nonce in all of the standard (CBC, CTR etc.) and AEAD (GCM, EAX etc.) modes without changing the key by specifying a null KeyParameter. e.g. cipher.init(new ParametersWithIV(null, ivBytes)) changes the IV, but leaves the key-dependent state alone. You can do the same thing through the JCE API by using RepeatedSecretKeySpec as the Key in Cipher.init(). The major cost in the Bouncy Castle GCM init is key based (GCM multiplier init and (e.g.) AES init) so doing this will avoid a lot of the cost of rotating the nonce. – archie Jul 23 '13 at 7:42
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Interpretting Standard Deviation of Random Effects in Sequential Mixed Effects Models
I am trying to ensure that my understanding of the random effects in Mixed Effects Models is correct, so I would like to share some R code and the standard deviations in the estimate of the random effect in sequential generalized logistic mixed effects regression models as well as my interpretation to double check with the Cross Validated community.
My understanding of fixed vs. random effects themselves is:
1. Fixed Effect – Measured effects for which intercepts of the observations will be estimated.
2. Random Effect – Considered unobserved and normally distributed random variables rather than unknown fixed parameters.
I will make a series of mixed effects models, each with an additional variable, as follows:
library("lme4")
library("titanic")
mod1 <- glmer(Survived ~ (1 | Embarked),
data = titanic_train,
family = binomial,
control = glmerControl(optimizer = "bobyqa"),
nAGQ = 1)
mod2 <- glmer(Survived ~ Pclass + (1 | Embarked),
data = titanic_train,
family = binomial,
control = glmerControl(optimizer = "bobyqa"),
nAGQ = 1)
mod3 <- glmer(Survived ~ Pclass + Sex + (1 | Embarked),
data = titanic_train,
family = binomial,
control = glmerControl(optimizer = "bobyqa"),
nAGQ = 1)
Here "Survived" is the outcome of interest, and I have three models:
1. "Embarked" as random effect
2. "Embarked" as random effect, Pclass as fixed effect
3. "Embarked" as random effect, Pclass and Sex as fixed effects
Now, if I check the standard deviations of the mixed effect (Embarked), I see that they decrease with each additional variable added:
> summary(mod1)$$varcor Groups Name Std.Dev. Embarked (Intercept) 0.37618 > summary(mod2)$$varcor
Groups Name Std.Dev.
Embarked (Intercept) 0.30105
> summary(mod3)$varcor Groups Name Std.Dev. Embarked (Intercept) 0.19804 Would it be correct to say that as the standard deviation decreases, it is implied that the covariates being added to the model are more sufficiently explaining the variation in the outcome as compared to the random effects’ estimates? Or stated differently, the mixed effects begin to appear more "similar" as subsequent covariates are added because the covariates being added explain more of the variation in outcome than do the mixed effects' estimates? The opposite interpretation being that if the standard deviation increased the covariates being added would explain the variation in the outcome less. If someone could answer these questions, especially with the help of formal logic, I would really appreciate it. 1 Answer The issue is complicated because your model is logistic. Under normal circumstances such as in a linear regression, most things you say would apply. Focusing on the linear model, I say most because adding variables should not increase random intercept variance even if the variables are mediocre predictors. The random intercept variance can go up very slightly but it shouldn't be by much. But with logistic regression, the case is not necessarily so. I'll make some claims then explain why at the end. If you add variables that explain the outcome better to a multilevel logistic regression, the variance of the random intercept will increase. However, if that variable also accounts for the differences between the embarks, then the random intercept variance may decrease. If that variable in no way accounts for any differences between embarks but explains the outcome better, the variance of the random intercept will definitely go up. An example is a variable that you have centered using the mean of each embark on that variable such that it doesn't vary across embarks. This is because the error variance is fixed to$\pi^2/3$, such that any improvements to the model will reflect in increased random intercept variance, unless such improvements simultaneously explain differences between embarks thus reducing the random intercept variance. I hope this makes some sense. Replying to your comments about the ICC. Since you are using R, check out the MuMIn package which has an R-squared glmm function. This should allow you to calculate$R^2\$ as defined by Nakagawa and Schielzeth http://dx.doi.org/10.1111/j.2041-210x.2012.00261.x Theirs is a relatively simple approach that takes the different sources of variance into consideration (fixed effects, random effects, logistic error) so that one can compare across models with varying fixed effects.
• Given that, could you comment on whether or not Intraclass Correlation would be a more appropriate statistic? – Edward Tyler Oct 3 '18 at 20:30
• @EdwardTyler added a paragraph to the answer to address this. – Heteroskedastic Jim Oct 3 '18 at 20:44
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# Important
This slide deck was created for use in a controlled environment, during a talk. It works best with (modern) Firefox OSX at 1024x768. Use Ctrl + {A, S, Z, X} in the live code examples to expose the inner & outer radii sizes of the top corners. The demos were live coded, so these slides are a bit pointless if you never watched the talk.
# Hi, I’m Lea
CSS WG Invited Expert, O’Reilly author (ETA Q4 2014), Ex-W3C staff. Made Prism, Dabblet, -prefix-free & more
And I’m Mr. Border-Radius.
You don’t think I’m cool, but you’re wrong.
# Browsersupport
Unprefixed since 5.0,
with `-webkit` since 1.0
Unprefixed since 4.0,
with `-moz-` since 2.0
Unprefixed since 10.5, no support in Mini
Unprefixed since 9.0
Unprefixed since 5.0 (4.0 in iOS),
with `-webkit-` since 3.1 (3.2 in iOS)
Unprefixed since 2.2,
with `-webkit-` since 2.1
That’s all. KTHXBAI!
w3.org/TR/css3-background/#corners
# Different radius per corner
`border-radius:` `20px` `100px``;`
```border-top-left-radius: 20px;
border-bottom-right-radius: 20px;```
```border-top-right-radius: 100px;
border-bottom-left-radius: 100px;```
`border-radius:` `20px` `100px` `50px``;`
`border-top-left-radius: 20px;`
```border-top-right-radius: 100px;
border-bottom-left-radius: 100px;```
`border-bottom-right-radius: 50px;`
`border-radius:` `20px` `100px` `50px` `0``;`
`border-top-left-radius: 20px;`
`border-top-right-radius: 100px;`
`border-bottom-right-radius: 50px;`
`border-bottom-left-radius: 0;`
# What you specify is not always what you get
“When the sum of any two adjacent border radii exceeds the size of the border box, UAs must proportionally reduce the used values of all border radii until none of them overlap.”
# Or, as they say in Maths...
$r′ top-left = min ( r top-left , width × r top-left r top-left + r top-right )$
↑ Isn’t MathML awesome?
docs.webplatform.org/mathml
↑ This would look much better if MathML was supported :(
docs.webplatform.org/mathml
# What we just learned
• We can have different radii per corner, in clockwise order
• Separate properties for every corner are available, like border-top-right-radius
• If the sum of two adjacent radii exceeds the length of their side, they are reduced proportionally
½
¼
# A few more shapes
`border-radius:`
`20px` `200px` `0` `100px` `/` `40px` `100px` `50px``;`
`border-top-left-radius: 20px 40px;`
`border-top-right-radius: 200px 100px;`
`border-bottom-right-radius: 0 50px;`
`border-bottom-left-radius: 100px;`
# What we just learned
• We can have different horizontal and vertical radii
• Percentages in border-radius refer to the corresponding dimension
• border-radius can take up to 8 (!) lengths
• When either the vertical or the horizontal radius of a corner is 0, there is no rounding
# Borders & inner radius
“The padding edge (inner border) radius is the outer border radius minus the corresponding border thickness. In the case where this results in a negative value, the inner radius is zero.”
# Or, as they say in Maths...
$r inner = max ( 0 , r outer - border )$ $⇒$ $r outer ≤ r inner + border$
# What we just learned
• border-radius rounds the outer border edge. The inner border radius will be smaller.
• Even with a circular radius, the inner radius might be elliptical, if different border widths are involved.
• If the borders have different colors, the transition will be abrupt and diagonal.
# What we just learned
• box-shadow follows the curves of border-radius
• box-shadow spread adds to the border-radius
• outline is always rectangular
• You can emulate rounded outlines with `box-shadow: 0 0 0 <length> <color>;`
# What we just learned
• border-radius is animatable!
• We can animate between any two radii
• border-radius affects the hit-testing area
# border-radius and overflow
But the bravest man amongst us is afraid of himself. The mutilation of the savage has its tragic survival in the self-denial that mars our lives. We are punished for our refusals. Every impulse that we strive to strangle broods in the mind and poisons us. The body sins once, and has done with its sin, for action is a mode of purification. Nothing remains then but the recollection of a pleasure, or the luxury of a regret. The only way to get rid of a temptation is to yield to it. Resist it, and your soul grows sick with longing for the things it has forbidden to itself, with desire for what its monstrous laws have made monstrous and unlawful. It has been said that the great events of the world take place in the brain. It is in the brain, and the brain only, that the great sins of the world take place also. You, Mr. Gray, you yourself, with your rose-red youth and your rose-white boyhood, you have had passions that have made you afraid, thoughts that have fined you with terror, day-dreams and sleeping dreams whose mere memory might stain your cheek with shame—’”
— Oscar Wilde, The Picture of Dorian Gray
# What we just learned
• `border-radius` does not affect text-wrapping
• `overflow: hidden` follows the rounding, but clips text
• Padding can help
• CSS Shapes Level 2 will solve this, but clumsily
# Scripting
• var mrBR = document.querySelector('.mr-border-radius');
• getComputedStyle(mrBR).borderTopLeftRadius;
• "100%"
• "360px 280px"
• getComputedStyle(mrBR).borderTopRightRadius;
• "80px"
• "53.3333px"
# What we just learned
• Incompatibilities in `getComputedStyle()` & `border-radius`
• Firefox converts percentages to pixels, all other browsers report percentages
• When the radius doesn’t fit, Firefox reports scaled down sizes, all other browsers report specified sizes
# Until then... (scoop)
HELP!!!11 I think I just forgot everything!
# Making of
• This slide deck was entirely built with open web technologies!
• Slideshow framework: github.com/LeaVerou/CSSS
• The illustrations are built with SVG
• Mr. Border-radius’ face is animated with SMIL
• The equations displayed were built with MathML
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## FAQ#
This assignment has an FAQ page.
The due date for this project is Sunday, 07/03 at midnight (Note: this is extended from Friday, 7/01). You cannot use slip days on projects, though you can turn in up to 2 days late for partial credit. If submitting with a partner, one of you should submit and add the other to your submission.
## Introduction #
In Project 1, we will build implementations of a “Double Ended Queue” using both lists and arrays in a package that other classes can use. The project is roughly split into three parts: the testing portion, the data structure portion and the application portion.
In the test writing part of the project, you will write unit tests to set goals and expectations for your finished project. You will fill in the two provided testing files, LinkedListDequeTest.java and ArrayDequeTest.java, and both will use the JUnit testing suite.
In the data structure part of the project, you will create two Java files: LinkedListDeque.java and ArrayDeque.java, with public methods listed below. You will be verifying the correctness of these data structures yourself using the randomized and timing test skills you gained from Lab 3.
In the application part of this project, you’ll create a Java file MaxArrayDeque.java as well as use your package to ultimately implement a sound synthesizer capable of playing music from Guitar Hero. You must test your MaxArrayDeque, but we’ll provide the tests for sound synthesizer.
We will provide relatively little scaffolding. In other words, we’ll say what you should do, but not how.
Additionally, we will be enforcing style. You must follow the style guide or you will lose points on the autograder.
## Getting the Skeleton Files and Working with a Partner (Applicable if you have a partner) #
As with all projects, we recommend doing pair programming for as much of the project as possible. This means that you and your partner should be on a Zoom call, where one person is typing and both are collectively deciding what to write. Some benefits of pair programming are:
• Both partners are on the same page and understand all parts of the project
• You are much more likely to catch bugs or mistakes when there are two sets of eyes
• You can discuss how to proceed, so you won’t get stuck as often
• You avoid merge conflicts.
The point about merge conflicts is especially important to avoid hassle with Git! To avoid merge conflicts, we recommend that you always clearly communicate with your partner what parts of the project you are working on, and avoid working on the same thing on each of your local computers. An example of a good workflow might be:
1. Both partners hop on a Zoom call
2. Partner1 pulls the skeleton code.
3. Both partners start thinking about what to write for the project. Partner1 types it down, acting as the scribe.
4. After working for an hour, Partner1 has to go to dinner. Partner1 pushes the code to the remote partner repository (nicknamed origin) and leaves the Zoom call.
5. Partner2 is bored and wants to keep working, so they pull from the partner repository (nicknamed origin). This gets Partner2 all the work that they just did on the Zoom call.
6. Partner2 works on ArrayDeque for a while, and lets Partner1 know.
7. After dinner, Partner1 also wants to work. They know Partner2 is currently making changes to ArrayDeque, so they decide to avoid making changes there to prevent merge conflicts. Instead, Partner1 works on LinkedListDeque.
8. Partner2 decides to go to bed. They push their changes to the remote.
9. Partner1 also finishes working on LinkedList. They try to push to the remote, but realize that the remote has changes they don’t have, so they must pull first. Partner1 pulls, and Git is able to automatically merge. Now, Partner1’s computer has the most up-to-date changes to ArrayDeque from Partner2, as well as Partner1’s changes to LinkedListDeque. Partner1 can push to the remote, and now the remote is fully up to date!
To do this, head to the folder containing your copy of your partner repository. For example, if your partnership is ‘p101’, then head to the ‘sp22-p101’ folder (or any subdirectory).
Make sure you’ve added the skeleton remote using the command git remote add skeleton https://github.com/cs61bl/skeleton-su22.git
To make sure you have the latest copy of the skeleton files, use the command:
git pull skeleton main
If you’re using a newer version of git, you might need to run:
git pull skeleton main --allow-unrelated-histories
You should now see a proj1 directory appear with two folders:
proj1
├── deque
└── ArrayDequeTest.java
└── Deque.java
└── gh2
├── GuitarHeroLite.java
├── GuitarPlayer.java
├── GuitarString.java
├── TTFAF.java
└── TestGuitarString.java
If you get some sort of error, STOP and either figure it out by carefully reading the git guide or seek help at Lab or Ed. You’ll potentially save yourself a lot of trouble vs. guess-and-check with git commands. If you find yourself trying to use commands recommended by Google like force push, don’t. Don’t use force push, even if a post you found on Stack Overflow says to do it!
The only provided files in the skeleton are the empty testing files as well as some skeleton for the second part of this project located in the gh2 folder (guitar hero 2).
Before we get into the details of the Deque API and the implementation requirements, let’s briefly talk about packages and why we are using them in this project.
### Packages #
Part of this project is using packages to separate logic and functionality. At the end of the project, you’ll have two packages: the deque package that provides an implementation of the Deque data structure, and the gh2 package that implements a synthesizer used to play guitar hero. You should already see folders with these names in the starter code, and your job is to implement them. Let’s look at the specifics for what a package really is.
A package is a collection of Java classes that all work together towards some common goal. We’ve already seen packages in CS 61B without knowing it. For example, org.junit is a package that contains various classes useful for testing, including our familiar Assert class, which contains useful static methods like assertEquals. In other words, when we saw org.junit.Assert.assertEquals, the org.junit was the package name, Assert was the class name, and assertEquals was the method name. We call org.junit.Assert.assertEquals the “canonical name” of the method, and we call assertEquals the “simple name” of the method.
When creating a package, we specify that code is part of a package by specifying the package name at the top of the file using the package keyword. For example, if we wanted to declare that a file is part of the deque package, we’d add the following line to the top of the file.
package deque;
If a programmer wanted to use a class or method from our deque package, they would have to either use the full canonical name, e.g. deque.ArrayDeque, or alternately use import deque.ArrayDeque, at which point they could just use the simple name ArrayDeque. So import statements just allow you to use the simple name of a class/method.
Typically, package names are the internet address of the entity writing the code, but backwards. For example, the JUnit library is hosted at junit.org, so the package is called org.junit.
Why are packages useful? It all boils down to that word “canonical”. As long as no two programmers use the same package name for their package, we can freely use the same class name in several different contexts. For example, there might exist a class called com.hrblock.TaxCalculator, which is different from com.turbotax.TaxCalculator. Given the requirement to either use the full canonical name or to use an import, this means we’ll never accidentally use one class when we meant to use the other.
Conceptually, you can think of packages as being similar to different folders on your computer. When you are building a large system, it is a good idea to organize it into different packages.
From this point forwards, most of our code in CS 61B will be part of a package.
With that out of the way, let’s talk about the methods that a Deque should have.
## The Deque API #
The double ended queue is very similar to the SLList and AList classes that we’ve discussed in class. Here is a definition from cplusplus.com.
Deque (usually pronounced like “deck”) is an irregular acronym of double-ended queue. Double-ended queues are sequence containers with dynamic sizes that can be expanded or contracted on both ends (either its front or its back).
Specifically, any deque implementation must have exactly the following operations:
• public void addFirst(T item): Adds an item of type T to the front of the deque. You can assume that item is never null.
• public void addLast(T item): Adds an item of type T to the back of the deque. You can assume that item is never null.
• public boolean isEmpty(): Returns true if deque is empty, false otherwise.
• public int size(): Returns the number of items in the deque.
• public void printDeque(): Prints the items in the deque from first to last, separated by a space. Once all the items have been printed, print out a new line.
• public T removeFirst(): Removes and returns the item at the front of the deque. If no such item exists, returns null.
• public T removeLast(): Removes and returns the item at the back of the deque. If no such item exists, returns null.
• public T get(int index): Gets the item at the given index, where 0 is the front, 1 is the next item, and so forth. If no such item exists, returns null. Must not alter the deque!
In addition, we also want our Deques to implement the special method:
• public boolean equals(Object o): Returns whether or not the parameter o is equal to the Deque. o is considered equal if it is a Deque and if it contains the same contents (as goverened by the generic T’s equals method) in the same order.
Your deques should accept any generic type (not just integers). For information on creating and using generic data structures, see lecture 5. Make sure to pay close attention to the rules of thumb on the last slide about generics.
So… we’ve defined a bunch of methods that any Deque should have. There are two specific ways we want you to implement a Deque (one powered by a Linked List, and the other by an array), but ultimately, they’ll have the same methods and external behavior. Have we learned about any programming tools that could enable us to do this? If you said, “Of course, silly, that sounds like an interface”, then you would be correct (and we would be silly)!
We have provided an empty file for the Deque interface. Recall that an interface does NOT typically provide implementations for any methods. Instead, it just lets you define what methods a class that implements the Deque interface must have.
### 0. Testing #
In agreement with the principles of test driven development, you will first write tests for the two Deque implementations, and then write the Deque implementations to pass the tests you wrote. Write JUnit tests that confirm your Deque has the expected behavior defined in the section above. Your two testing files, LinkedListDequeTest.java and ArrayDequeTest.java will likely be identical except for the dynamic type of the Deques being created. We recommend writing at least one test for every method described in the Deque API above, e.g. a test that targets the addFirst method.
The compiler will be upset if you write tests referencing classes/methods that haven’t been defined anywhere, so you’ll have to define the methods of the Deque interface. Write in the method signatures for all the methods described in the section above, and you should be good to start writing your tests.
Note: In the Deque interface, give isEmpty() a default implementation, which returns true if the size() is 0. Since your LinkedListDeque and ArrayDeque implement the Deque interface, given the default isEmpty() implementation, now you won’t have to define isEmpty in the LinkedListDeque and ArrayDeque classes!
You’ll fail all your tests, since you haven’t written your LinkedListDeque and ArrayDeque to do anything yet. This is the intention: in test driven development, you set the goals/expectations first and slowly build up your code base until you can pass the tests.
This portion of the project is meant to ensure that you learn to write your own tests, instead of just relying on the autograder. In the real world, there is no autograder to tell you if your code works! Additionally, running your own tests locally is much faster, easier, and doesn’t use up autograder tokens.
NOTE: To enforce the idea of Test-Driven Development (TDD), we will not be answering questions in project party/lab/ticketing system unless you have completed writing the tests in LinkedListDeque.java and ArrayDequeTest.java
### 1. Linked List Deque #
As your first deque implementation, you’ll build the LinkedListDeque class, which will be Linked List based.
Your operations are subject to the following rules:
Any reference to constant time simply means that the operation should not depend on the size of the deque (regardless of whether the deque has 10 elements or 100000, it should take the same amount of time to run.
• add and remove operations must not involve any looping or recursion. A single such operation must take “constant time”. This means that you cannot use loops that go over all/most elements of the deque.
• get must use iteration, not recursion.
• size must take constant time.
• Iterating over the LinkedListDeque using a for-each loop should take time proportional to the number of items.
• Do not maintain references to items that are no longer in the deque. The amount of memory that your program uses at any given time must be proportional to the number of items. For example, if you add 10,000 items to the deque, and then remove 9,999 items, the resulting memory usage should amount to a deque with 1 item, and not 10,000. Remember that the Java garbage collector will “delete” things for us if and only if there are no pointers to that object.
Implement all the methods listed above in “The Deque API” section. Add @Override tags to each method that overrides a Deque method.
In addition, you also need to implement:
• public LinkedListDeque(): Creates an empty linked list deque.
• public T getRecursive(int index): Same as get, but uses recursion.
You may add any private helper classes or methods in LinkedListDeque.java if you deem it necessary. If you do, please add helpful javadoc comments for your and your TAs sake.
While this may sound simple, there are many design issues to consider, and you may find the implementation more challenging than you’d expect. Make sure to consult the lecture on doubly linked lists, particularly the slides on sentinel nodes: two sentinel topology, and circular sentinel topology. I prefer the circular approach. You are not allowed to use Java’s built in LinkedList data structure (or any data structure from java.util.*) in your implementation and the autograder will instantly give you a 0 if we detect that you’ve imported any such data structure.
### 2. Array Deque #
Before you start ArrayDeque, please peruse these proj1 tips slides to get a sense of how to best approach the Array Deque!
As your second deque implementation, you’ll build the ArrayDeque class. This deque must use arrays as the core data structure.
For this implementation, your operations are subject to the following rules:
• add and remove must take constant time, except during resizing operations.
• get and size must take constant time.
• The starting size of your underlying array should be 8. At any given moment, the current size of the array should be proportional to the number of items.
• The amount of memory that your program uses at any given time must be proportional to the number of items. For example, if you add 10,000 items to the deque, and then remove 9,999 items, you shouldn’t still be using an array of length 10,000ish. For arrays of length 16 or more, your usage factor should always be at least 25%. This means that before performing a remove operation that will bring the number of elements in the array under 25% the length of the array, you should resize the size of the array down. For smaller arrays, your usage factor can be arbitrarily low.
Implement all the methods listed above in “The Deque API” section. Add @Override tags to each method that overrides a Deque method.
In addition, you also need to implement:
• public ArrayDeque(): Creates an empty array deque.
You may add any private helper classes or methods in ArrayDeque.java if you deem it necessary.
You will need to somehow keep track of what array indices hold the Deque’s front and back elements. We strongly recommend that you treat your array as circular for this exercise. In other words, if your front item is at position zero, and you addFirst, the new front should loop back around to the end of the array (so the new front item in the deque will be the last item in the underlying array). This will result in far fewer headaches than non-circular approaches. See the project 1 demo slides for more details.
Correctly resizing your array is very tricky, and will require some deep thought. Try drawing out various approaches by hand. It may take you quite some time to come up with the right approach, and we encourage you to debate the big ideas with your fellow students or TAs. Make sure that your actual implementation is by you alone.
MaxArrayDeque is out of scope for this project (6/29). Feel free to continue on to the Guitar Hero portion of the spec.
## MaxArrayDeque (OPTIONAL) #
After you’ve fully implemented your ArrayDeque and tested its correctness, you will now build the MaxArrayDeque. A MaxArrayDeque has all of the methods that an ArrayDeque has, but it also has 2 additional methods and a new constructor:
• public MaxArrayDeque(Comparator<T> c): creates a MaxArrayDeque with the given Comparator.
• public T max(): returns the maximum element in the deque as governed by the previously given Comparator. If the MaxArrayDeque is empty, simply return null.
• public T max(Comparator<T> c): returns the maximum element in the deque as governed by the parameter Comparator c. If the MaxArrayDeque is empty, simply return null.
The MaxArrayDeque can either tell you the max element in itself by using the Comparator<T> given to it in the constructor, or an arbitrary Comparator<T> that is different from the one given in the constructor.
Do not override the equals(Object o) method of this class.
If you find yourself starting off by copying your entire ArrayDeque implementation in a MaxArrayDeque file, then you’re doing it wrong. This is an exercise in clean code, and redundancy is one our worst enemies when battling complexity! For a hint, re-read the second sentence of this section above.
There are no runtime requirements on these additional methods, we only care about the correctness of your answer. Sometimes, there might be multiple elements in the MaxArrayDeque that are all equal and hence all the max: in in this case, you can return any of them and they will be considered correct.
You should write tests for this part as well! They do not need to be nearly as robust as your randomized and timing tests you created for the two Deque implementations above since the functionality you’re adding is fairly simple. You’ll likely be creating multiple Comparator<T> classes to test your code: this is the point! To get practice using Comparator objects to do something useful (find the maximum element) and to get practice writing your own Comparator classes. You will not be turning in these tests, but we still highly suggest making them for your sake.
You will not use the MaxArrayDeque you made for the next part. It is its own isolated exercise.
## Guitar Hero #
In this part of the project, we will create another package for generating synthesized musical instruments using the deque package we just made. We’ll get the opportunity to use our data structure for implementing an algorithm that allows us to simulate the plucking of a guitar string.
### The GH2 Package #
The gh2 package has just one primary component that you will edit:
• GuitarString, a class which uses an Deque<Double> to implement the Karplus-Strong algorithm to synthesize a guitar string sound.
We’ve provided you with skeleton code for GuitarString which is where you will use your deque package that you made in the first part of this project.
## GuitarString #
We want to finish the GuitarString file, which should use the deque package to replicate the sound of a plucked string. We’ll be using the Karplus-Strong algorithm, which is quite easy to implement with a Deque.
The Karplus-Algorithm is simply the following three steps:
1. Replace every item in a Deque with random noise (double values between -0.5 and 0.5).
2. Remove the front double in the Deque and average it with the next double in the Deque (hint: use removeFirst) and get()) multiplied by an energy decay factor of 0.996 (we’ll call this entire quantity newDouble). Then, add newDouble to the back of the Deque.
3. Play the double (newDouble) that you dequeued in step 2. Go back to step 2 (and repeat forever).
Or visually, if the Deque is as shown on the top, we’d remove the 0.2, combine it with the 0.4 to form 0.2988, add the 0.2988, and play the 0.2.
You can play a double value with the StdAudio.play method. For example StdAudio.play(0.333) will tell the diaphragm of your speaker to extend itself to 1/3rd of its total reach, StdAudio.play(-0.9) will tell it to stretch its little heart backwards almost as far as it can reach. Movement of the speaker diaphragm displaces air, and if you displace air in nice patterns, these disruptions will be interpreted by your consciousness as pleasing thanks to billions of years of evolution. See this page for more. If you simply do StdAudio.play(0.9) and never play anything again, the diaphragm shown in the image would just be sitting still 9/10ths of the way forwards.
Complete GuitarString.java so that it implements steps 1 and 2 of the Karplus-Strong algorithm. Note that you will have to fill you Deque buffer with zeros in the GuitarString constructor. Step 3 will be done by the client of the GuitarString class.
Make sure to open your project with Maven, as usual, otherwise IntelliJ won’t be able to find StdAudio.
For example, the provided TestGuitarString class provides a sample test testPluckTheAString that attempts to play an A-note on a guitar string. If you run the test should hear an A-note when you run this test. If you don’t, you should try the testTic method and debug from there. Consider adding a print or toString method to GuitarString.java that will help you see what’s going on between tics.
Note: we’ve said Deque here, but not specified which Deque implementation to use. That is because we only need those operations addLast, removeFirst, and get and we know that classes that implement Deque have them. So you are free to choose either the LinkedListDeque for the actual implementation, or the ArrayDeque. For an optional (but highly suggested) exercise, think about the tradeoffs with using one vs the other and discuss with your friends what you think the better choice is, or if they’re both equally fine choices.
#### GuitarHeroLite #
You should now also be able to use the GuitarHeroLite class. Running it will provide a graphical interface, allowing the user (you!) to interactively play sounds using the gh2 package’s GuitarString class.
The following part of the assignment is not graded.
Consider creating a program GuitarHero that is similar to GuitarHeroLite, but supports a total of 37 notes on the chromatic scale from 110Hz to 880Hz. Use the following 37 keys to represent the keyboard, from lowest note to highest note:
String keyboard = "q2we4r5ty7u8i9op-[=zxdcfvgbnjmk,.;/' ";
This keyboard arrangement imitates a piano keyboard: The “white keys” are on the qwerty and zxcv rows and the “black keys” on the 12345 and asdf rows of the keyboard.
The ith character of the string keyboard corresponds to a frequency of $$440 \cdot 2^{(i - 24) / 12}$$, so that the character ‘q’ is 110Hz, ‘i’ is 220Hz, ‘v’ is 440Hz, and ‘ ‘ is 880Hz. Don’t even think of including 37 individual GuitarString variables or a 37-way if statement! Instead, create an array of 37 GuitarString objects and use keyboard.indexOf(key) to figure out which key was typed. Make sure your program does not crash if a key is pressed that does not correspond to one of your 37 notes.
## Just For Fun: TTFAF #
Once you’re relatively comfortable that GuitarString should be working, try running TTFAF. Make sure your sound is on!
You can read the GuitarPlayer and TTFAF classes to figure out how they work. TTFAF in particular includes (as commented-out code) an example of how to use it another way.
## Even More Fun #
This part of the assignment is not graded and just for fun.
• Harp strings: Create a Harp class in the gh2 package. Flipping the sign of the new value before enqueueing it in tic() will change the sound from guitar-like to harp-like. You may want to play with the decay factors to improve the realism, and adjust the buffer sizes by a factor of two since the natural resonance frequency is cut in half by the tic() change.
• Drums: Create a Drum class in the gh2 package. Flipping the sign of a new value with probability 0.5 before enqueueing it in tic() will produce a drum sound. A decay factor of 1.0 (no decay) will yield a better sound, and you will need to adjust the set of frequencies used.
• Guitars play each note on one of 6 physical strings. To simulate this you can divide your GuitarString instances into 6 groups, and when a string is plucked, zero out all other strings in that group.
• Pianos come with a damper pedal which can be used to make the strings stationary. You can implement this by, on iterations where a certain key (such as Shift) is held down, changing the decay factor.
• While we have used equal temperament, the ear finds it more pleasing when musical intervals follow the small fractions in the just intonation system. For example, when a musician uses a brass instrument to play a perfect fifth harmonically, the ratio of frequencies is 3/2 = 1.5 rather than 27/12 ∼ 1.498. Write a program where each successive pair of notes has just intonation.
## Why It Works #
The two primary components that make the Karplus-Strong algorithm work are the ring buffer feedback mechanism and the averaging operation.
• The ring buffer feedback mechanism. The ring buffer models the medium (a string tied down at both ends) in which the energy travels back and forth. The length of the ring buffer determines the fundamental frequency of the resulting sound. Sonically, the feedback mechanism reinforces only the fundamental frequency and its harmonics (frequencies at integer multiples of the fundamental). The energy decay factor (.996 in this case) models the slight dissipation in energy as the wave makes a round trip through the string.
• The averaging operation. The averaging operation serves as a gentle low-pass filter (which removes higher frequencies while allowing lower frequencies to pass, hence the name). Because it is in the path of the feedback, this has the effect of gradually attenuating the higher harmonics while keeping the lower ones, which corresponds closely with how a plucked guitar string sounds.
To submit the project, add and commit your files, then push to your remote repository. Then on Gradescope go to the assignment and submit there. One partner should submit the project, and then add the other to the submission.
The entire project is worth 24 points, or 8% of your total grade.
• deque/LinkedListDequeTest.java: 1 point
• deque/ArrayDequeTest: 1 point
• deque/LinkedListDeque.java: 10 points
• deque/ArrayDeque: 10 points
• gh2/GuitarString: 2 points
Credits: RingBuffer figures from wikipedia. This assignment adapted from Kevin Wayne’s Guitar Heroine assignment.
## Tips #
• Check out the project 1 slides for some additional visually oriented tips.
• If you’re stuck and don’t even know where to start: One great first step is implementing SLList and/or AList. For maximum efficiency, work with a friend or two or three.
• Take things a little at a time. Writing tons of code all at once is going to lead to misery and only misery. If you wrote too much stuff and feel overwhelmed, comment out whatever is unnecessary.
• If your first try goes badly, don’t be afraid to scrap your code and start over. The amount of code for each class isn’t actually that much (my solution is about 130 lines for each .java file, including all comments and whitespace).
• For ArrayDeque, consider not doing resizing at all until you know your code works without it. Resizing is a performance optimization (and is required for full credit).
• Work out what your data structures will look like on paper before you try implementing them in code! If you can find a willing friend, have them give commands, while you attempt to draw everything out. Try to come up with operations that might reveal problems with your implementation.
• Make sure you think carefully about what happens if the data structure goes from empty, to some non-zero size (e.g. 4 items) back down to zero again, and then back to some non-zero size. This is a common oversight.
• Sentinel nodes make life much easier, once you understand them.
• Circular data structures may take a little while to understand, but make life much easier for both implementations (but especially the ArrayDeque).
• Consider a helper function to do little tasks like compute array indices. For example, in my implementation of ArrayDeque, I wrote a function called int minusOne(int index) that computed the index immediately “before” a given index.
• Consider using the Java Visualizer (which you installed in lab 1) to visualize your Deque as you step through with the debugger. The visualizer is an icon of a blue coffee cup with an eye, and is the tab next to the “Console” tab in the debugger panel). See the CS 61B plugin guide if you can’t figure out how to get the visualizer to show.
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Journal article Open Access
# On S, A, P, T, and R as comparative concepts for alignment typology
Haspelmath, Martin
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<dct:description><p>ABSTRACT:</p> <p>The terms S (intransitive), A, P (transitive), as well as T and R (ditransitive) have been used since the 1970s to allow linguists to characterize the differences between major alignment patterns such as accusative vs. ergative. They are often taken for granted, but a closer examination of the literature reveals that they can have three rather different meanings: In the Dixonian approach, they are used as universal syntactic functions based on transitivity; in the Comrian approach, they are seen as comparative concepts for the two arguments of a typical action clause, and in the Bickelian approach, they are taken as generalized semantic roles (Dowtyan proto-roles). In this paper, I explain the three approaches in some detail, and I argue that the Comrian approach is superior to the other two approaches. It is better than the Dixonian approach, because it does not take an undefined transitivity notion for granted but defines transitivity in terms of A and P. It is also better than the Bickelian approach, because the Dowtyan proto-roles were designed for the description of a particular class of English verbs; for cross-linguistic comparison, we need to limit ourselves to restricted (and semantically coherent) classes of verbs if we want to arrive at general statements.</p></dct:description>
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In what sense (if any) is Action a physical observable?
Is there any sense in which we can consider Action a physical observable? What would experiments measuring it even look like? I am interested in answers both in classical and quantum mechanics.
I ran across a physics textbook called "Motion Mountain" today, with volumes covering a broad swath of physics, written over the past decade by a dedicated German physicist with support from some foundations for physics outreach. So it appears to be a serious endeavor, but it's approach to many things is non-standard and often just sounds wrong to me. In discussing his approach to some topics the author says:
On action as an observable
Numerous physicists finish their university studies without knowing that action is a physical observable. Students need to learn this. Action is the integral of the Lagrangian over time. It is a physical observable: action measures how much is happening in a system over a lapse of time. If you falsely believe that action is not an observable, explore the issue and convince yourself - especially if you give lectures.
http://www.motionmountain.net/onteaching.html
Further on the author also discusses measurements of this physical observable, saying
No single experiment yields [...] action values smaller than hbar
So I think he does mean it literally that action is physically measurable and is furthermore quantized. But in what sense, if any, can we discuss action as an observable?
My current thoughts:
Not useful in answering the question in general, but hopefully explains where my confusion is coming from.
I will admit, as chastised in the quote, I did not learn this in university and actually it just sounds wrong to me. The textbook covers classical and quantum mechanics, and I really don't see how this idea fits in either.
Classical physics
The system evolves in a clear path, so I guess we could try to measure all the terms in the Lagrangian and integrate them along the path. However, multiple Lagrangians can describe the same evolution classically. The trivial example being scaling by a constant. Or consider the Lagrangian from electrodynamics which includes a term proportional to the vector potential, which is itself not directly measurable. So if Action was actually a "physical observable", one could determine the "correct" Lagrangian, which to me sounds like nonsense. Maybe I'm reading into the phrasing too much, but I cannot figure out how to interpret it in a manner that is actually both useful and correct.
Quantum mechanics
At least here, the constant scaling issue from classical physics goes away. However the way to use the Lagrangian in quantum mechanics is to sum over all the paths. Furthermore, the issue of the vector-potential remains. So I don't see how one could claim there is a definite action, let alone a measurable one. Alternatively we could approach this by asking if the Action can be viewed as a self-adjoint operator on Hilbert space ... but the Action is a functional of a specific path, it is not an operator that acts on a state in Hilbert space and gives you a new state. So at first blush it doesn't even appear to be in the same class of mathematical objects as observables.
Ultimately the comments that experiments have measured action and show it is quantized, make it sound like this is just routine and basic stuff I should have already learned. In what sense, if any, can we discuss action as a physical observable? What would experiments measuring it even look like?
• Related: physics.stackexchange.com/q/9686/2451, physics.stackexchange.com/q/41138/2451 and links therein. Jul 23, 2017 at 3:08
• Classically, if energy is measurable at all times then integral of energy over time is measurable. For a particle is integral of 1/2mv^2. I believe we should check the mathematical definitions in physics of action not deduce them only from a rant about teaching philosophy that pre-supposes we are familiar with the definition. Definition of action: en.m.wikipedia.org/wiki/Action_(physics) Stationary Action Principle: en.m.wikipedia.org/wiki/Stationary_Action_Principle , should be in the question Jul 28, 2021 at 22:03
• Also relevant... quora.com/Is-Motion-Mountain-Physics-a-good-reference-book Jul 29, 2021 at 2:53
The book propagates a myth.
Experiments measure angular momentum, not action - even though these have the same units. One finds empirically that angular momentum in any particular (unit length) direction appears in multiples of $$\hbar/2$$, due to the fact that its components generate the compact Lie group SO(3), or its double cover U(2).
That Planck's constant $$\hbar$$ is called the ''quantum of action'' is solely due to historical reasons. It does not imply that the action is quantized or that its minimal value is $$\hbar$$. Early quantum theories such as Bohr-Sommerfeld approximation used quantized action but this was an approximation to the more general quantization of Dirac, etc... One must additionally take care not to confuse the word action in "action-angle coordinates" with the action of variational calculus.
Indeed, the action of a system defined by a Lagrangian is a well-defined observable only in the very general and abstract sense of quantum mechanics, where every self-adjoint operator on a Hilbert space is called an observable, regardless of whether or not we have a way to measure it. The action of a system along a fixed dynamically-allowed path depends on an assumed initial time and final time, and it goes to zero as these times approach each other - this holds even when it is an operator. Hence its eigenvalues are continuous in time and must go to zero when the time interval tends to zero. This is incompatible with a spectrum consisting of integral or half integral multiples of $$\hbar$$.
• The value of the action between two time slices aka Hamilton's function is an observable on the phase space. Dec 5, 2020 at 8:09
• @Prof.Legolasov: How can it be observed? Dec 6, 2020 at 15:15
• it is a function on the phase space, therefore it can be observed by observing the coordinate and momentum and plugging them into the model-specific formula for the Hamilton-Jacobi function. In QM it becomes an operator that acts on the Hilbert space. Dec 12, 2020 at 7:18
• Arnold, can you answer the post below? It suggests that there is an error in your argument.
– user85598
Jan 26, 2021 at 6:28
• @Christian: By the way, the answer of Motion Mountain does not exhibit an error in my arguments but proves his contrary position by arguments from authority, which carry little substantial weight. Jan 26, 2021 at 21:16
In the language of the OP, action is a functional, since it is an integral of the Lagrangian... but over an arbitrary path. In other words, it is an abstract mathematical object, which has no counterpart in the real world.
This functional is then minimized in respect to all possible trajectories. In quantum mechanical terms the action along the optimal trajectory corresponds to the phase of a wave function, which is measurable (although defined up to a constant), e.g., in the experiments on Aharonov-Bohm effect, and any other interference experimente. The fact was recognized long before the advent of the path integrals - Landau&Livshitz derive quasiclassical approximation is an eiconal expansion of the phase of the wave function, which they openly call action.
• I believe youre supposed to take the actual path. It cannot be calculated if the path isnt known. Secondly, recall the principal of least action when calculating: en.wikipedia.org/wiki/Principle_of_least_action Classically, if energy is measurable at all times then integral of energy over time is measurable. For a particle is integral of 1/2mv^2. Jul 28, 2021 at 22:05
• @AlBrown Not sure what you disagree with, since you say yourself that action is known only for a specified path. It is like a function: $f(.)$ is an abstract object that cannot be measured, but for any given point $x$ the value of function at this point, $f(x)$ is a number that could be measurable. Jul 29, 2021 at 12:47
• That makes sense. I see Aug 1, 2021 at 22:28
One can measure an on-shell action by counting cycles and noting the phase within the cycle.
Details:
$$∂τ$$ is a proper time step of the system consisting of observable simultaneous components. By observing the components one can find repeated patterns. Basically all systems cycle.
$$τ=∫∂τ$$ is the constant information of the system or Hamilton's principal function.
$$0=\frac{dτ}{dt}=\frac{∂τ}{∂x}\;\frac{∂x}{∂t}+\frac{∂τ}{dt}\\ W=\frac{∂τ}{dt}=-\frac{∂τ}{∂x}\;\frac{∂x}{∂t}=-H\\ H=pẋ=mẋ²$$
Note, $$p = mẋ$$ is by observation, i.e. the physical content. Then it is assigned to $$\frac{∂τ}{∂x}$$ by definition, leading to $$H=mẋ²$$. $$∂τ=mẋ∂x$$ then says that $$ẋ$$ and $$∂x$$ contribute independently to a time step $$∂τ$$.
Splitting off some non-observable part of the system and associating it to the location of the observed part $$H(ẋ,x)=T(ẋ)+V(x)$$ half-half, makes $$T(ẋ)=mẋ²/2$$. Half-half is the usual but not mandatory choice for $$V$$.
$$W=-H$$ stays constant. It is the cycle's information divided by the period time. $$I=∮(∂τ/∂t)dt=∮(-H)t=Wt$$. One cannot minimize $$∫Wdt$$ because it increases monotonously, counting until the system ceases to exist.
$$L(x)=mẋ²+W=mẋ-H$$ oscillates and returns to 0 in a cycle. $$J=∫Ldt$$ returns to the same value after one or many cycles. Minimizing this produces conditions to attribute observables to the same system time step (the equations of motion).
$$0 = \frac{δJ}{δx} = \frac{1}{δx}∫\left(δx\frac{∂L}{∂x}+δẋ\frac{∂L}{∂ẋ}\right)dt = \frac{1}{δx}∫δx\left(\frac{∂L}{∂x}-\frac{d}{dt}\;\frac{∂L}{∂ẋ}\right)dt \\ \frac{∂L}{∂x} = \frac{d}{dt}\;\frac{∂L}{∂ẋ} \\ F=ṗ$$
One can measure an action satisfying the equations of motion by counting cycles and noting the phase within the cycle.
Measuring our every-day time is also via measuring action.
Comparing system changes to our time unit motivates energy $$W$$. $$W=\frac{∂τ}{dt}$$ is system time divided by our time, which is frequency times a factor to keep the units consistent.
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### ABOUT THAT UNDER CONTROL INFLATION
Posted on 16th October 2012 by Administrator in Economy |Politics |Social Issues
Today’s CPI misinformation report dutifully reports that inflation has been virtually non-existent over the last 12 months. The charts below beg to differ. But Bernanke doesn’t think food, energy and copper are that important to the average schmuck. With annual increases ranging from 4% to 35% for things we need everyday, how could the CPI only show annual inflation of 2.0%? We know from prior posts that 24% of the CPI is made up of owners equivalent rent. I keep reading that rents are skyrocketing by 10% and home prices are increasing. So, why isn’t this being heavily reflected in the CPI? The drones at the BLS say owners equivalent rent is only up 2.1% in the last year. The people running this country have no problem lying out of both sides of their mouth, saying housing is recovering strongly but their is no inflation in housing or rent.
I’ll tell you why the numbers don’t make sense. Because the Federal Government and the drones at the BLS are lying about the true inflation figures. They use their little regression models to manipulate the data and report tame inflation, when anyone with half a brain that fills up their tank, pays their utility bills, or goes grocery shopping knows that inflation is running north of 5%. The BLS is actually reporting that your food costs have only gone up by 0.8% in the last year.
Even the commodities below that show modest year over year changes should worry you.
• Heating oil is up 9% YTD and natural gas has surged 21% since May, just as we enter what is expected to be a colder than normal winter. Those senior citizens who are getting a $15 increase in their SS checks per month will just have to bundle up. • Unleaded gas is up 19% YTD and 2012 has seen the HIGHEST average price in the history of the U.S. • The huge increases in wheat, soybeans and corn due to the drought have resulted in some of the lowest inventories in history. The true impact of these price increases will really hit in 2013. Farmers have had to slaughter their cattle and hogs earlier to save on feed costs and this has resulted in meat prices temporarily declining. Prices for meat will soar next year as their is less supply. The other side effect will be unrest around the world, as food costs account for 50% of the budgets for poor people around the world. The weightings in the CPI calculation are a joke. They have the balls to tell you that motor fuel only makes up 5% of your costs and food at home less than 9% of your costs. Let’s examine those assumptions. The median household income is$50,000. A family of four with both parents working would drive on average 12,000 miles per year for each of their two cars. That is 24,000 miles per year at 20 mpg equaling 1,200 gallons of gas used per year. At $3.80 per gallon, that would be an annual cost of$4,560. That would be 9.1% of your costs for a normal family. That is 80% more than the BLS weighting for motor fuel.
A normal family of four, based on my grocery expenditures of $150 to$200 per week, would spend $7,800 to$10,400 per year for food at home. Even using the low figure, it comes to 15% for a median income family. That is 67% higher than the BLS weighting. I would urge you to put your own circumstances into these equations and figure out if the BLS is full of shit. Thinking is essential to defeating the powers that be.
# WHEAT
1. ThePessimisticChemist says:
I’ve talked my wife into looking at houses out in the country, with enough ground to have a veggie garden and hopefully some chickens.
I’d love to have enough room to have a jersey cow in addition to that, but I think I might be reaching.
Like or Dislike: 4 0
16th October 2012 at 9:50 am
2. Eddie says:
Reach for the cows, and if you miss them, at least you’ll be likely to land in….,well, compostable organic matter.
Like or Dislike: 4 0
16th October 2012 at 10:02 am
3. ThePessimisticChemist says:
@Eddie – I think I could talk her into snagging a property with the correct amount/quality of land for the setup I envision, but I’m not sure I would have the time to take care of everything properly.
Man, I wish I made a hair more, then I would just hire a farmhand out for a set amount of cash each month and produce enough eggs/milk for BOTH of our families lol
Like or Dislike: 3 0
16th October 2012 at 10:05 am
Except For Food And Gas, September Inflation Barely Higher
Submitted by Tyler Durden on 10/16/2012 08:45 -0400
September core CPI, ex such trivial, hedonically displacable items as food and energy (remember: when in doubt, just nibble on your obsolete first generation iPad, for which you waited hours in line – cause Bill Dudley said so) rose a tiny 0.1%, on expectations of a 0.2% pick up. Of course, for those lucky few who still get to eat and/or have a job to drive to, inflation rose by 0.6% in September from August, higher than expectations of a 0.5% increase. Luckily, in America the intersection of the Venn Diagrams for those who i) eat and ii) drive is so small it is barely worth mentioning…
Like or Dislike: 1 0
16th October 2012 at 10:06 am
——————————————————————————–
Posted 2012-10-16 08:44
by Karl Denninger
CPI: Nobody Buys Food Or Energy
There’s no problem with food and energy prices going up…. right?
The Consumer Price Index for All Urban Consumers (CPI-U) increased 0.6 percent in September on a seasonally adjusted basis, the U.S. Bureau of Labor Statistics reported today. Over the last 12 months, the all items index increased 2.0 percent before seasonal adjustment.
For the second month in a row, the substantial increase in the all items index was mostly the result of an increase in the gasoline index, which rose 7.0 percent in September after increasing 9.0 percent in August. The other major energy indexes increased in September as well.
That’ll leave a mark.
Let’s have a look inside at the unadjusted figures.
Notables are Dairy, fruits and vegetables, up 0.5% and 0.4% respectively, while meats and cereals were each down a similar amount. Non-alcoholic beverages were up 0.7% on the month, which is a curious change.
Energy commodities were up in a seirous way, other than piped gas and electricity; all sorts of fuels were up 4.1% (again, unadjusted.) And all items less food and energy, unadjusted, was up 0.3%, not 0.1%, which is a 3.7% annualized rate of increase in core.
Apparel was up 4.1% while used vehicles were down 2.1%. Demand problem? Hmmm….
Finally, physicians and hospitals were up 0.4 and 0.5%, respectively, on the month.
If you want to see real stunners, look in the detail tables. Women’s clothing was up a shocking 24.2% on the month for outerwear and 13.4% for dresses (!) Girls wasn’t much better, up 10.1%. It appears that whatever the “new fashion” game may be for this fall the clothing designers are screwing women in a big way; my recommendation would be to not bite on that bait.
Another interesting internal indication in the data is that car and truck rental prices were down over 7% on the month; this may be an indication of travel softness. These figures have been weak for the last year, and this is a rather notable secondary indication on business travel in particular.
Like or Dislike: 1 0
16th October 2012 at 10:07 am
The PC,
No need to move to the country (well, except for the cow;)).
Just buy a house on a large enough lot and go ahead and have your garden and chickens.
We live in a very nice neighborhood and people are doing it.
There’s no HOA as of now, but we do have convenants, which they are obviously choosing to ignore.
One neighbor has complained to the developer, to no avail, and they’re his damn covenants. Ha!
Like or Dislike: 3 0
16th October 2012 at 10:12 am
7. Eddie says:
When you think about it, lying about inflation is the most important job that government statisticians have…because it’s an absolutely crucial part of the plan to destroy the dollar without causing a run on the currency.
This was the mistake that the Fed made in the 1970s, not controlling media spin. Inflation was constant headline news. They have learned how to control the stats and the media.
They are right now, today, relentlessly shorting gold and silver in the paper markets.
It’s a perfect storm of manipulation, misinformation and deception. Most people are fooled.
Well-loved. Like or Dislike: 6 0
16th October 2012 at 10:20 am
8. ThePessimisticChemist says:
“The PC,
No need to move to the country” – MM
=================================
Unfortunately the amount of money we would need to spend in town to get even a decent sized lawn is pretty large compared to how much we could get out in the countryside.
You get about triple the land, with the same size/quality house, for the same price.
The main thing is commuting. We have really gotten use to zero commute and she doesn’t want to go back to burning up gas again.
Like or Dislike: 1 1
16th October 2012 at 10:24 am
9. Persnickety says:
@TPC: garden and chickens are practical even with a postage stamp lawn (though city laws or the evil HOA may beg to differ). A 1/2 acre is more than plenty for those uses, and a 1/4 acre would satisfy most people given the time commitment.
As for the cow, well… can you use 5 to 15 gallons of milk A DAY? People who want home dairy, which is great btw, generally get dairy goats – think about 1 gal of milk/day, drink some and make cheese with the rest. That’s much more practical. As a bonus, if you get kicked you get a mild bruise, not a weeklong hospital stay.
Sounds like you are somewhere super-urban – are there any gardening or organic food co-ops or community gardens that you could join/use? If I were in the city and wanted to stay there, that’s what I would look at doing. Beats commuting. If there isn’t one already, maybe look at starting one.
Like or Dislike: 3 0
16th October 2012 at 10:50 am
10. Persnickety says:
{Rant mode: on}
{Begin unfiltered reality}
I’ve been living out in farm country for about six years now, dragged by my horse-fascinated bride. I’ve learned a number of things that I will now share with you so you can make a better evaluation of food and living options:
Raising your own meat – chickens and rabbits are very practical, with rabbits more economical. You could do goats or sheep with an acre plus and some commitment. Forget beef cattle. Really. They are huge and dangerous, cost a lot to maintain, damage your fences, and are too big for you to slaughter yourself. Once you take them somewhere, you will then need to find a place to store 500-700lbs of beef. That will entirely fill 1 to 1.5 large chest freezers. We’ve done this twice, and each time we were frantically contacting relatives to give away what we couldn’t store, AND we had to get dry ice to get the now-full freezer to cool back down, because the overload of 400lbs of new material at just 25-30 degrees F. made our nearly new freezer overheat – it couldn’t cool that much down quickly enough.
Dairy – dairy goats. Forget dairy cows. If you had any idea the amount of work that goes into maintaining dairy cows you would be astonished that cow milk is even available for sale. It’s only widespread because it lends itself to resource-intensive industrial-scale production, plus the government subsidies, plus the American belief that milk is an essential part of the daily diet for anyone under 18.
Growing food: think vegetables and whatever you might grow in your garden. You’re growing tomatos and zucchini because it’s easy to grow and harvest by hand. Notice that you aren’t growing wheat or field corn? There’s a reason for that. They also require large fertilizer inputs. Oh, and look up “threshing” if you’re not already familiar with that. It can be done by hand, much as castles can be built by hand, but having a combine ($200k+ for a machine that gets used 100 hours a year, maybe) is the only fast way to do it, and not practical on less than 80 acres, minimum. {Rant mode: off} {End unfiltered reality} Note: I am not anti-meat or dairy, and in fact will probably have both twice today. I just have learned that there’s a huge difference between the food that is provided by our industrial agribusiness and the food you can realistically grow at home without major farm machinery. Well-loved. Like or Dislike: 9 0 16th October 2012 at 11:06 am 11. Stucky says: The Bureau of Lying Shitheads releases these lies for two reasons; 1) Get Obama reelected 2) 57 million get Social Security. Annual increases are tied in to inflation. So, now they can perpetuate the SS ponzi scheme by giving SS recipients a 1.5 increase this year. Let those oldfuks eat cat food!! \end:sarcasm/ They ain’t fooling nobody …. at least those of us, like TBPers, who have at least two synapse connections. Which, of course, leaves out the 50% Obama loving FSA fuckwads. Like or Dislike: 3 0 16th October 2012 at 11:17 am 12. Stucky says: An amazing coincidence …. or a BLS sack ‘o shit conspiracy? For November unemployment rates on Presidential election years going back to FDR ….. EVERY SINGLE TIME when a Democrat was in the White House, unemployment was DOWN from the term before, and every time a Republican was in the White House (sans Reagan), unemployment had gone UP since the previous election. Like or Dislike: 0 0 16th October 2012 at 11:31 am 13. Eddie says: Persnickety You are the man! I’d say everything you said makes absolute perfect sense. To that I’d add….forget about being able to do any of that without being on site daily. You just can’t live in the city and viisit the rural property once a week and get anything done..It just snowballs out of control. Well-loved. Like or Dislike: 5 0 16th October 2012 at 11:34 am 14. matt says: Inflation has only risen on energy and food for now, water and oxygen coming soon. Like or Dislike: 4 0 16th October 2012 at 11:47 am 15. ThePessimisticChemist says: “Dairy – dairy goats. Forget dairy cows. If you had any idea the amount of work that goes into maintaining dairy cows you would be astonished that cow milk is even available for sale. It’s only widespread because it lends itself to resource-intensive industrial-scale production, plus the government subsidies, plus the American belief that milk is an essential part of the daily diet for anyone under 18. If I had a dairy cow it would be providing for my family and someone else’s, else I would probably not even bother with my own dairy. You are spot on with the beef cow, those things require a lot of work, and again its something you either split with another family, or you have 6 kids because thats what it will take to consume all that food. Like or Dislike: 4 0 16th October 2012 at 1:32 pm 16. Outtahere says: Wherever you live, rabbits are the way to go. Low maintenance, high volume of reproduction, highest protein meat available, rabbits eat most anything green (yard clippings included) and there’s no waste with a rabbit. You can eat the meat, sell the skin, use the manure for an excellent fertilizer and also use the entrails for fertilizer. You can sell or barter the extras that you don’t eat or put in the freezer. It’s a win win with rabbits! Well-loved. Like or Dislike: 5 0 16th October 2012 at 1:39 pm 17. Administrator says: Inflation: Washington is Blind to Main Street’s Biggest Concern Peter Schiff Euro Pacific Capital, Inc. Posted Oct 16, 2012 Journalists, politicians and economists all seem to agree that the biggest economic issue currently worrying voters is unemployment. It follows then that most believe that the deciding factor in the presidential race will be the ability of each candidate to convince the public that his policies will create jobs. It seems that everyone got this memo… except the voters. According to the results of a Fox News poll released last week (a random telephone sample of more than 1,200 registered voters), 41% identified “inflation” as “the biggest economic problem they faced.” This is nearly double the 24% that named “unemployment” as their chief concern. For further comparison, 19% identified “taxes” and 7% “the housing market” as their primary concern. A full 44% of women, who often do more of the household shopping and would therefore be more sensitive to price changes, identified rising prices as their primary concern. My most recent video blog addresses this topic in detail. While these statistics do not surprise me, they should shock the hell out of the establishment. According to the Federal Reserve, inflation is not a concern at all. Time after time, in front of Congress and the press, Fed Chairman Ben Bernanke has said that inflation is contained and that it is below the Fed’s “mandated” rate of inflation (whatever that may be.) The Bureau of Labor Statistics is saying the same thing. The measures they use to monitor inflation, such as the Consumer Price Index (CPI) and the Personal Consumption Expenditure (PCE), show annual inflation well below 2%. In fact, the GDP price deflator used by the Commerce Department to calculate the second quarter’s 1.3% annual growth rate assumed annual inflation was running at just 1.6%. In fact, Bernanke thinks inflation is so low that he is actually worried about deflation, which he believes is a more dangerous issue. As a result, he is recommending policies that look to raise the inflation rate, not just to combat the phantom menace of deflation but to boost the housing market and reduce unemployment. He mistakenly believes these problems are the ones that concern Americans the most. If inflation really is as subdued as the government claims, how is it that so many people are concerned? It’s not as if the media or political candidates are fanning the fears of rising prices. In fact, given the media’s preoccupation with the housing market, the fact that nearly seven times as many Americans worry more about rising food prices than falling home prices shows just how large the inflation problem must be. Yet most economic observers continue to swallow the government’s inflation propaganda hook, line and sinker. In fact, although the Fox poll came out last week, I did not read or hear a single story on this topic, even from Fox news itself, which appears to not have noticed the significance of its own data. For years my critics have always attempted to discredit my inflation fears by pointing to government statistics showing low rates. However, I have long maintained that such statistics under-report inflation, and the results of this poll seem to confirm my suspicion. There are only two possible ways to explain the disconnect. Either the government is correct and consumers are worried about a non-existent problem, or the consumers’ concerns are real and the government’s statistics are not. From my perspective, it seems that it is far more likely that consumers are in the right. If so, we are in a lot of trouble. If annual inflation is actually higher than 3%, which would certainly be the case if consumers are so worried about it, then we are already in recession. Had government used a 3% inflation deflator (rather than the 1.6% that they actually used) to calculate 2nd quarter GDP, then growth would have been reported at negative .1% rather than the positive 1.3%. I believe that if the government used more accurate inflation data over the past several years, it is possible that we would have seen no statistical recovery from the recession that began in the fourth quarter of 2007. This would help explain why the “recovery” has failed to create jobs or lift personal incomes. The Fed’s zero percent interest rate policy is predicated on the assumption that there is currently no inflation. If this is not accurate, then they are making a major policy mistake. The Fed is easing when it should be tightening. If inflation is such a major concern now, imagine how much bigger the problem will become once the Fed achieves its goal of pushing the rate higher. More importantly, how much tighter will future monetary policy have to be to put the inflation genie back in her bottle? If inflation becomes so virulent before the Fed realizes its mistake, then it may be forced to raise interest rates significantly. U.S. national debt is projected to reach$20 trillion within a few years. As a result, a 10% interest rate (which would be needed to combat 1970′s style inflation) will require the U.S. government to pay about $2 trillion per year in interest on the national debt. This will absolutely upend all economic projections. Since 10% interest rates will likely crush the economy, not to mention the banks and the real estate market, tax revenues will plunge and non-interest government expenditures will go through the roof. Assuming we try to borrow the difference, annual budget deficits could go much, much higher from the already ridiculously high levels that they have reached during President Obama’s term. Annual deficits of$2 trillion, $3 trillion, or even$4 trillion, would result in a sovereign debt crisis that would force the Federal Government to either default on its obligations or inflate them away. Given the tendency for politicians to prefer the latter, voters who think rising prices are a problem now should just wait until they see what is waiting down the road!
Like or Dislike: 1 0
16th October 2012 at 1:43 pm
18. Kill Bill says:
Meh, just buy a house with a pool and raise catfish or tilapia.
Like or Dislike: 1 0
16th October 2012 at 1:47 pm
19. ThePessimisticChemist says:
“It’s a win win with rabbits!”
Yeah, I don’t think my wife would be ok with that. They are too close to the “cuddle” zone for her comfort.
Chickens though? I’ll send her out to collect eggs just once and she’ll never have trouble with taking off their heads again.
Like or Dislike: 4 0
16th October 2012 at 1:52 pm
20. Eddie says:
Common fallacies:
1. Belief that inflation and deflation cannot coexist at the same time.
2. Belief that hyperinflation is preceded by a slow rise in inflation.
3. Belief that a smart guy like Peter Schiff can win his party’s nomination against a wrestling promoter with no political experience.
Well-loved. Like or Dislike: 5 0
16th October 2012 at 1:53 pm
21. Kill Bill says:
In fact, Bernanke thinks inflation is so low that he is actually worried about deflation, which he believes is a more dangerous issue -admin
I think Bernanke is worried about deflation…in toxic asset prices, so what if more wealth needs to be extracted from the sheople in higher costs for fuel and food.
Like or Dislike: 2 0
16th October 2012 at 2:59 pm
The PC,
You are right. We (my dh) gave up 10 acres for a 1/2-acre lot. I still don’t understand it. But here we are. He cut his commute in half and I know that is great for him, so I’m ok with it (at least until the SHTF!)
Persnickety,
You are right also. Rabbit is delicious! Very much like chicken. When I was a child, my mom prepared rabbit for us every time my dad and brothers went hunting. Fried and then “smothered” in the oven or rabbit and dumplings. Yum! I never learned to clean them though…that was the men’s work :-/
After raising our own beef for a few years, dh and I were coming to the same conclusion about the beef cattle. Maybe a little risky at our middle-age and very pricey meat if you end up in the hospital. The largest animal we ever had was a 1200# mama-cow. But she was young and feisty. She unnerved me more than once (and I grew up on a farm!)
My parents made a very nice living for 25+ years on a dairy farm back when that was actually possible. It is a tremendous amount of work and takes some serious committment. Whether it’s one cow or 40, they must be milked daily. Days off? Forget about it unless you’ve got someone to take over for you.
And farming? Yup. You’ve got that down too.
Eddie,
We still have our rural property and just mowing the grass and keeping things tidy is almost more than dh has time for. Once upon a time, we considered having a garden there though we live about 30 minutes away. We can see now how nearly impossible that would be. It sure sounded like a good idea when we were brainstorming though.
Like or Dislike: 0 0
16th October 2012 at 3:06 pm
23. Willy2 says:
1. Stop using corn to make ethanol to be blended with gasoline.
2. Stop eating meat. Then grain prices will come down. It’s also good to reduce obesitas among US citizens. And too many US citizens are obese/overweight.
3. And Congress still thnks the reported inflation is too high. They want a new calculalation that would say that inflation is even lower.
4. Since 1981 the average yearly increase of wages has been below the real inflation. That means that workers have seen their real purchasing power shrink for 30 years. But employers shoot themselves in the foot as well because by reducing wages they curtail the purchasing power of their customers and that hurts profits.
Like or Dislike: 0 0
16th October 2012 at 5:01 pm
24. ThePessimisticChemist says:
“he largest animal we ever had was a 1200# mama-cow. But she was young and feisty. She unnerved me more than once (and I grew up on a farm!)”
I’m rather ashamed to say it, but my object of choice in the farm lot was a “walking tall” style board that I had reinforced at the handle side with duct tape.
I got injured by hogs, cattle and horses for years before I started carrying that thing. Never had a lick of trouble afterwards.
I did accidentally kill a hog with it once though :-/
Like or Dislike: 0 0
16th October 2012 at 10:26 pm
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# [FOM] consistency of NF?
Randall Holmes holmes at diamond.boisestate.edu
Fri Feb 25 15:39:34 EST 2005
Dear FOM'ers,
I am looking for refutations of a purported proof of the consistency
of NF, which is found at the head of the Current Research section of
my web page
http://math.boisestate.edu/~holmes/holmes/nfconsistency.ps
This is a quite brief argument, but it does require one to read a (very
nice) paper of Marcel Crabb\'e:
Marcel Crabb\'e, The
Hauptsatz for comprehension, a semantic proof'', {\em Mathematical
Logic Quarterly\/}, 40 (1994), 481-489.
in which he proves cut-elimination for NF with no extensionality axiom.
It appears to me at the moment that with a slight additional twist this
can be turned into an NF consistency proof.
While I believe this at the moment, I certainly don't quite meta-believe
it, so I await refutation with interest.
--Randall Holmes
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## Convergence Test
Convergence insufficiency is usually diagnosed in school-age children and adolescents. We will illustrate how partial sums are used to determine if an infinite series converges or diverges. Geometric series. About Convergence; What's in Convergence? Convergence Articles; Images for Classroom Use. ratio test for convergence worksheet. He will do this by asking you to focus on a target held say 70cms from your eyes and then move the target closer to your eyes. Note that at the endpoints of the interval, the ratio test fails. Get the free "Convergence Test" widget for your website, blog, Wordpress, Blogger, or iGoogle. The concern is whether this iteration will converge, and, if so, the rate of convergence. edu Introduction Gauss-Seidel method is an advantageous approach to solving a system of simultaneous linear equations because it allows. Apr 27, 2007 · A consistent test for the functional form of a regression based on a difference of variance estimators Dette, Holger, The Annals of Statistics, 1999 Moderate deviations of minimum contrast estimators under contamination Inglot, Tadeusz and Kallenberg, Wilbert C. Ask the patient to follow your finger as you bring it toward the bridge of his nose. Here are examples of each case: Example 5 Determine whether converges or diverges. People can pass the standard eye exam even if they have CI. Halmos Photograph Collection; Other Images; Critics Corner; Quotations; Problems from Another Time; Conference Calendar; Guidelines for Convergence Authors; MAA FOCUS; Math Horizons. A note about the geometric series Before we get into today's primary topic, I have to clear up a little detail about the geometric series. Convergence of Fourier Series Some Definitions A function $$f\left( x \right)$$ defined on an interval $$\left[ {a,b} \right]$$ is said to be piecewise continuous if it is continuous on the interval except for a finite number of jump discontinuities (Figure $$1$$). To distinguish between these four intervals, you must check convergence at the endpoints directly. convergence follows from the root test but not from the ratio test. Another test for convergence or divergence of a series is called the Integral Test. We will now look at a useful theorem that we can apply in order to determine whether a sequence of positive real numbers converges. How to know which convergence tests to apply for a series Given P an, if you are asked to compute the value of the series if it converges, here are some hints: • First thing first, the limit of a serie is the value. Infinite series whose terms alternate in sign are called alternating series. The convergence test is applied to the matrix equation, AX=B stored in the LinearSOE. For multiple sums, convergence tests are performed for each independent variable. Integral test. The test is a general case of Bertrand's test, the root test, Gauss's test, and Raabe's test. 6 {Ratio Test, Root Test, Absolute Convergence Fall 2010 1 / 10. 00 diopters, right eye is leading). A person can pass the 20/20 test and still have convergence insufficiency. com FREE DELIVERY and Returns possible on eligible purchases. May 21, 2013 · Root Test. Usually the root or ratio test works best for this part. Find the interval of convergence for ∞ n=0 (x−3)n n. Convergence Insufficiency (CI) is characterized by a decreased ability to converge the eyes and maintain binocular fusion while focusing on a near target. Tips & Tricks: Convergence and Mesh Independence Study The previous posts have discussed the meshing requirements that we need to pay attention to for a valid result. Blogs, articles, and analyst reports have noted that the acceleration from the digital revolution is upending long-established industries. The convergence test is applied to the matrix equation, AX=B stored in the LinearSOE. This test can apply to any series and should be the first test used in determining the convergence or divergence of a series. Problems 1-38 from Stewarts Calculus, page 784 X n2 1 n2 +. The commentary on multi-industry convergence is vast. Find more Mathematics widgets in Wolfram|Alpha. Chapter 5 Sequences and Series of Functions In this chapter, we define and study the convergence of sequences and series of functions. The alternating series test (also known as the Leibniz test), is type of series test used to determine the convergence of series that alternate. Here are some important facts about the convergence of a power series. The symptoms of convergence insufficiency may include:. This page was last edited on 20 September 2019, at 11:05. 6) I Alternating series. The Chicken Littles are correct, we have a problem with 12 marker STR results. As a child, expl Play Convergence. Generally we use the Ratio Test to determine the divergence/convergence of series containing factorials, exponents, and other more complex terms. Sep 09, 2019 · The results of the test should be noted for example, NPC 7cm, CRP 12 cm Jump convergence. This causes double vision and great fatigue when trying to focus on printed material at reading distance. Use the ratio test to show that the Taylor series centered at 0 for sin(x) converges for all real numbers. 2 +8𝑛𝑛−1. What is convergence insufficiency? Convergence insufficiency occurs when the eyes are unable to work together during close tasks like reading, writing, and using a computer. Whilst it is not possible to isolate completely reflex convergence, the Capobianco test, which employs a deep red filter placed before one eye to produce 'partial dissociation', is purported to provide a measure of reflex convergence with little or no voluntary input (Capobianco, 1952). Chapter 5 Sequences and Series of Functions In this chapter, we define and study the convergence of sequences and series of functions. A power series converges absolutely in a symmetric interval about its expansion point, and diverges outside that symmetric interval. This test is detailed by working through several examples. Jun 23, 2016 · Primarily the focus is explaining convergence insufficiency and accomodative insufficiency. Here are some important facts about the convergence of a power series. Alternating series and absolute convergence (Sect. This page was last edited on 20 September 2019, at 11:05. The commentary on multi-industry convergence is vast. Unfortunately some improper integrals fails to fall under the scope of these tests but we will not deal with them here. What is a convergence test? Answer: This is a test that the optometrist does in order to test the ability of your eyes to come together effectively as you focus close to. Jul 15, 2017 · To diagnose convergence insufficiency, your eye doctor might: Take a medical history. 6 Absolute Convergence and the Ratio Test Absolute Convergence. The tests of convergence are very useful tools in handling such improper integrals. Convergence corresponds to a small 1. Next: Convergence of Infinite Sequences Example Our next task is to establish, given an infinite sequence, whether or not it converges. retinal convergence: the sharing of a single nerve fibre by several rods in the retina of the vertebrate EYE. This condition causes one eye to turn outward instead of inward with the other eye creating double or blurred vision. Series Convergence Tests Math 122 Calculus III D Joyce, Fall 2012 Some series converge, some diverge. The nearpoint of convergence was also measured using a penlight for 10 repetitions. Convergence theory presumes that as nations move from the early stages of industrialization toward becoming fully industrialized, they begin to resemble other industrialized societies in terms of societal norms and technology. Master of Science in Mathematics Lecture Notes. A test has convergent validity if it has a high correlation with another test that measures the same construct. Convergence Insufficiency and the Standard 20/20 Eye Test It is important to understand that a finding of 20/20 vision does not mean perfect or even good enough vision. All structured data from the main, Property, Lexeme, and EntitySchema namespaces is available under the Creative Commons CC0 License; text in the other namespaces is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. EX 4 Show converges absolutely. Semantic Scholar extracted view of "Modernization, Gender Role Convergence and Female Crime: A Further Test" by Timothy F. Convergence Disorder or Convergence Insufficiency is the principal cause of strain in the eyes, double vision or diplopia, headaches and blurred vision. You can use the Ratio Test (and sometimes, the Root Test) to determine the values for which a power series converges. The geometric series and the ratio test Today we are going to develop another test for convergence based on the interplay between the limit comparison test we developed last time andthe geometric series. The test is a general case of Bertrand's test, the root test, Gauss's test, and Raabe's test. Integral test. Harold's Series Convergence Tests Cheat Sheet 24 March 2016 1 Divergence or nth Term Test Choosing a Convergence Test for Infinite Series. Let : [, ∞) → + be a non-negative and monotonically decreasing function such that () =. We motivate and prove the Alternating Series Test and we also discuss absolute convergence and conditional convergence. Get the free "Infinite Series Analyzer" widget for your website, blog, Wordpress, Blogger, or iGoogle. Root Test Example (4 n 5 5 n 6) n n 1 f ¦ Test for convergence Lets evaluate the limit, L =Lim (a n) 1 n n o f Lim n o f ((4 n 5 5 n 6) n) 1 n Lim n o f 4 n 5 5 n 6 4 5 1 By the root test, since L<1, our series will converge. Convergence & Convergence Insufficiency Convergence is the coordinated movement and focus of our two eyes inward. The Worth 4 Dot test is indicated when stereopsis is below 40 secondsof arc. Absolute and conditional convergence Remarks: I Several convergence tests apply only to positive series. In fact, if the series is only conditionally convergent, then both the Ratio and Root Test will turn out to be. During this test, you're asked to read letters on an. If lim n!1 a n 6= 0, then the series diverges by the n-th Term Test (Vanishing Test). For an integer N and a continuous function f(x) that is defined as monotonic and decreasing on. The Integral Test. Convergence, Divergence, Pupillary Reactions and Accommodation of the Eyes from Faradic Stimulation of the Macaque Brain'f' ROBERT S. I Few examples. How is convergence insufficiency (CI) diagnosed? Eye specialists called optometrists or ophthalmologists diagnose CI. If the sequence of these partial sums {S n } converges to L, then the sum of the series converges to L. System response (stress, deformation) will converge to a repeatable solution with decreasing element size. This test can apply to any series and should be the first test used in determining the convergence or divergence of a series. The series converges. Tips & Tricks: Convergence and Mesh Independence Study The previous posts have discussed the meshing requirements that we need to pay attention to for a valid result. It is not di cult to prove Leibniz’s test. Geometric series. Cauchy root test With the default setting Method -> Automatic , a number of additional tests specific to different classes of sequences are used. Get the free "Convergence Test" widget for your website, blog, Wordpress, Blogger, or iGoogle. It is also used in the differential diagnosis of unilateral decreased VA. Note that at the endpoints of the interval, the ratio test fails. Use only the Divergence Test to determine if the statement is true, false, or can't be decided yet. The latest Tweets from Convergence Con (@conedtech): "Day 2 begins with another amazing breakfast. p-Series Convergence The p-series is given by 1/n p = 1/1 p + 1/2 p. Another test for convergence or divergence of a series is called the Integral Test. There is one important, and easy to understand result about uniform convergence which we need, but did not discuss. Sep 01, 2011 · Read "Predictors of orbital convergence in primates: A test of the snake detection hypothesis of primate evolution, Journal of Human Evolution" on DeepDyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. If lim n!1 a n 6= 0, then the series diverges by the n-th Term Test (Vanishing Test). This test is designed to check the driver's "internal clock. Si, GaAs, Ge, when you include only the valence electrons a cutoff of ~15-20. Reflex convergence. A person can pass the 20/20 test and still have convergence insufficiency. At Ookla, we are committed to ensuring that individuals with disabilities can access all of the content at www. mathematical criterion about whether a series converges or not. Convergence LMS - Deliver cutting-edge training with our flagship software and eLearning platform. SEE ALSO: Convergence Tests , Raabe's Test. Hartnagel et al. Perform more functions depending on your program. Remark on uniform convergence of series 1 Comparison test for uniform convergence In the introductory notes we discussed uniform convergence and norm. Let ρ n = |a n+1/a n| and ρ = lim n. retinal convergence: the sharing of a single nerve fibre by several rods in the retina of the vertebrate EYE. The series can be compared to an integral to establish convergence or divergence. Find more Mathematics widgets in Wolfram|Alpha. Instruct the patient to focus on the object and then slowly move the object closer to the patient at a steady rate, stopping 1 to 2 inches away from the patient's nose. You can use the Ratio Test (and sometimes, the Root Test) to determine the values for which a power series converges. The nearpoint of convergence was also measured using a penlight for 10 repetitions. Convergence skills are learned and developed during our early years. Get the free "Infinite Series Analyzer" widget for your website, blog, Wordpress, Blogger, or iGoogle. MACD represents the convergence and divergence of two moving averages. metropolitan cities. Sep 24, 2014 · And, a judge authorizing a search warrant to draw and test blood for the presence of drugs is going to give a lack of convergence weight among other observations of the presence of marijuana. class: center, middle, inverse, title-slide # Productivity Differences and Convergence Clubs in Latin America ### Carlos Mendez. If you find a series divergent by this method, you need not continue testing! If the series converges, you must proceed to one of the other tests we will discuss. ESMA is an authority of the. If lim n!1 n p ja nj= L = 1, then the test is inconclusive. Convergence of integration - test. If S¯ converges then S converges (absolutely). This condition causes one eye to turn outward instead of inward with the other eye creating double or blurred vision. Visit our Portrait Gallery & our Paul Halmos Photo Collection. Let ρ n = |a n+1/a n| and ρ = lim n. Infinite Series: Root Test For Convergence The root test may be used to test for convergence of an infinite series. The test was devised by the 19th-century German mathematician Peter Gustav Lejeune Dirichlet. Direct Comparison Test. A series P a n is called conditionally convergent if it is con-. I Integral test, direct comparison and limit comparison tests,. Apr 27, 2007 · A consistent test for the functional form of a regression based on a difference of variance estimators Dette, Holger, The Annals of Statistics, 1999 Moderate deviations of minimum contrast estimators under contamination Inglot, Tadeusz and Kallenberg, Wilbert C. It is easily demonstrated by having one eye fixate from a far point to a near point along its line of sight, while the other eye is occluded. Convergence corresponds to a small 1. As additional test it can be used for the patients with signs of the convergence insufficiency. We will use the comparison test to conclude about the convergence of this series. Therefore, one typically applies it for series that look divergent right from the start. Review your knowledge of the various convergence tests with some challenging problems. The picture shows. Should I use $$\frac{4n}{n^4}$$ which is $$\frac{4}{n^3},$$ but then I have to show convergence of \frac{4}{n^3}. Is this lack of convergence thing real, or just an unreliable indicator used by officers to justify investigations and arrests?. Understanding Drug Field Sobriety Evaluations. This test, according to Wikipedia, is one of the easiest tests to apply; hence it is the first "test" we check when trying to determine whether a series converges or diverges. The geometric series is used in the proof of Theorem 4. Move the pen torch rapidly between the two pupils, shining the light for three seconds in each eye. Convergence in distribution di ers from the other modes of convergence in that it is based not on a direct comparison of the random variables X n with X but rather on a comparison of the distributions PfX n 2Agand PfX 2Ag. The Chicken Littles are correct, we have a problem with 12 marker STR results. We will now look at a useful theorem that we can apply in order to determine whether a sequence of positive real numbers converges. Instruct the patient to focus on the object and then slowly move the object closer to the patient at a steady rate, stopping 1 to 2 inches away from the patient's nose. Convergence corresponds to a small 1. If you are not doing that, then maybe the title of this thread should be renamed. Convergence Insufficiency is a condition where both eyes do not aim at the same spot at close range, such as for reading. Anomalies of vergence. Effects of prism-induced, accommodative convergence stress on reading comprehension test scores: Journal of the American Optometric Association Vol 59(6) Jun 1988, 440-445. Convergent series, divergent series, power series, power series convergence, nth partial sum, remainder of a series, series rules, series. A vulnerability in the TFTP service of Cisco Network Convergence System 1000 Series software could allow an unauthenticated, remote attacker to retrieve arbitrary files from the targeted device, possibly resulting in information disclosure. I can't find any easily accessible (ie: online or in a popular book) source with a proof of Kummer's Test. Rods are used particularly in low illumination when the stimulus of light on a single rod may be insufficient to generate an ACTION POTENTIAL in the NEURONE. So how to conduct the convergence test or mesh refinement study in finite element. -Afterwards, carry out the sums and series a few steps as shown at the left. Before we do so, we must first prove the following lemma. edu Introduction Gauss-Seidel method is an advantageous approach to solving a system of simultaneous linear equations because it allows. It is the hope that an iteration in the general form of will eventually converge to the true solution of the problem at the limit when. Mar 27, 2017 · Convergence: Mesh convergence determines how many elements are required in a model to ensure that the results of an analysis are not affected by changing the size of the mesh. Cause and impact of the ITCZ. The statement clearly true for n=2. The book represents the 12 animals that highlight the chinese zodiacs. If you're behind a web filter, please make sure that the domains *. A pattern exists where the fractions will cross out and what is left is 1-1/(n+2). Part 1: (45 pts) Test for convergence. Convergence of Fourier Series Some Definitions A function $$f\left( x \right)$$ defined on an interval $$\left[ {a,b} \right]$$ is said to be piecewise continuous if it is continuous on the interval except for a finite number of jump discontinuities (Figure $$1$$). I Few examples. X's examination revealed visual-motor problems, double vision, and a condition called convergence insufficiency. For policymakers it is a policy goal and challenge. This leads to a new concept when dealing with power series: the interval of convergence. Comparasion Test:. Chapter 5 Sequences and Series of Functions In this chapter, we define and study the convergence of sequences and series of functions. It is one of the most commonly used tests for determining the convergence or divergence of series. 6 Absolute Convergence and the Ratio Test Absolute Convergence. You can use the Ratio Test (and sometimes, the Root Test) to determine the values for which a power series converges. In this paper we test for the relevance of financial market characteristics in explaining this divergence in the catching-up process in Europe and Asia. When you are doing a Mesh Convergence Study, you don't change any geometry. Halmos Photograph Collection; Other Images; Critics Corner; Quotations; Problems from Another Time; Conference Calendar; Guidelines for Convergence Authors; MAA FOCUS; Math Horizons. Convergence of Gauss-Seidel Method Jamie Trahan, Autar Kaw, Kevin Martin University of South Florida United States of America [email protected] Here are some important facts about the convergence of a power series. tending to move toward one point or to approach each other : converging; exhibiting convergence in form, function, or development…. Learn exactly what happened in this chapter, scene, or section of Economic Growth and what it means. Convergence insufficiency is usually diagnosed in school-age children and adolescents. Harold's Series Convergence Tests Cheat Sheet 24 March 2016 1 Divergence or nth Term Test Choosing a Convergence Test for Infinite Series. Korvax Convergence Cube is a curiosity of Korvax origin and one of the trade commodities. We have seen that the integral test allows us to determine the convergence or divergence of a series by comparing it to a related improper integral. Lack of Convergence and Romberg Balance Test. An infinite sequence (a n) is called convergent if limit n tends to infinity a n exists and is finite. The integral test for convergence of an infinite series is explained. For multiple sums, convergence tests are performed for each independent variable. This involves using the limit of the absolute value of the ratio of the n + 1 term to the n term as n. A series P a n is called conditionally convergent if it is con-. Jump convergence is a test of the maximum amount a patient can converge comfortably in free space through the prism. Tests for convergence and divergence are methods to determine the convergence or divergence of infinite series. Mar 27, 2018 · This calculus 2 video tutorial provides a basic introduction into series. Techniques for the Finding Interval of Convergence. ∑n^(5)/5^(n)^(4), n=1 to infinity ρ=limn→∞||a[n+1]/a[n]||=. In our increasingly uncertain world, bringing together the best and the brightest of the public and private sectors has never been more important. At Ookla, we are committed to ensuring that individuals with disabilities can access all of the content at www. CODA performs convergence diagnostics and statistical and graphical output analyses. A routine eye exam with the familiar 20/20 eye chart does not diagnose CI. Dirichlet's test, in analysis (a branch of mathematics), a test for determining if an infinite series converges to some finite value. Convergent Series: A series is convergent if the sequence of its partial sums converges. 7 For a series P 1 n=1 a n = a 1 + a 2 + a 3 + , determine if it converges toward a limit as we add more terms, or diverges (often to 1). My role was to build and maintain an E2E automated test suite for a large scale application used to report energy trades to EU regulatory bodies Key Achievements: • Reduced the time taken to test and deploy each monthly release by 50% by writing clean and robust automated tests • Completed a C# training course outside of. This leads to a new concept when dealing with power series: the interval of convergence. The picture shows. Second, find out the behavior of the series at each of the two endpoints, c - R and c + R. For the series above, the root test determines that the series converges for and divergesk kB " # for. That means a study on how the mesh, specifically the element size affects the results. If either test can be applied to the series, enter CONV if it converges or DIV if it diverges. Is this lack of convergence thing real, or just an unreliable indicator used by officers to justify investigations and arrests?. The ratio test is quite useful for determining the interval of convergence of power series, along the lines of the above example. The 20-20 passing grade simply means that a person can see clearly with at least one eye at the distance of 20 feet only. Get the free "Convergence Test" widget for your website, blog, Wordpress, Blogger, or iGoogle. iii) if ρ = 1, then the test is inconclusive. Convergence insufficiency is usually diagnosed in school-age children and adolescents. If you are not doing that, then maybe the title of this thread should be renamed. The test states that if you take the limit of the general term of the series and it does not equal to 0, then the series diverge. 12, which is known as the ratio test. And so we know this thing converges and we see that actually these two series combined meet all of the constraints we need for the comparison test. Series that are absolutely convergent are guaranteed to be convergent. $\endgroup$ - coffeemath Dec 10 '13 at 14:54. Patients with convergence insufficiency often leverage accommodation to drive convergence. I Absolute convergence test. -Take the limit of the remainder to see if the series converges or diverges. 6 Absolute Convergence and the Ratio Test Absolute Convergence. Equipment: The Dot Card or Brock String. With the geometric series, if r is between -1 and 1 then the series converges to 1 ⁄ (1 – r). The book represents the 12 animals that highlight the chinese zodiacs. For robustness sake, we have also used a panel stationarity test that. In our increasingly uncertain world, bringing together the best and the brightest of the public and private sectors has never been more important. The alternating series test (also known as the Leibniz test), is type of series test used to determine the convergence of series that alternate. However,. Tips & Tricks: Convergence and Mesh Independence Study The previous posts have discussed the meshing requirements that we need to pay attention to for a valid result. ESMA is an authority of the. Series Convergence Tests Math 122 Calculus III D Joyce, Fall 2012 Some series converge, some diverge. Convergent series, divergent series, power series, power series convergence, nth partial sum, remainder of a series, series rules, series. Summary of Convergence estsT for Series estT Series Convergence or Divergence Comments n th term test (or the zero test) X a n Diverges if lim n !1 a n 6= 0 Inconclusive if lim a n = 0. Convergence of Gauss-Seidel Method Jamie Trahan, Autar Kaw, Kevin Martin University of South Florida United States of America [email protected] Alternating series and absolute convergence (Sect. Halmos Photograph Collection; Other Images; Critics Corner; Quotations; Problems from Another Time; Conference Calendar; Guidelines for Convergence Authors; MAA FOCUS; Math Horizons. Dirichlet's test, in analysis (a branch of mathematics), a test for determining if an infinite series converges to some finite value. The 20-20 passing grade simply means that a person can see clearly with at least one eye at the distance of 20 feet only. net or Speedtest apps, please email [email protected] Therefore, one typically applies it for series that look divergent right from the start. Remark on uniform convergence of series 1 Comparison test for uniform convergence In the introductory notes we discussed uniform convergence and norm. A geometric series X1 n=0 arn converges when its ratio rlies in the interval ( 1;1), and, when it does, it converges to the. Test smooth pursuit by having the patient follow an object moved across their full range of horizontal and vertical eye movements. Statistical Convergence and Convergence in Statistics 3 branches of mathematics, such as, theory of metric and topological spaces, studies of convergence of sequences and functions, in the theory of linear systems, etc. Cauchy convergence test of a sequence. Confirm your already scheduled appointment. Infinite series whose terms alternate in sign are called alternating series. The proof is similar to the one used for real series, and we leave it for you to do. It is easily demonstrated by having one eye fixate from a far point to a near point along its line of sight, while the other eye is occluded. Convergence insufficiency is a condition in which your eyes are unable to work together when looking at nearby objects. The series converges. It explains the difference between a sequence and. The test states that if you take the limit of the general term of the series and it does not equal to 0, then the series diverge. This condition causes one eye to turn outward instead of inward with the other eye creating double or blurred vision. tending to move toward one point or to approach each other : converging; exhibiting convergence in form, function, or development…. If you can define f so that it is a continuous, positive, decreasing function from 1 to infinity (including 1) such that a[n]=f(n), then the sum will converge if and only if the integral of f from 1 to infinity converges. Apr 16, 2019 · The divergence test is a test on divergence, and nothing more, so it is a rather basic test. Could we also just solve for N by arriving at the inequality n > something and setting N = something? This gives a different expression for N, but then when you work backwards you arrive at the same inequality as above, so I'm thinking this approach might also work. Oct 30, 2019 · PathSensors, Inc. You originally asked about a Mesh Convergence Study. It is an area of high pressure. View Test Prep - Convergence Test Practice Problems from MATH 1552 at Georgia Institute Of Technology. Convergence of Fourier Series Some Definitions A function $$f\left( x \right)$$ defined on an interval $$\left[ {a,b} \right]$$ is said to be piecewise continuous if it is continuous on the interval except for a finite number of jump discontinuities (Figure $$1$$). The nearpoint of convergence was also measured using a penlight for 10 repetitions. •Test is abnormal if eyes slip off target or reports blurred vision and target motion •Repeat in vertical direction -Dynamic Visual Acuity Test (DVAT) •Using Snellen eye chart the patient reads the lowest line within their comfort •20 deg of head turns R/L are performed at 120 bpm. Conditional Convergence. An infinite sequence (a n) is called convergent if limit n tends to infinity a n exists and is finite. It is important to remember that your solution is the numerical solution to the problem that you posed by defining your mesh and boundary conditions. P 1 n=4 1diverges, so P 1 n=4 3 diverges. A comprehensive vision evaluation by an eye doctor who tests binocular (two-eyed) vision and who can refer or provide for in-office vision therapy is recommended for all individuals who do reading and deskwork -- particularly students of any age. The series can be compared to an integral to establish convergence or divergence. guarantee convergence because it is only based on observations from the chain. Certain SolutionAlgorithm objects require a ConvergenceTest object to determine if convergence has been achieved at the end of an iteration step. py: Python script for visualization of convergence results. As mesh elements decrease in size but increase in quantity, the computational requirements to solve a given model increase. edu Introduction Gauss-Seidel method is an advantageous approach to solving a system of simultaneous linear equations because it allows. These econometric methods are applied to analyze convergence in cost of living indices among 19 U. Oct 30, 2019 · PathSensors, Inc. If you're seeing this message, it means we're having trouble loading external resources on our website. This leads to a new concept when dealing with power series: the interval of convergence. gnu: Gnuplot script for visualization of convergence results. com for assistance. X's examination revealed visual-motor problems, double vision, and a condition called convergence insufficiency. Tests for convergence and divergence are methods to determine the convergence or divergence of infinite series. Loading Convergence. Convergence LMS - Deliver cutting-edge training with our flagship software and eLearning platform. In general there are no real firm answers on this. Since 0<1 (in this example the limit does not depend on the value of x), the series converges for all x. The test was first published by Jean le Rond d'Alembert and is sometimes known as d'Alembert's ratio test or as the Cauchy ratio test. Note the convergence of the eyes and pupillary constriction. Unfortunately some improper integrals fails to fall under the scope of these tests but we will not deal with them here. The steps are identical, but the outcomes are different!. Review of Series and Series Tests: (a) Let a n = n 3n+ 1. The physical meaning of this quantity depends on the integrator and constraint handler chosen. Tests for Series Convergence Geometric Series: A series of the form will converge if and only if <1. Solved Problems for Series: Testing convergence. Iteration is a common approach widely used in various numerical methods. P 1 n=4 1diverges, so P 1 n=4 3 diverges. This is always the sort of information that k kB the root test provides: " # RADIUS OF CONVERGENCE Let be a power series. There are many different ways to define the convergence of a sequence of functions, and different definitions lead to inequivalent types of convergence. Convergence objects inserted under an environment that is referenced by an Initial Condition object or a Thermal Condition load object, will invalidate either of these objects, and not allow a solution to progress. Speed of visual acuity for vertical eye-movements with a change of accommodation: Perceptual and Motor Skills Vol 69(3, Pt 1) Dec 1989, 751-754. Alternating series and absolute convergence (Sect. If lim n!1 a n 6= 0, then the series diverges by the n-th Term Test (Vanishing Test). Next: Convergence of Infinite Sequences Example Our next task is to establish, given an infinite sequence, whether or not it converges. Convergence insufficiency is usually diagnosed in school-age children and adolescents. Reflex convergence. Accessed on: 2019-12-01 08:04:09. Techniques for the Finding Interval of Convergence. How to use convergent in a sentence. Is there a convergence eye test?.
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# Beamer Presentations with Org Mode
Jun 1, 2020
When we moved all courses online for the semester because of the COVID-19 pandemic, I found myself having to make many more presentation slide-decks than I normally do, especially when I normally don’t do any at all. Keynote produces some beautiful slides, but it requires using the mouse more than I like. I wanted something that I could use to make slides with just a few keystrokes. I settled on exporting an Org mode document to Beamer slides. This is an explanation of the challenges that I had and the solutions that I found.
## Beamer 🔗
The Beamer class is a tool used to produce presentations with LaTeX. It provides the same advantages for presentations that standard LaTeX provides for other document types, including elegant mathematical formulas, transportability between computing platforms, and the ability focus on content when writing. There are disadvantages, though. Writing LaTeX can be cumbersome, since it is not a simple markup language. After gaining a certain proficiency, it is possible to write it fairly easily. Even so, it is never as simple as using something like Markdown.
## Solution 🔗
The solution that I found was Org mode, a very powerful system that includes a simple markup language and the ability to export to LaTeX. Producing the slide deck was easy, and there were many great examples online. Producing an article was also easy, again with great examples. Producing both a slide presentation and an article, using the same contents file should have been easy, since the official guide to Beamer explains how to do that in LaTeX. Unfortunately, I couldn’t find much guidance on how to do this with Org Mode. The only article I could find was this on the official Org mode site . It seemed too complicated to me, and I was tempted to just go back to directly writing LaTeX. I decided to try anyway, but I quickly ran into a series of problems:
1. The Org Beamer export automatically inserted a title slide with \maketitle. Unfortunately, using the same content for the presentation and notes requires using the ignorenonframetext switch in the document class declaration. Since the export didn’t place the command in a frame, there was no title slide.
2. I attempted to fix this by beginning the contents file with a title-less frame that included the \maketitle command. That generated a title page, but it came after the table of contents slide.
3. The article export did not treat headings correctly, and failed to recognize the Beamer-specific commands.
This was enough to make me want to scrap the project, especially when I looked at the site linked above. The author fixed the title issue by hacking the Beamer export file, something that I certainly didn’t want to do. So, as is often the case, after hours of searching for solutions, I began to wonder if the solution could be much simpler than I (or anyone else, it seems) was thinking.
## Details 🔗
I’ll spare any reader the record of attempts, mistakes, other attempts, more mistakes, etc., and just get to the final workflow. I wrote a small script (in my case a function in Fish) that creates three files. One for the presentation, one for the article, and one for shared content. After creating the files, it opens a Dired buffer of the relevant folder in Emacs.
For those who use the Fish shell, here is the function:
function lecture
touch {$argv}.org echo -e '#+startup: beamer' \n'#+TITLE: ' \n'#+AUTHOR: Dr. Ridenour' \n'#+DATE: ' >>{$argv}.org
touch {$argv}-beamer.org cat /Users/rlridenour/Dropbox/emacs/beamer/lecture-beamer.org >{$argv}-beamer.org
echo -e '#+include: "'{$argv}'.org" :minlevel 1' >>{$argv}-beamer.org
touch {$argv}-notes.org cat /Users/rlridenour/Dropbox/emacs/beamer/lecture-notes.org >{$argv}-notes.org
echo -e '#+include: "'{$argv}'.org" :minlevel 1' >>{$argv}-notes.org
dired
end
So, entering “lecture kant” in the shell will open a Dired buffer containg the files kant.org, kant-beamer.org, and kant-notes.org.
### Contents File 🔗
The contents file is a standard org file. Initially, I had it containing nothing in the header, except for possibly one line containing #+startup: beamer, which makes it easier to insert some Beamer-specific commands. After getting tired of entering the same data twice in the other files, I wondered if shared header information could just be placed in the contents file. Occasionally things work exactly how hoped they would, so no the function adds the following lines at the top of the contents file:
#+startup: beamer
#+TITLE:
#+AUTHOR: Dr. Ridenour
#+DATE:
You will need to decide what heading level will designate a slide. Don’t worry, you’ll still be able to use that heading level in the article, as I’ll explain later. I use h3, so slides begin with a line like this in the contents file:
*** Slide Title
To add notes, you need to structurally separate the note content, which should only be printed on the article, from the preceding slide. To do this, add another h3 heading (I creatively title it “Notes”) with instructions to ignore the heading:
*** Notes :B_ignoreheading:
:PROPERTIES:
:END:
Any text that follows will only appear in the article, not in the presentation. This does not have to be done for every successive note paragraph, it only needs to be done after a slide. So, any paragraphs that are in the scope of an h1 or h2 heading won’t need that.
### Presentation File 🔗
The magic happens with two small files. The first is the presentation file. At the top, put your preferred Beamer export header, but be sure to include #+LaTeX_CLASS_options: [ignorenonframetext] and #+OPTIONS: toc:nil. The latter is to ensure that Beamer export doesn’t make the contents slide before the presentation title slide. Then, make the title page like this:
***
\maketitle
If you want a table of contents slide, you can do the same thing except use \tableofcontents. Finally, include the contents file with this line:
#+include: "contents.org" :minlevel 1
For the article, use your preferred header with all of the packages declared, but be sure to add this line: #+LaTeX_HEADER: \usepackage{beamerarticle}. After the export header lines, include the contents file, again using #+include: "contents.org" :minlevel 1. When exporting, be sure to export with the one of the Beamer-specific exports. Otherwise, things just won’t look right.
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# Has anyone developed a technique to generate a polytope given (possibly redundant) inequality constraints? [closed]
I've found a few papers that deal with removing redundant inequality constraints for linear programs, but I'm just trying to find the vertices for a feasible region, given a set of inequality constraints.
For instance, if I have:
$$0x_1 + x_2 \leq -1\\ 0x_1 - x_2 \leq -1\\ -x_1 + 0x_2 \leq -2\\ x_1 + 0x_2 \leq -2$$
I'd like to generate a 2x1 rectangle. But, I also want to be able to handle the situation where I have, for instance:
$$0x_1 + x_2 \leq -1\\ 0x_1 - x_2 \leq -1\\ -x_1 + 0x_2 \leq -2\\ x_1 + 0x_2 \leq -2\\ x_1 + 0x_2 \leq -6$$
Where the last constraint is clearly redundant.
I haven't been able to find much in my paper search - any suggestions?
• Aug 22 '15 at 3:34
• The question I cited as duplicate deals with the dual, but equivalent, problem. Aug 22 '15 at 3:36
Suppose you have a system $Ax \leq b$, where $A$ is $m \times n$. Suppose also that $A$ has rank $n$, otherwise the feasible region has no vertices. To find all vertices, all you have to do is consider all subsets $I \subseteq \{ 1, \ldots, m \}$ with $n$ elements. Each such set $I$ gives a subsystem of $Ax \leq b$, lets call it $A_I x \leq b_I$, consisting of those rows with indices in $I$. Notice $A_I$ is a square $n \times n$ matrix; if it is nonsingular, then you can find $x_0$ such that $A_I x_0 = b_I$. Now, if $A x_0 \leq b$, then $x_0$ is a vertex of your polyhedron, and every vertex can be generated thus.
Notice this is independent of whether your system has redundancy or not. Of course, if you have a redundant system, then the above brute-force algorithm will take longer to run for no good reason.
To identify redundant inequalities, you can do as follows. You start with your full system $Ax \leq b$, and you pick some inequality $a^T x \leq \beta$ from it. Then you remove $a^T x \leq \beta$ from the system, obtaining the system $A'x \leq b'$. Then you solve the linear programming problem $\max\{ a^T x : A'x \leq b' \}$. If the optimal value is greater than $\beta$, then the inequality you removed was not redundant. If the optimal value is $\leq \beta$, then the inequality was redundant, and you can repeat the procedure with the smaller system by picking a new inequality, etc.
To solve an LP like the one above, you can use standard software like GLPK, which is freely available and can be called from C, C++, Python, etc.
Also take a look at the Sage (http://sagemath.org) mathematics software, which has tools to compute vertices of polyhedra given an inequality description (try the Polyhedron class, it is very easy to use!). Take a look also at polymake (http://www.polymake.org).
Hope this helps!
• I should be a bit clearer - I don't have an objective function, I'm just looking to prune inequalities. Should I just pick a general objective function and repeatedly solve the LP as you described? Aug 24 '15 at 15:22
• @Nick Sweet: Notice that the LP I described $\max\{ a^T x : A'x \leq b' \}$ has as objective function the left-hand side of one of the inequalities of the original system ($a^T x \leq \beta$) that you are trying to prune. Aug 25 '15 at 14:14
• My mistake - I missed that aspect. I do worry that if I have $n$ inequalities, I'll have to solve $n$ LPs, though. Aug 25 '15 at 20:12
• Well, you already want to find all vertices anyway, so solving $n$ LPs is not that bad. You could think of working with the dual, because then to obtain each LP you have to set a variable to zero and change the right-hand side. A smart solver might use this to save some work... Aug 27 '15 at 0:11
The magic words are "Motzkin's double description method"
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Dissemination of IT for the Promotion of Materials Science (DoITPoMS)
# Optical Microscopy (all content)
Note: DoITPoMS Teaching and Learning Packages are intended to be used interactively at a computer! This print-friendly version of the TLP is provided for convenience, but does not display all the content of the TLP. For example, any video clips and answers to questions are missing. The formatting (page breaks, etc) of the printed version is unpredictable and highly dependent on your browser.
## Aims
On completion of this teaching and learning package you should:
• Appreciate the differences between reflected-light and transmitted-light microscopes.
• Understand the use of polarised light in a transmission microscope.
• Be able to set up and use a microscope to study a range of specimens.
• Understand the steps required to prepare metallographic, ceramic and polymer specimens.
## Introduction
Optical microscopes have a wide variety of applications; they are very powerful tools for inspecting the microstructure of a great range of materials. It is important to use the appropriate mode for the specimen, choosing from reflected-light or transmitted-light modes.
Reflected-light microscopy is used for a range of materials, including metals, ceramics and composites. Contrast between different regions when viewed in reflected light can arise from variations in surface topography and differences in reflectivity (e.g. of different phases, different grain orientations, or boundary regions). These features are revealed by a series of specimen preparation techniques which, when carried out with care, can produce useful, high quality images.
Transmission mode can be used when the specimen is transparent. The specimen is usually in the form of a thin slice (e.g. tens of microns thick). Contrast arises from differences in the absorption of light through different regions. This method is used for the examination of minerals and rocks, as well as glasses, ceramics and polymers. In addition, the transmission mode can often be further enhanced with use of polarised light.
Polarised light microscopy is a specialised use of the transmission mode, and contrast is due to differences in birefringence and thickness of the specimen. This can allow the observation of grains, grain orientation and thickness.
## For Metals
When preparing samples for microscopy, it is important to produce something that is representative of the whole specimen. It is not always possible to achieve this with a single sample. Indeed, it is always good practice to mount samples from a material under study in more than one orientation. The variation in material properties will affect how the preparation should be handled, for example very soft or ductile materials may be difficult to polish mechanically.
### Cutting a specimen
It important to be alert to the fact that preparation of a specimen may change the microstructure of the material, for example through heating, chemical attack, or mechanical damage. The amount of damage depends on the method by which the specimen is cut and the material itself.
Cutting with abrasives may cause a large amount of damage, whilst the use of a low-speed diamond saw can cause fewer problems. There are many different cutting methods, although some are used only for specific specimen types.
### Mounting
Mounting of specimens is usually necessary to allow them to be handled easily. It also minimises the amount of damage likely to be caused to the specimen itself.
The mounting material used should not influence the specimen as a result of chemical reaction or mechanical stresses. It should adhere well to the specimen and, if the specimen is to be electropolished (an electrolytic process) or examined under a Scanning Electron Microscope, then the mounting material should also be electrically conducting.
Specimens can be hot mounted (at around 200 °C) using a mounting press, either in a thermosetting plastic (e.g. phenolic resin), or a thermosoftening plastic (e.g. acrylic resin). If hot mounting will alter the structure of the specimen a cold-setting resin can be used, e.g. epoxy, acrylic or polyester resin. Porous materials must be impregnated by resin before mounting or polishing, to prevent grit, polishing media or etchant being trapped in the pores, and to preserve the open structure of the material.
A mounted specimen usually has a thickness of about half its diameter, to prevent rocking during grinding and polishing. The edges of the mounted specimen should be rounded to minimise the damage to grinding and polishing discs.
A diagram of a mounted specimen
### Grinding
Surface layers damaged by cutting must be removed by grinding. Mounted specimens are ground with rotating discs of abrasive paper flushed with a suitable coolant to remove debris and heat, for example wet silicon carbide paper. The coarseness of the paper is indicated by a number: the number of grains of silicon carbide per square inch. So, for example, 180 grit paper is coarser than 1200.
The grinding procedure involves several stages, using a finer paper (higher number) for each successive stage. Each grinding stage removes the scratches from the previous coarser paper. This is more easily achieved by orienting the specimen perpendicular to the previous scratches, and watching for these previously oriented scratches to be obliterated. Between each grade the specimen is washed thoroughly with soapy water to prevent contamination from coarser grit present on the specimen surface. Typically, the finest grade of paper used is the 1200, and once the only scratches left on the specimen are from this grade, the specimen is thoroughly washed with water, followed by alcohol and then allowed to dry.
It is possible to determine the start point for grinding using the following empirical relationship where the width of the largest scratch is measured under a microscope:
This prevents putting more damage into the sample than already exists; the coarsest grades of paper are often not useful.
Cleaning specimens in an ultrasonic bath can also be helpful, but is not essential.
The series of photos below shows the progression of the specimen when ground with progressively finer paper.
Copper specimen ground with 180 grit paper Copper specimen ground with 400 grit paper Copper specimen ground with 800 grit paper Copper specimen ground with 1200 grit paper
### Polishing
Polishing discs are covered with soft cloth impregnated with abrasive diamond particles and an oily lubricant. Particles of two different grades are used : a coarser polish - typically with diamond particles 6 microns in diameter which should remove the scratches produced from the finest grinding stage, and a finer polish – typically with diamond particles 1 micron in diameter, to produce a smooth surface. Before using a finer polishing wheel the specimen should be washed thoroughly with warm soapy water followed by alcohol to prevent contamination of the disc.
Copper specimen polished to 6 micron level Copper specimen polished to 1 micron level. Ideally there should be no scatches after polishing, but it is often hard to completely remove them all.
Mechanical polishing will always leave a layer of disturbed material on the surface of the specimen, if the specimen is particularly susceptible to mechanical damage (or excessive force is used in the grinding and polishing stages) debris can become embedded in the surface and plastic deformation may exist below the surface. Electropolishing or chemical polishing can be used to remove this, leaving an undisturbed surface.
### Etching
Etching is used to reveal the microstructure of the metal through selective chemical attack. It also removes the thin, highly deformed layer introduced during grinding and polishing.
In alloys with more than one phase, etching creates contrast between different regions through differences in topography or reflectivity. The rate of etching is affected by crystallographic orientation, the phase present and the stability of the region. This means contrast may arise through different mechanisms – therefore revealing different features of the sample.
In all samples, etchants will preferentially attack high energy sites, such as boundaries and defects.
The specimen is etched using a reagent. For example, for etching stainless steel or copper and its alloys, a saturated aqueous solution of ferric chloride, containing a few drops of hydrochloric acid is used. This is applied using a cotton bud wiped over the surface a few times (Care should be taken not to over-etch - this is difficult to determine, however, the photos below may be of some help). The specimen should then immediately be washed in alcohol and dried.
Following the etching process there may be numerous small pits present on the surface. These are etch pits caused by localised chemical attack and, in most cases, they do not represent features of the microstructure. They may occur preferentially in regions of high local disorder, for example where there is a high concentrationof dislocations.
If the specimen is over etched, ie. etched for too long, these pits tend to grow, and obscure the main features to be observed. If this occurs it may be better to grind away the poorly etched surface and re-polish and etch, although it is important to remember what features you are trying to observe – repeatedly grinding a very thin sample may leave nothing to see.
Etched copper specimen Over etched copper specimen
Ideally the surface to be examined optically should be flat and level. If it is not, the image will pass in and out of focus as the viewing area is moved across the surface. In addition, it will make it difficult to have the whole of the field of view in focus - while the centre is focused, the sides will be out of focus. By using a specimen levelling press (shown below) this problem can be avoided, as it presses the mounted specimen into plasticene on a microscope slide, making it level. A small piece of paper or cloth covers the surface of the specimen to avoid scratching.
Specimen levelling press
## Ceramics
### Thin Sections
To prepare ceramic specimens for transmitted light microscopy, a thin slice, approximately 5 mm thick, is cut using a diamond saw or cutting wheel. One surface is then lapped using liquid suspensions of successively finer silicon carbide powders. Between stages in the process the specimen must be thoroughly cleaned. After final washing and drying the ground surface is bonded to a microscope slide with resin. A cut off saw is used on the exposed face to reduce the thickness to about 0.7 mm. The specimen is then lapped to take it to the required thickness – usually about 30 µm, although some ceramic specimens are thinned to as little as 10 µm, due to their finer grain size. The slide is checked for thickness under the microscope, and then hand finished. The slide is then covered with a protective cover slip.
### Lapping
The lapping process is an alternative to grinding, in which the abrasive particles are not firmly fixed to paper. Instead a paste and lubricant is applied to the surface of a disc. Surface roughness from coarser preparation steps is removed by the micro-impact of rolling abrasive particles.
### Polished sections
These differ from ordinary thin sections in that the upper surface of the specimen is not covered with a cover slip, but is polished. Care must be taken to prevent the specimen breaking. Sections may be examined using both transmitted and reflected light microscopy, which is particularly useful if some constituents are opaque.
## Polymers
### Thin sections
Thin sections of organic polymers are prepared from solid material by cutting slices using a microtome – a mechanical instrument used for cutting thin sections. They must be cut at a temperature below the glass transition temperature of the polymer. A cut section curls up during cutting and must be unrolled and mounted on a microscope slide and covered with a cover slip. A few drops of mounting adhesive are used and must be compatible with the specimen. As always the mounting temperature must not affect the microstructure of the specimen.
The thickness of cut slices of polymer tend to lie in the range 2 to 30 µm depending on the type of material.
Harder polymers can be prepared in the same way as thin ceramic specimens.
### Polished sections
These are prepared in the same way as metallographic specimens. Elastomers are more difficult to polish than thermosetting polymers and require longer polishing times. Lubricants used during polishing must not be absorbed by the specimen.
As crystalline regions are attacked more slowly than amorphous ones, etching of polymer specimens can produce contrast revealing the polymer structure.
## Using the Reflection Microscope
Looking down a reflection microscope we see the light reflected off a sample. Remember that contrast can arise in different ways. Below is a diagram of a reflection light microscope; roll the mouse over the labelled parts to see a description.
## Using the Transmission Microscope
Transmission light microscopes are used to look at thin sections – the specimen must transmit the light. Here is a diagram of a transmission microscope; roll the mouse over the labelled parts to see a description.
For further information see Tint Plates.
## Using Microscopes
Both types of microscope are used in very similar ways, here are some guidelines as to how to set up a specimen to be observed:
• The specimen is mounted and placed on the stage; begin by slowly increasing the power of the light source until there is a bright spot visible on the sample (without looking down the eye piece)
• With the lowest magnification lens in place focus using the coarse focus knob: without looking down the microscope, lower the objective lens close to the specimen surface, and then use the coarse focus knob to slowly raise it until the circle of light on the specimen appears reasonably sharp. Now, looking through the eyepiece, adjust the coarse focus control. When looking down the eyepiece and using coarse focus, you should only ever adjust so as to move the sample away from the objective.
• The eye piece distance (for binocular microscopes) should be adjusted to a comfortable separation and, looking through the eye pieces, use the fine focus knob to bring the image to a sharp focus.
• The image should be focussed to the non-adjustable eyepiece and then the other changed such that it is also in focus.
• To increase the magnification, slide the rotatable nosepiece around (ensuring the lens does not touch the specimen) and then re-focus using the fine focus (it should take very little adjustment!).
Once a representative area is found, and focused a digital camera can be used to take a photo and a sketch can be made. It is important, even in cases where there is access to microscope cameras, to make labelled sketches of important aspects of the field of view. Remember that a sketch does not have to be a copy of what you see, but should include the key aspects of the microstructure.
## Scale bars
Observations under a microscope are of no value if there is no scale accompanying them, so it is very important to understand the scale. All sketches should have scale bars and microscope camera software often allows a scale bar to be added before saving the image (given the right information about magnification).
The easiest way of measuring the size of a feature under a microscope is to relate it to the size of the field of view. The simplest way of achieving this is to measure the size of the field of view at a low magnification, and then scale the size appropriately as the magnification is increased. The field of view can be measured approximately by looking at a ruler under the lowest magnification lens.
Accuracy can be improved by using a graticule. A graticule is a slide with a very fine grating which, if metric, will usually measure 1 mm across, and is divided into 100 segments, i.e. each segment is 10 µm across. This allows much greater accuracy in measuring the field of view, and so greater accuracy in measuring features.
Metric graticule in polarised light
On some microscopes, a scale bar is superimposed on one of the eyepieces, which can be used to further improve the accuracy of measuring feature sizes. The scale bar can be calibrated by observing either a graticule or a ruler at a low magnification. For example, if 1 division is equivalent to 20 mm with a ×5 magnification lens, each division is equivalent to 2 mm with a ×50 magnification lens. By measuring a feature using the scale in the eyepiece, the actual size of the feature can be calculated by knowing the width of the divisions in the eyepiece. The scale bar on the eyepiece is particularly useful because it can be rotated, and so both widths and lengths can be measured without rotating the specimen.
## Polarised light
Visible light is a form of electromagnetic radiation, with electric and magnetic vectors oscillating perpendicular to the direction of propagation. Usually the oscillations are in any direction perpendicular to the direction of propagation; polarised light is light with oscillations in a few, restricted, directions.
Polarised light has a variety of uses including in Polaroid sunglasses, which block out light of a certain polarisation, removing much of the glare from the ground. The following flash animation gives an introduction to how polarised light is used in microscopy.
As can be seen, two polarising films at 90° to one another transmit no light; inserting an optically anisotropic material between the polarisers can result in the light vector being rotated.
When light enters an optically anisotropic material, the light vectors are polarised in two Permitted Vibration Directions (PVDs). The difference in refractive index of these directions results in a retardation of one ray with respect to the other; the rays propagate at different speeds within the material and exit with a phase difference between them. This causes one light frequency (i.e. one wavelength) to show destructive interference, and that wavelength of light is lost. Other wavelengths will constructively interfere (to different extents), so different colours are seen, depending on the retardation. This is called birefringence.
A quartz wedge under crossed polars shows how the observed colour changes as the retardation increases. In the photo below, the wedge increases in thickness from left to right. As the thickness increases, the retardation also increases. The quartz wedge below shows a range of birefringent colours as its thickness varies. The relation between retardation, birefringence and thickness can be seen on a Michel-Levy chart.
The retardation also depends on the orientation of the optical axes of the material relative to the polarised light (so rotating the stage may change the colour).
The arrangement of the crossed polars also allows for the insertion of plates at 45° to the planes of polarisation. These are used to enhance the contrast in a specimen. For further effects, it is also often possible to rotate one of the polarisers if crossed polars are not to be used.
When observing a specimen, differences in birefringence allow phases and grains to be identified. For example, different grain orientations may exhibit differences in birefringence and this will cause them to appear a different colour.
The series of photos below shows the difference in the appearance of some glass ceramic specimens as different plates are inserted.
Glass ceramic transmission microscope image made with unpolarised light. Glass ceramic transmission microscope image made with polarised light. See also the DoITPoMS Micrograph Library entry Glass ceramic transmission microscope image made with polarised light and quarter wave plate. Glass ceramic transmission microscope image made with polarised light and full wave plate. See also the DoITPoMS Micrograph Library entry
Optically anisotropic materials aligned with one of the permitted vibration directions parallel to the direction of the polarised light vector appear ‘in extinction’ (i.e. black) between crossed polars.
## Resolution and Imaging
The limit of resolution (or resolving power) is a measure of the ability of the objective lens to separate in the image adjacent details that are present in the object. It is the distance between two points in the object that are just resolved in the image. The resolving power of an optical system is ultimately limited by diffraction by the aperture. Thus an optical system cannot form a perfect image of a point.
For resolution to occur, at least the direct beam and the first-order diffracted beam must be collected by the objective. If the lens aperture is too small, only the direct beam is collected and the resolution is lost.
Consider a grating of spacing d illuminated by light of wavelength λ, at an angle of incidence i.
The path difference between the direct beam and the first-order diffracted beam is exactly one wavelength, λ. So,
d sin i + d sin α = λ
where 2α is the angle through which the first-order beam is diffracted. Since the two beams are just collected by the objective, i = α, thus the limit of resolution is,
$${d_{\min }} = {\lambda \over {2\sin \alpha }}$$
The wavelength of light is an important factor in the resolution of a microscope. Shorter wavelengths yield higher resolution. The greatest resolving power in optical microscopy requires near-ultraviolet light, the shortest effective visible imaging wavelength.
### Numerical Aperture
The numerical aperture of a microscope objective is a measure of its ability to resolve fine specimen detail. The value for the numerical aperture is given by,
Numerical Aperture (NA) = n sin α
where n is the refractive index and equal to 1 for air and α is the half angle subtended by rays entering the objective lens.
Numerical aperture determines the resolving power of an objective, the higher the numerical aperture of the system, the better the resolution.
Low numerical aperture Low value for a Low resolution High numerical aperture High value for a High resolution
### Airy Discs
When light from the various points of a specimen passes through the objective and an image is created, the various points in the specimen appear as small patterns in the image. These are known as Airy discs. The phenomenon is caused by diffraction of light as it passes through the circular aperture of the objective.
Airy discs consist of small, concentric light and dark circles. The smaller the Airy discs projected by an objective in forming the image, the more detail of the specimen is discernible. Objective lenses of higher numerical aperture are capable of producing smaller Airy discs, and therefore can distinguish finer detail in the specimen.
The limit at which two Airy discs can be resolved into separate entities is often called the Rayleigh criterion. This is when the first diffraction minimum of the image of one source point coincides with the maximum of another.
Unresolvable Rayleigh Criterion Resolvable
Circular apertures produce diffraction patterns with circular symmetry. Mathematical analysis gives the equation,
$${d_{\min }} = {\lambda \over {2\sin \alpha }}$$
θR is the angular position of the first order diffraction minimum (the first dark ring)
λ is the wavelength of the incident light
d is the diameter of the aperture
From the equation it can be seen that the radius of the central maximum is directly proportional to λ/d. So, the maximum is more spread out for longer wavelengths and/or smaller apertures.
The primary minimum sets a limit to the useful magnification of the objective lens. A point source of light produced by the lens is always seen as a central spot, and second and higher order maxima, which is only avoided if the lens is of infinite diameter. Two objects separated by a distance less than θR cannot be resolved.
## Summary
• The optical microscope is a very useful tool for the observation of materials and can be used to gain valuable information about a large variety of specimens. Some knowledge of the material and the information that is required is essential to determine the best techniques to employ when preparing and examining specimens.
• Sample preparation is a critical part of microscopy, as this determines the quality of the images produced. Many techniques, when correctly applied to a specimen, can enhance the information present.
• One of the limitations of the optical microscope is that of resolution. High resolution imaging is more commonly carried out in a scanning electron microscope (SEM).
• In addition, for 'transparent' specimens, in particular those of anisotropic materials, polarised light microscopy can offer large benefits, with high contrast possible.
## Questions
### Deeper questions
The following questions require some thought and reaching the answer may require you to think beyond the contents of this TLP.
1. How should the initial focusing of the microscope be done?
a With the coarse focus moving the lens towards the specimen. b With the fine focus moving the lens towards the specimen. c With the coarse focus moving the lens away from the specimen. d With the fine focus moving the lens away from the specimen.
2. The specimen preparation is important in metallurgy because:
a A poorly prepared specimen can damage the microscope. b A poorly prepared specimen will distract from features on the specimen. c Only a well prepared specimen will reflect light. d A poorly prepared specimen will corrode, and the resulting images will be misleading.
3. When increasing the magnification on the microscope, which of the following occurs?
a The depth of field increases. b The resolution limit decreases. c The visible area decreases. d The contrast increases.
4. When the aperture stop is made smaller, which of the following occur?
Yes No a The depth of field increases. Yes No b The resolution decreases. Yes No c The contrast increases. Yes No d The brightness increases.
5. The red tint plate (also known as a full wave sensitive tint plate) increases the contrast in a polarised light microscope because:
a Our eyes are more sensitive to red light, so it is easier to see the light and dark areas when there is a red tint plate. b The red tint plate only lets a small window of wavelengths through, and so increases the birefringence. c The red tint plate displaces the ordinary and the extraordinary beams by an extra wavelength, so that small differences in birefringence cause large differences in colour. d The red tint plate increases the differences in birefringence in the material so that the different grain directions cause a greater difference in colour than in just the polarised light.
6. Contrast in reflected microscopy tends to be caused by:
a Variations in topography and differences in reflectivity of areas. b Only differences in reflectivity of areas. c Only topography. d Variations in thickness of the specimen.
7. If a graticule is observed under the 10x lens of a microscope so that the diameter of the field of view is from 150 μm to 450 μm on the graticule, what is the width of one lamella when it takes 15 lamellae to fill the field of view when viewed under the 50x lens.
## Going further
### Books:
R.C. Gifkins, Optical microscopy of metals, Pitman, 1970
R Haynes, Optical microscopy of materials, Kluwer Academic Publishers, 1984
Eugene Hecht, Optics, Addison Wesley, 2001
### Websites:
Molecular Expressions Microscopy Primer - a site with a lot of information about microscopy including pages on polarised light microscopy
## Tint plates
From left to right: Quarter wave plate; full wave sensitive tint plate; quartz wedge
### Full wave sensitive tint plate (also known as red tint plate)
A sensitive tint plate can be used to introduce colour contrast in polarised light images, and consists of a slice of birefringent material, usually gypsum, mica or quartz.
The slice is cut or cleaved parallel to the optic axis of the crystal, to such a thickness that the O-rays and E-rays for green light (λ = 540 nm) are out of phase by exactly one wavelength. The analyser therefore extinguishes green light, but permits other wavelengths to pass through to some extent.
When using white light this causes the field of view to appear red (white light minus green light). Isotropic, non-birefringent materials also appear red.
The sensitive tint plate increases the observed birefringence by one wavelength. The path difference between the O-rays and E-rays emerging from an anisotropic crystal adds or subtracts from this single wavelength path difference. Individual grains appear to exhibit differences in colour, depending on their composition and orientation.
### Quarter wave plate
A quarter wave plate is made form a flake of mica that is cleaved to such a thickness that the O-rays and E-rays emerge a quarter of a wavelength out of phase. This corresponds to a pale grey interference colour.
This plate is especially useful for examining specimens showing bright interference colours, because they are moved only a short distance along the scale. The plate can be used to enhance the contrast between regions of the specimen.
### Quartz wedge
The quartz wedge is cut so that it varies in thickness from about 0.01 mm to about 0.08 mm and covers several orders of retardation colours. As the wedge is inserted into the slot in the microscope it produces progressively higher retardations, and the position at which complete extinction occurs is noted.
Michel Levy produced a colour chart which plots the thickness of an isotropic specimen, its birefringence and its retardation in nanometres. Once two of these variables is known, the third can be easily determined.
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# Merge TEX
## How to merge TEX files in C#
GroupDocs.Merger allows developers to merge TEX files when it’s needed to organize multiple TEX files into single document or send fewer attachments etc. And you can do this without any third-party software or manual work involved. With GroupDocs.Merger it is possible to combine TEX documents of any size and structure - all text, images, tables, graphs, forms and other content will be preserved.
The following example demonstrates how to merge TEX files with several lines of C# code:
• Create an instance of Merger class and pass source TEX file path as a constructor parameter. You may specify absolute or relative file path as per your requirements.
• Add another TEX file to merge with Join method. Repeat this step for other TEX documents you want to merge.
• Call Merger class Save method and specify the filename for the merged TEX file as parameter.
``````// Load the source TEX file
using (Merger merger = new Merger(@"c:\sample1.tex"))
{
// Add another TEX file to merge
merger.Join(@"c:\sample2.tex");
// Merge TEX files and save result
merger.Save(@"c:\merged.tex");
}
``````
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My Math Forum Matrice - Eigenvalue and its eigenvectors
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Linear Algebra Linear Algebra Math Forum
October 20th, 2015, 08:15 AM #1
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Joined: Oct 2015
From: Poland
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Matrice - Eigenvalue and its eigenvectors
I have a matrix - given in the picture I uploaded.
They give the information that 2(1.5+0.5a) is an eigenvalue for the vector A. I now need to find the eigenvectors for that exact eigenvalue. I can't really figure it out, but i thought about using this:
(A−λE)x = 0
But I'm getting some complicated shit because I don't have the values for the eigenvector, I'm just calling the vector <v1,v2,v3> and multiply it onto the A-lambda matrix
Anybody who can figure this out?
Attached Images
matrice.png (2.4 KB, 1 views)
October 22nd, 2015, 12:59 PM #2 Math Team Joined: Jan 2015 From: Alabama Posts: 3,102 Thanks: 850 Tedious but straight forward. Given that 2(1.5+ 0.5a)= 3+a is an Eigen value then $\displaystyle A- \lambda E= \begin{bmatrix} 1+ a & 1+ a & 1+ a \\ 1+ a & 1+ a & 1+ a \\-3(1+ a) & -3(1+ a) & -3(1+ a)\end{bmatrix}$. So $\displaystyle (A- \lambda E)(v_1, v_2, v_3)= $$\displaystyle \begin{bmatrix}(1+ a)(v_1+ v_2+ v3) \\ (1+ a)(v_1+ v_2+ v_3) \\ -3(1+ a)(v_1+ v_2+ v_3)\end{bmatrix}$$\displaystyle = \begin{bmatrix}0 \\ 0 \\ 0\end{bmatrix}$. So we must have $\displaystyle (1+ a)(v_1+ v_2+ v_3)= 0$, $\displaystyle (1+ a)(v_1+ v_2+ v_3)= 0, -3(1+ a)(v_1+ v_2+ v3)= 0$. If a= -1, so that 1+ a= 0, that is true for all vectors. If a is not -1, then $\displaystyle v_1+ v_2+ v_3= 0$. We can solve for any one of those in terms of the other two. For example $\displaystyle v_3= -v_1- v_2$ so we can write $\displaystyle (v_1, v_2, v_3)= (v_1, v_2, v_1- v_2)= (v_1, 0, -v_1)+ (0, v_2, -v_2)= v_1(1, 0, -1)+ v_2(0, 1, -1)$. Do you see a basis for this Eigen-space?
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# Sentence Examples with the word angular velocity
By adjusting the right ascension of the plane ABC and rotating the axis with the angular velocity of the sun, it follows that BC will be the direction of the solar rays throughout the day.
Lines of lanthanum and carbon which are believed to belong to a low level showed systematically smaller angular velocity than the average.
Required the relation between the velocity of translation 02 of W and the angular velocity af of the differential barrel.
View more
According to numerous observations made at Cape Thorsden, the apparent angular velocity of arcs increases on the average with their altitude.
Tait that a similar representation of the type (30) is obtained if we replace the circle by an equiangular spiral described, with a constant angular velocity about the pole, in the direction of diminishing radius vector.
We therefore have the fundamental theorem that the angular velocity of the body around the centre of attraction varies inversely as the square of its distance, and is therefore at every point proportional to the gravitation of the sun.
Msh2(n-a); (3) so that this ellipse can be rotating with this angular velocity R for an instant without distortion, the ellipse a being fixed.
Hence also, in any pair of circular wheels which rotate continuously for one revolution or more, the ratio of the numbers of teeth and its reciprocal the angular velocity ratio must be expressible in whole numbers.
Further, by causing the hour circle, and with it the polar axis, to rotate by clockwork or some equivalent mechanical contrivance, at the same angular velocity as the earth on its axis, but in the opposite direction, the telescope will, apart from the effects of refraction, automatically follow a star from rising to setting.
Du and the work performed in a unit cf time in overcoming friction, when the angular velocity is a, is af f pr.
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Version: V1
Using the Router#
We recommend using the router for swapping to and from different assets, which has different swapping functions for ETH, tokens, and tokens that have fee-on-transfer features.
Before executing the swap, it is recommended that an external price source is used to fix the minimum output tokens receivable when selling a fixed amount of tokens, or maximum amount of input tokens to be used when purchasing a fixed amount of tokens.
Your smart contract should also:
1. Have enough ETH / tokens when executing the swap
2. Be able to receive ETH, when it is the destination address for swapping to ETH
3. Granted approval to the router when swapping from tokens
It is necessary to specify which pools are to be used for the token swap. Read more about fetching pool addresses before proceeding.
Since each pool represents a token pair, it stands to be the case that the poolsPath specified for token swaps must be 1 size smaller than path, ie. poolsPath.length = path.length - 1. Refer to the examples below.
Examples#
We will cover 3 scenarios:
1. Swap 1 ETH for DAI
2. Swap USDC to obtain 1 ETH, with WBTC as an intermediary
3. Swap 1 CORE (fee-on-transfer token) for USDT
1 ETH -> DAI#
swapExactEthForTokens#
A common error when swapping from ETH is forgetting to actually send ETH when calling the function. The value field should match the amount of ETH sent.
IERC20[] memory path = new address[](2);
path[0] = dmmRouter.weth();
path[1] = dai; // assuming dai is specified as IERC20
// value = 1 ETH, alternatively use msg.value
dmmRouter.swapExactEthForTokens{value: 1e18}(
amountOutMin, // should be obtained via a price oracle, either off or on-chain
poolsPath, // eg. [eth-dai-pool]
path,
msg.sender,
block.timestamp
);
USDC -> WBTC -> 1 ETH#
Transferring tokens#
Before swapping, the contract should be in possession of USDC. The caller can either send the tokens beforehand, or give allowance to the contract to call the transferFrom method. The short code snippet below showcases the latter.
uint256 amountIn = 50 * 10 ** usdc.decimals();
require(usdc.transferFrom(msg.sender, address(this), amountIn), 'transferFrom failed');
Granting Approval#
The next step is then to give the router some USDC allowance.
require(usdc.approve(address(dmmRouter), amountIn), 'approve failed');
path and poolsPath#
Since the intended token path is usdc -> wbtc -> eth, path = [usdc, wbtc, weth]. We also need to specify the usdc-wbtc and wbtc-eth pools to be used. Deciding which pools to use can be found in this section. Eg. poolsPath=[usdc-wbtc-pool, wbtc-weth-pool], where usdc-wbtc-pool and wbtc-weth-pool are the pool addresses to be used.
swapExactTokensForETH#
Note that if the destination address is a contract, it should have the receive() external payable { ... } or fallback function declaration in order to receive ETH.
IERC20[] memory path = new address[](3);
path[0] = usdc; // assuming usdc is specified as IERC20
path[1] = wbtc; // assuming wbtc is specified as IERC20
path[2] = dmmRouter.weth();
dmmRouter.swapExactTokensForEth(
amountOutMin, // should be obtained via a price oracle, either off or on-chain
poolsPath, // eg. [usdc-wbtc-pool, wbtc-weth-pool]
path,
msg.sender, // has to be able to receive ETH
block.timestamp
);
1 CORE -> USDT#
This example is similar to the previous example of USDC -> WBTC -> ETH. The only exception is that CORE is a fee-on-transfer token, and thus requires special handling.
swapExactTokensForTokensSupportingFeeOnTransferTokens#
We assume that the previous steps of transferring tokens and token approval to the router has been performed.
IERC20[] memory path = new address[](2);
path[0] = core; // assuming core is specified as IERC20
path[1] = usdt; // assuming usdt is specified as IERC20
dmmRouter.swapExactTokensForTokensSupportingFeeOnTransferTokens(
amountOutMin, // should be obtained via a price oracle, either off or on-chain
poolsPath, // eg. [core-usdt-pool]
path,
msg.sender,
block.timestamp
);
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Department of
# Mathematics
Seminar Calendar
for events the day of Thursday, February 24, 2005.
.
events for the
events containing
Questions regarding events or the calendar should be directed to Tori Corkery.
January 2005 February 2005 March 2005
Su Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa
1 1 2 3 4 5 1 2 3 4 5
2 3 4 5 6 7 8 6 7 8 9 10 11 12 6 7 8 9 10 11 12
9 10 11 12 13 14 15 13 14 15 16 17 18 19 13 14 15 16 17 18 19
16 17 18 19 20 21 22 20 21 22 23 24 25 26 20 21 22 23 24 25 26
23 24 25 26 27 28 29 27 28 27 28 29 30 31
30 31
Thursday, February 24, 2005
12:00 pm in 464 Loomis,Thursday, February 24, 2005
#### Higgs bundles and Calabi-Yau threefolds
###### Tony Pantev (U Penn, Math)
1:00 pm in Altgeld Hall 347,Thursday, February 24, 2005
#### The intersection form and geodesic currents on free groups (continued)
###### Ilya Kapovich [email] (UIUC)
Abstract: The notion of a geometric intersection number between free homotopy classes of closed curves on surfaces plays a pivital role in Thurston's treatment of the Teichmuller space and of the dynamics of surface homeomorphisms. In particular, Bonahon proved that this notion extends to a symmetric and bilinear notion of intersection number between two geodesic currents on a hyperbolic surface. We investigate to what extend these ideas are applicable in the free group context. Thus we define and study an Out(F_n)-equivariant "intersection form" on the product of the (non-projectivized) Culler-Vogtmann outer space and the space of geodesic currents on a free group. We also find an obstruction, arising from non-symmetric behaviour of generic stretching factors of free group automorphisms, to the existence of a symmetric notion of an intersection number between two geodesic currents on a free group. Time permitting, we will discuss an embedding of the Culler-Vogtmann outer space into the projectivized space of geodesic currents, based on the use of Patterson-Sullivan conformal densities.
1:00 pm in 241 Altgeld Hall,Thursday, February 24, 2005
#### Digital expansions, exponential sums and central limit theorems.
###### Michael Drmota (Technical University, Wien)
Abstract: The purpose of this talk is to present recent results on the distribution of the values of the ($q$-ary) sum of digits function $s_q(n)$, where we mainly focus on two problems, the joint distribution $(s_{q_1}(n),s_{q_2}(n))$ of two different expansions and on the distribution of the sum-of-digits function of squares: $s_q(n^2)$.
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# What is a gravity gradient?
The statement: 'For most of that day, Mir remained in a "gravity gradient," (sic) which basically means that the most massive part of Mir naturally pointed toward Earth.' is in https://history.nasa.gov/SP-4225/nasa4/nasa4.htm .
I don't understand this at all. Why would the most massive part of Mir be naturally pointed toward Earth? We know Aristotle was wrong in saying that a more massive object has a greater acceleration when in free fall. So what is going on?
Wikipedia https://en.wikipedia.org/wiki/Gravity-gradient_stabilization makes no mention of a most massive part in its very short article about it.
My question is, what does it mean to be 'in a "gravity gradient"'?
– uhoh
Apr 25 '21 at 11:56
• @uhoh Definitely helpful. Apr 25 '21 at 14:43
• Tides. See the story en.wikipedia.org/wiki/Neutron_Star_(short_story) Apr 26 '21 at 12:48
Due to the fact that gravity follows an inverse square law with distance. So an object twice as far away from the centre of another object will only feel half of the gravitational force.
So for a long tubular space station pointing at the centre of the Earth the end nearest the Earth will feel a greater force of gravity than the end furthest from the Earth. The effect is tiny, but over time can create a noticeable effect on space craft orientation. The effect is the same one that would cause "spaghettification" of anything in close proximity to a black hole (although obviously dozens of orders of magnitude less intense).
Having the "heavy end" down minimises the rotational inertia of the object.
• Equally important (see my link in comment on question), orbital speed (for stable orbit) varies with altitude as a result of the gravity equation, which enhances the "stability" of the dense portion hanging downwards. Apr 26 '21 at 12:51
• "Due to the fact that gravity follows an inverse square law with distance. So an object twice as far away from the centre of another object will only feel half of the gravitational force." I think you have a typo here. Apr 26 '21 at 17:17
• "Having the "heavy end" down minimises the rotational inertia of the object." How does that work? Apr 26 '21 at 17:19
• I have added a diagram - hope that helps Apr 26 '21 at 18:05
• Interesting way to look at the situation with moment of inertia and rotational energy from revolution in orbit! Nice! Apr 26 '21 at 20:46
This is actually quite interesting problem, which has a few levels to it:
• In first order approximation, there is a tendency to align the orbiting body vertically with its mass extending as much as possible either above or below the orbit without difference if it is up or down. This is the effect described in the linked wikipedia article.
• But there is higher-order effect too, which adds some minuscule preference to orientation with largest extending mass downwards.
• Last but not least, this is not limited to orbiting bodies. You will get exactly same behavior (with some minor differences in factors) for vertical free fall. So even freely falling objects will tend to align itself vertically! (Even without effect of Earth rotation, but OTOH in most cases the duration of any realistic free fall will be too short for this to be noticeable.)
In order to analyze the situation, one need to consider non-inertial coordinate frame. Either rotating in case of space station or constantly accelerating in case of vertical free fall. In both situations there will be gravity force + an inertial force. Both of them acts proportional to the mass and at all single points of the body, so it is easy to add them together in each point.
For a space station, the inertial force is centrifugal one. Proportional to angular velocity of the orbit (constant, let's not make things more complicated by elliptical orbits) and distance from Earth centre. Gravity is inversely proportional to square of this distance. Total force on any point with mas $$m_x$$ will be then: $$F_x = F_G - F_C = m_x \left[{G\cdot M_E \over r_x^2} - \omega^2\cdot r_x\right]\,,$$ $$r_x$$ is distance from earth centre to the respective point, $$\omega$$ is angular speed of reference frame rotation, $$G$$ gravitational constant and $$M_E$$ Earth mass. Positive force points inwards into Earth centre.
Let's pick an reference frame which revolves with such angular speed that resulting force will be zero at some radius $$r$$ and split $$r_x$$ into this "mean" radius $$r$$ where forces are in balance and distance $$a_x$$ from this altitude ($$r_x = r + a_x$$): $$F_x = m_x \cdot G \cdot M_E \left[{1\over (r+a_x)^2} - {r + a_x\over r^3}\right]\,.$$
Note that resulting force points upwards for positive $$a_x$$ and downwards for negative $$a_x$$. So what is holding a space station in its orbit then? As soon as some perturbation shifts it a bit lower, it starts accelerating down to the ground. Or does it? Turns out that there is one more inertial force in the rotating frame, namely Coriolis force, which acts on objects which move inside the reference frame.
So if object starts to "fall down" from its orbit, this extra velocity results in Coriolis force accelerating it in direction of frame rotation and this tangential component in turn induces Coriolis force upwards. At the end this object starts "orbiting" a point in rotating reference frame located at original radius $$r$$. What does it mean? Well, an elliptical orbit (maybe there is something to epicycles at the end :) ). But to not complicate things further, we will analyze only static configurations in perfectly circular orbit (any mas below equilibrium height pulling down will be compensated by other mass above this height pulling up) where Coriolis force equals zero.
Staying with gravitational and centrifugal forces only, let's integrate the expression with respect to $$a_x$$. The result will be potential energy of a point mass in this pseudopotential created by combination of gravitational and centrifugal force with respect to distance from nominal orbit radius: $$E_x=-{3G\cdot M_E\over 2r^3}\cdot m_x\cdot a_x^2 \cdot {r+a_x/3\over r+a_x}\,.$$ Zero energy level was set for zero $$a_x$$, that is at the "nominal" orbit radius. Any mass above or below it will have negative potential energy.
Note that last fraction will be really close to 1 for $$a_x \ll r$$ (size of station compared to distance to Earth centre) and remaining term is a quadratic potential without any preference for up or down ($$a_x$$ is squared).
So the lowest energy is with as much of the mass sticking as much up or down as possible, but without any preference for up or down.
The $${3G\cdot M_E\over 2r^3}$$ equals numerically $$2\,\rm \mu J/(kg\cdot m^2)$$. To get a feeling how much it is: it is similar to a rotational energy of object revolving around its own axis with period about 1 hour. So this first-order tidal stabilization is strong enough to have noticeable effect on such slow movement (roughly independently of size or mass of spacecraft/spacestation).
The up and down preference comes from the last term. It can be approximated as $$1-2a_x/3r$$, so same mass sticking the same distance above $$r$$ will have lower energy than when placed same distance below (the the energy is negative, so bigger coefficient means lower energy).
In avoid forces pushing object away from circular orbit mass distribution needs to stay such that total energy is a its maximum (zero derivative) with respect to vertical shift. Any upside down turn needs to be balanced by realigning center of gravity. Generally, "heavier" part here needs to be interpreted in terms of moments of inertia or, for the last fraction, third power of distance from center of mass. So it there is not much specific to be stated for generic object without taking its detailed geometry into account.
Playing with some specific numbers for two small spheres in vertical position either heavier down or up results in relative differences in energy at order of 0.1‰ to 0.01‰ preferring heavier at the bottom. So difference is really small, but existing.
And to get back to free fall. (Lets neglect the Earth rotation -- either free falling at the pole or high from the space without any tangential velocity -- to get the other extreme case). Here the inertial force is caused by constant acceleration by gravity and it is simply equal to acceleration of coordinate frame times mass independent of $$a_x$$: $$F_{x,fall} = F_G - F_A = m_x \left[{G\cdot M_E \over (r+a_x)^2} - {G\cdot M_E \over r^2}\right]\,,$$ After integrating with respect to $$a_x$$ we get a similar expression for energy as above: $$E_{x,fall}=-{G\cdot M_E\over r^3}\cdot m_x\cdot a_x^2 \cdot {1\over 1+a_x/r}\simeq-{G\cdot M_E\over r^3}\cdot m_x\cdot a_x^2 \cdot \left(1 - {a_x\over r}\right)\,.$$ So the quadratic potential is in this situation weaker by factor 1.5 compared to orbiting case and higher order up--down preference factor is higher by the same factor, otherwise no difference.
• I don't understand why there is a down vote on this answer. It's really better to leave a comment if one downvotes. Sure there can be lots of reasons why that's not convenient. But it's really better. Apr 26 '21 at 1:28
• "So even freely falling objects will tend to align itself vertically! ". I wonder whether nonfreely falling objects might also, e.g. a metal rod sinking in water, or falling in air? And what about a wooden rod floating upwards in water? Or rod shaped helium balloon floating upwards in air? Apr 26 '21 at 17:35
• @kimholder Agreed. Apr 26 '21 at 17:36
• "In order to analyze the situation, one need to consider inertial coordinate frame. Either rotating in case of space station or constantly accelerating in case of vertical free fall." Is there a typo here? Apr 26 '21 at 17:39
• @MatthewChristopherBartsh of course, it should be non-inertial there, thanks! (Too many inertial forces non-inertial frames in the text :) ... sorry) Apr 26 '21 at 20:34
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### METHOD OF MOMENTS:
The R implementation can be found in the gmm package.
Once decided that a sample comes from a normal distribution, the parameters $$\mu$$ and $$\sigma$$ can be estimated by either the method of moments or the generalized method of moments.
In the method of moments the two first moments of the population:
$$\mathbb E[X]=\mu$$ and $$\mathbb E[(x-\mu)^2]= \sigma^2$$
can be estimated with a system of two equations and two unknowns:
$$\mathbb E[X-\hat\mu]=\frac{1}{n}\sum_{i=1}^n x_i - \hat \mu = \mu$$
and
$$\mathbb E[(X- \hat \sigma)^2] = \frac{1}{n}\sum_{i=1}^n (x_i-\hat\mu)^2=\sigma^2$$
However, more moments can be included in a weighted fashion:
The third moment (skewness) is:
$$\mathbb E[(x-\mu)^3]$$, which it is zero in the population.
This is estimated in the sample by $$\frac{1}{n}\sum_{i=1}^n (x_i-\hat\mu)^3=0.$$
The fourth moment (kurtosis) is:
$$\mathbb E[(x-\mu)^4]$$, which is $$3\sigma^4.$$
This is estimated in the sample by $$\frac{1}{n}\sum_{i=1}^n (x_i-\hat\mu)^4 = 3\sigma^4.$$
At this point we look at cost functions for each one of these moment estimators:
$$\hat g_1=\frac{1}{n}\sum_{i=1}^n x_i-\hat\mu$$
$$\hat g_2=\frac{1}{n}\sum_{i=1}^n (x_i-\hat\mu)^2-\sigma^2$$
$$\hat g_3=\frac{1}{n}\sum_{i=1}^n (x_i-\hat\mu)^3$$
$$\hat g_4=\frac{1}{n}\sum_{i=1}^n (x_i-\hat\mu)^4-\sigma^2-3\sigma^4$$
We place these estimated cost functions into a vector:
$$\hat g=\begin{bmatrix}\hat g_1\\ \hat g_2\\ \hat q_3\\ \hat g_4\end{bmatrix}$$
and we calculate the cost function:
$\hat g\quad \begin{bmatrix}\text{WEIGHTS}\end{bmatrix}\quad\hat g$
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# Evolution of spectral and transport quantities with doping in the SU(2) theory of cuprates
Abstract : Recent transport experiments in the cuprate superconductors linked the opening of the pseudogap to a change in electronic dispersion [S. Badoux et al., Nature 531, 210 (2015)]. Transport measurements showed that the carrier density sharply changes from $x$ to $1+x$ at the pseudogap critical doping, in accordance with the change from Fermi arcs at low doping to a large hole Fermi surface at high doping. The SU(2) theory of cuprates shows that antiferromagnetic short range interactions cause the arising of both charge and superconducting orders, which are related by an SU(2) symmetry. The fluctuations associated with this symmetry form a pseudogap phase. Here we derive the renormalised electronic propagator under the SU(2) dome, and calculate the spectral functions and transport quantities of the renormalised bands. We show that their evolution with doping matches both spectral and transport measurements.
Document type :
Preprints, Working Papers, ...
Cited literature [28 references]
https://hal-cea.archives-ouvertes.fr/cea-01567560
Contributor : Emmanuelle de Laborderie Connect in order to contact the contributor
Submitted on : Monday, July 24, 2017 - 10:00:31 AM
Last modification on : Friday, January 7, 2022 - 3:51:44 AM
### File
1704.06557.pdf
Files produced by the author(s)
### Identifiers
• HAL Id : cea-01567560, version 1
• ARXIV : 1704.06557
### Citation
Corentin Morice, Xavier Montiel, Catherine Pépin. Evolution of spectral and transport quantities with doping in the SU(2) theory of cuprates. 2017. ⟨cea-01567560⟩
### Metrics
Les métriques sont temporairement indisponibles
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# Asymptotic integral expansion of $\int_0^{\infty} t^{3/4}e^{-x(t^2+2t^4)}dt$ for $x \to \infty$
I'm still having a little trouble applying Laplace's method to find the leading asymptotic behavior of an integral. Could someone help me understand this? How about with an example, like:
$$\int_0^{\infty} t^{3/4}e^{-x(t^2+2t^4)}dt$$ for $x>0$, as $x\rightarrow\infty$.
The basic idea is that the maximum contribution of the integral comes from a neighborhood of $t=0$, and near there we have $t^2+2t^4 \approx t^2$. This problem is particularly nice because we can do everything explicitly. I'll do the calculation in two steps (two changes of variables) to illustrate what's going on.
Start with the change of variables $t^2+2t^4 = s^2$, where $s \geq 0$. This gives
$$t=\frac{1}{2}\sqrt{-1+\sqrt{1+8s^2}},$$
so that the integral becomes
$$\int_0^{\infty} t^{3/4}e^{-x(t^2+2t^4)}\,dt = \int_0^\infty s^{3/4} f(s) e^{-xs^2}\,ds,$$
where
$$f(s) = \frac{2^{1/4}s^{1/4}}{\sqrt{1+8s^2}\left(-1+\sqrt{1+8s^2}\right)^{1/8}} = 1 - \frac{15}{4}s^2 + \frac{713}{32}s^4 + \cdots.$$
Now we can make the second change of variables $s^2 = r$ to put the integral into a form where we can directly apply Watson's lemma. Indeed, this gives
$$\int_0^\infty s^{3/4} f(s) e^{-xs^2}\,ds = \int_0^\infty r^{-1/8} g(r) e^{-xr}\,dr,$$
where
$$g(r) = \frac{1}{2}f\left(\sqrt{r}\right) = \frac{1}{2} - \frac{15}{8}r + \frac{713}{64}r^2 + \cdots.$$
Finally
\begin{align*} \int_0^\infty r^{-1/8} g(r) e^{-xr}\,dr &\approx \sum_{n=0}^{\infty} \frac{g^{(n)}(0) \Gamma(n+7/8)}{n! x^{n+7/8}} \\ &= \frac{1}{2}\Gamma\left(\frac{7}{8}\right) x^{-7/8} - \frac{15}{8}\Gamma\left(\frac{15}{8}\right)x^{-15/8} + O\left(x^{-23/8}\right) \end{align*}
as $x \to \infty$, by Watson's lemma.
• Can you elaborate on the advantages of splitting the change of variables in two steps? Mar 20, 2018 at 22:05
• @ThomasAhle there are no mathematical advantages. Mar 21, 2018 at 0:40
In summary, just look for the minimum of the function $2t^4+t^2$ which gives the location of the major contribution to the integral. The minimum of $2t^4+t^2$ is attained at the point $t=0$. So, we have
$$\int_0^{\infty} t^{3/4}e^{-x(t^2+2t^4)}dt \sim \int_0^{\infty} t^{3/4}e^{-xt^2}dt = \frac{\Gamma \left( {\frac {7}{8}} \right)}{2 {x}^{ {\frac {7}{8}} }\, }.$$
To evaluate the last integral use the change of variables $u=xt^2$ to transform the integral to the gamma function.
Note: If instead of the function $t^2+2t^4$, you have $g(t)$ and attains its minimum at the point $0$, then just use the Taylor series to get the leading term.
• In the first step, how did you drop the $t^2$ portion?
– Alex
Jan 5, 2013 at 16:23
• @Alex: It is corrected. Jan 5, 2013 at 22:12
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# Tag Info
50
You're right - this election is looking a bit ridiculous right now. Not that I don't believe most of the candidates are sincere in their desire to help... But this site has over 4 years of history, nuanced policies have grown up, and expecting someone who hasn't spent a lot of time here to even know about them - much less uphold them - is kinda silly. The ...
50
I agree. Right now there is a smallish list of words that will cause a warning to pop up when they appear in short question titles . From Shog9's answer the regex is ^.{0,30}(^|\W)(anyone|difficult|doubt|easy|hard|help|interesting|please|problem|query|question|someone|stuck|very)(\W|$).{0,30}$ (Essentially a title containing at least one of the listed ...
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How can I choose the preferred website to put my questions? I once read something along the lines "If you have to ask whether to ask on MO or math.se, likely you should ask it on math.se." I think this is a good rule of thumb. What if someone writes his or her question down on the not preferable website? If you post on MO and they think it is ...
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It's... uh... been a while since this request was made. We require a few things before turning a switch like this - it's pretty harmful to small sites to have what is a large chunk of potential askers to be blocked. But... Maths isn't a small site, and it hasn't been one for a loooong time. So when this request was first passed into my radar two years ago, ...
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To quote Asaf Karagila What a terrible terrible ... terrible terrible idea. (I might have opted for "awful" instead, to squeeze a couple more in there, but the sentiment is the same.) If you want to pay people to answer your questions, there are already sites which provide this service. So what benefits would it have here? None that I can see. In ...
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Probably a "Senator" badge, in the gold badges category. Anyone have other ideas for the name?
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I think this would be a very difficult task. In the unambiguous cases, adding math formatting doesn't add much value to the post. Added value begins where the math formatting clarifies math which is hard to read in ASCII representation, and that usually requires some decisions: Whether 1/2a is $\frac12a$ or $\frac1{2a}$ Whether a lone a or I is a variable ...
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Not only does the community not have much basis to evaluate moderator candidates, but we then must elect them for life. I think it would be helpful if moderators were initially elected to a probationary term of fairly short duration (say, 3 months). At the end of this term, the community could evaluate their work (based on either detailed or aggregate data ...
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You can write "they". I don't think that the feature request is very good, and this discussion might help to shed some light on the topic. (In a nutshell, the presence of female users online is still often accompanied by harassment.) Moreover, much like many users won't fill in their real name, there's no reason to believe that people would bother to fill ...
28
Your post leaves many important details open. Some questions follow: How are the answers? Are the answers that are being upvoted any good? If they're not good, then this is an issue indeed. If the answers are good -- could these upvotes be deserved? So: are you sure this is an incident of mutual tactical serial upvoting? Is the symbiotic relationship of ...
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This is a poll. Please vote this answer up if you think there should be a delay between the deadline for nominations and the beginning of voting. This has been mentioned in several comments; I am posting it as an answer just so that it can be put to a vote.
27
Some notes in the form of an answer. Moderators cannot invalidate votes. Any votes. Ever. (And those downvotes I received on that answer were certainly not appropriate! Why can't I invalidate them!? Why!?) We're trusted with a lot of information and tools, but SE still likes to keep certain things out of our reach. Invalidation of votes is either done ...
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My personal experience and big sentiment is that yes, the level of Math.SE is falling. It would be great if "relatively advanced but not research level questions" got the same attention as it used to be when I started here and fell in love with Math.SE quickly. While it is true that they are drowned in the ever-growing flood of no-effort no-motivation solve-...
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The procedure used by the Crusade of answers, which specializes in trying to reduce the unanswered questions queue, is a synthesis of the aforementioned strategies: Consider if there is a legitimate reason to vote to move/delete/close the question. If not, ask the commenter using the @ feature to convert his/her comment to an answer. If there's no response ...
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Ok, there's a warning now: This is triggered by a title matching ^.{0,30}(^|\W)(anyone|difficult|doubt|easy|hard|help|interesting|please|problem|query|question|someone|stuck|very)(\W|$).{0,30}$ ...so it heavily favors short titles - Tricky integration/functions problem would've warned, while Least squares problem equivalent to solving Poisson problem for ...
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proof-theory If your question is not about the area of mathematic logic called proof theory, do not use the proof-theory tag. Questions requesting proofs for a particular statement, or about how/why a particular proof of a statement works, should be tagged with the mathematical subject the statement lies in. This tag is almost universally used ...
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Moderators are not everywhere. We are only aware of flags if you raise them. (I've deleted the posts before you even finished composed your Meta posting here... your "this user" link is already dead.) What you can do to help: Flags Ahoy!!! For obvious spam posts, instead of voting to close, or flagging as off-topic, you should just flag it as Spam. ...
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I think this comes back to the fundamental question of how people here envision the purpose of this site, and if the history of such debates is any indication, addressing the issue will lead to acrimony, drama, accusations, frustration, and other uncivil behavior on the part of certain individuals...over what I would like to remind people is ultimately just ...
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How about this as a feature request? It is certainly fine that a person have privacy if suspended. But, if they want to run for moderator, then they must surrender that privacy and it must be made known of their past suspensions, including some detail as to what happened and why they were suspended. If they don't want people to know about their ...
22
One should first note that only a few StackExchange sites have MathJax capabilities. Add in the fact that such questions appear only rarely and it is probably unlikely that SE will implement a change to require titles to have non-$\LaTeX$ elements anytime soon. There is, of course, a simple work-around that requires no changes to the SE system: When ...
22
In the past, the SE network has had issues with some people excessively bugging others about what they perceived as an insufficient accept rate. (Those links are all from just a single quick search on meta.SE.) This eventually got to the point where, to make it stop, the accept rate was removed from user profiles, and comments mentioning "accept rate" are ...
21
The goal of answers is not only to help the original poster; otherwise there'd be little reason to keep questions around after they've been answered. Answers can also help other people looking at the question, either while it's still on the front page or if they find the question via a search engine. When asking a question, should there be a place to ...
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I interpret this in two ways: Do not ask well-known open problems as questions. It's not practical to ask MSE to prove the Riemann hypothesis for you. You're allowed to ask difficult, research level questions, but they shouldn't be attempts to "stump MSE" with famous conjectures. (Of course, if your question unexpectedly turns out to be equivalent to ...
21
The toggle button functionality is more or less available through the MathJAX extension action. Though it is more suitable for displaying alternate math formulas instead of displaying a paragraph of text. Personally, I can't think of any good way to utilize this functionality. \require{action} \toggle{ \begin{array}{cl} & \bbox[2pt,color:red;border-...
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Sometimes a "yes" or a "yes you did it right" is all that's necessary; my own question history has a few of these. But many of the questions that tend to be answerable with "yes" are relatively simple questions, and it is my broad belief that these users could benefit from some additional insight that could relate to the question. Personally, if I find that ...
21
Complementing Alexander's answer, a close reason of "lack of research effort" is, in my opinion, intrinsically bad for multiple reasons: it is impossible to ascertain how much research effort there was, "effort" is not a cop out for a poor question, it often gets dragged down to debates about how often students have "no clue on where to start" etc. If ...
Only top voted, non community-wiki answers of a minimum length are eligible
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J Plant Ecol ›› 2015, Vol. 8 ›› Issue (1): 17-29.
• Research Articles •
### Biodiversity change in heathland and its relationships with shifting local fire regimes and native species expansion
Nancy Shackelford1,*, Michael Renton1,2, Michael P. Perring1, Kristine Brooks3 and Richard J. Hobbs1
1. 1 School of Plant Biology, The University of Western Australia (M090), 35 Stirling Highway, Crawley, Western Australia 6009, Australia; 2 Centre of Excellence for Climate Change, Forest and Woodland Health, Murdoch University, Murdoch, Western Australia 6150, Australia; 3 Department of Environment and Conservation, Great Southern District Office, PO Box 100, Narrogin, Western Australia 6312, Australia
• Received:2013-10-19 Accepted:2014-05-04 Published:2015-01-22
• Contact: Shackelford, Nancy
Abstract: Aims Understanding the relationships among disturbance, invasion and species change is essential for effective management of many systems. We investigated relationships among fire history, invasion by a native tree species, Allocasuarina huegeliana, and diversity change to understand the potential drivers of plant community alteration in a complex and biodiverse system.
Methods We used plant species surveys from 1983 and 2011 to quantify species loss/gain and thence compositional changes. Additionally, we surveyed population densities of the invasive species and collated long-term fire history data for each site. General linear models and non-parametric models were used to assess the strength of relationships between the three variables of interest.
Important findings Within the last 30 years, ~11% of the plant species richness was lost from the reserve. At an individual site level, we found only a 4% average decrease in overall plant species richness, but large species losses and gains that imply considerable compositional shifts. Though such shifts might be expected over 30 years, many of the gained species were common, potentially opportunistic species, while those lost were often locally rare woody perennials. In addition, gained species tended to be expanding their recorded range westward suggesting that they may be responding to the regional drying climate. The relationship between invasion density and species loss was strong over all spatial scales. We identified a potential state change to dominance by the native invasive particularly as high densities prevented species gain at the site scale. In these extreme cases of high invasive density and high biodiversity loss, we argue that there may be a need to directly address the expanding native population.
Aims Understanding the relationships among disturbance, invasion and species change is essential for effective management of many systems. We investigated relationships among fire history, invasion by a native tree species, Allocasuarina huegeliana, and diversity change to understand the potential drivers of plant community alteration in a complex and biodiverse system.
Methods We used plant species surveys from 1983 and 2011 to quantify species loss/gain and thence compositional changes. Additionally, we surveyed population densities of the invasive species and collated long-term fire history data for each site. General linear models and non-parametric models were used to assess the strength of relationships between the three variables of interest.
Important findings Within the last 30 years, ~11% of the plant species richness was lost from the reserve. At an individual site level, we found only a 4% average decrease in overall plant species richness, but large species losses and gains that imply considerable compositional shifts. Though such shifts might be expected over 30 years, many of the gained species were common, potentially opportunistic species, while those lost were often locally rare woody perennials. In addition, gained species tended to be expanding their recorded range westward suggesting that they may be responding to the regional drying climate. The relationship between invasion density and species loss was strong over all spatial scales. We identified a potential state change to dominance by the native invasive particularly as high densities prevented species gain at the site scale. In these extreme cases of high invasive density and high biodiversity loss, we argue that there may be a need to directly address the expanding native population.
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# Mar 24 2021 Crest Factor Reduction I
CFR is a technique used to reduce the PAPR of the transmitted signals. This a key module in LTE/NR radio. In this serious of post, let's see how a CFR is implemented in digital circuits.
## CFR Concept
In LTE/NR telecommunications, OFDM is used as the method to encoding digital data to RF waveform. One of the disadvantage if OFDM is high (Peak to Average Power Ratio) PAPR. Crest Factor Reduction (CFR) is a technique used to reduce PAPR of transmitted signal so that power amplifier (PA) can operate more efficiently.
To know the necessary and consideration of CFR, we need to know the typical characteristic of an RF (PA).
### RF Power Amplifier
For an ideal PA, the input and output has a linear relationship: that is, if input power of PA increased 1 dB, output power of PA will also increase by 1 dB. But ability of PA is not infinity. As the input power level increase, there will be a power level that the output of PA won't increase that much with input. That is, the output power starts to saturate. Eventually, the slope of the input vs output power becomes zero. The power level at which this happen is known as saturate point. The output power of saturate point is $$P_{saturation}$$.
To use this PA to amplify RF signal, usually we will have some back off from saturation power (for example, 1 dB, which is called $$P_{1dB}$$). Below the back off level is the operation range. If more linearity is wanted, we can back off more, if you need more output power, you need back off less. Peak power of output is $$P_{peak}$$. Average output power $$P_{average}$$ can be calculated by:
\begin{equation*} P_{average} = P_{saturation} - P_{backoff} - {PAPR} \end{equation*}
You can see that PAPR of signal affects average power of PA output. Signal with smaller PAPR can be amplified to higher power. If the output power is a fixed requirement, you can operate PA with lower saturation level (less supply voltage). In fact, efficiency of PA is a complex thing, but we can have a quick evaluation that operating at nonlinear range will be more efficient than operating at linear range. So at both case, it will be better efficient during transmit.
### Functionality of CFR
We know that PAPR has impact on PA's efficiency. So CFR is introduced to reduce the PAPR of signal. But how? The easiest way to reduce PAPR of signal is to clip the peak of the signal. We will have a threshold, which is target PAPR we want. If we see the signal's instance power is larger than threshold, we clip the excess part. Result a PAPR reduced signal, but causes error comparing to the original signal. That's the cost of CFR. Also we need to know that for RF equipment, the Adjacent Channel Leakage Ratio (ACLR) is a key requirement. This requires your signal should be "smooth" enough so that you will not interfere your neighborhood. This is another consideration CFR should take.
### Placed in System
Usually, CFR module is placed at the position before DAC to alter the signal to transmit:
## CFR Methods
In practice, there are different CFR algorithms, such as:
• Hard Clipping
• Peak Cancellation
• Peak Windowing
• Noise Shaping
I will have a brief introduction based on the complexity of each method:
### Hard Clipping
Hard Clipping is very simple, it's the way I mentioned in Functionality of CFR. The clipping mathematical equation is as follows:
\begin{equation*} x_{clipped}[n] = c[n] \cdot x[n] \end{equation*}
\begin{equation*} c[n] = \begin{cases} \frac{A}{|x[n]|} & |x[n]| \ge A\\ 1 & |x[n]| < A \end{cases} \end{equation*}
where $$A$$ is the clipping threshold. However, in digital circuits, division and multiplication is expensive. The clipping is usually archived by a subtraction. The structure is as follows:
The Polar Clipping block will extract the part exceeds threshold and it will be subtracted from original signal with matched delay. The purpose of the delay is to compensate for all the latencies generated during the detection and extraction of the clip sequence. The main problem of this algorithm is the clipping cause some sharp corner during the point where signal go through the threshold. The sharp corner will cause some out-of-band noise and impact adjacent channel.
We will not go much detail here since we will introduce the implementation in next post.
### Noise Shaping
The Noise Shaping (NS) algorithm, sometimes referred to as Peak Filtering (PF), consists of extracting the part of the input signal whose magnitude exceeds the threshold, then filtering it and finally subtracting it from a properly delayed version of the original signal itself. Compared with hard clipping, the difference is the noise shaping filter. The filter should have a better shape on adjacent channel, so the filter is designed based on the specified type of signal.
The problem of NS CFR is that after the filtering operation it self, it is possible that some peaks will be created again (the so called peak regrowth phenomenon). So cascading several stages of NS maybe necessary.
### Peak Windowing
The Peaking Windowing (PW) algorithm is based on multiplying the signal with an attenuation window rather than adding a correction to the signal. When a peak is detected in the input signal, a set of coefficients is loaded and scaled to ensure that the peaks will be attenuated to the desired level (threshold). The coefficients (the window) is multiplied to the peak using a FIR.
The advantage of the algorithm is that the window amplitude changes smoothly, then not much out-of-band emission is expected to appear. But on the other hand, the windowing method lack the knowledge about the exact frequency characteristics. This make it harder to guarantee a required specified OOB performance.
### Peak Cancellation
The algorithm of Peak Cancellation (PC-CFR) is simple. When a peak is detected in the input signal, it does not filter the clip error sequence. PC-CFR only isolates a single input element sample among those identified as peaks. It cancels them individually by subtracting properly shaped cancelling pulses from the signal, one for each peak. The pules are designed to have the same spectrum as the signal.
It aims to strike a balance between the out-of-band emission and in-band waveform quality when compressing the signal to a target PAPR.
## Comparing of Methods
The system-level performance of Peak Cancellation (PC-CFR) is shown to be better than other methods with save overall cost. This is the compare result of different CFR Method from Xilinx:
In next post of this series, I'll introduce more detailed implementation of a real CFR module.
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# Why do we have to take torque due to pseudo force for an accelerating axis?
For e.g.
Here, A,B are strings. A was cut. The rod is of length l. Acceleration of center of mass = a and angular acceleration of the rod is alpha.
Now if I want to find the net torque about the point of intersection of B and the rod(which is accelerating towards the right with $l/2 sin37 \theta$), why do I have to apply a pseudo force on the c.o.m and observe it from that accelerating frame to equate net torque = $I\alpha$ (including the torque due to pseudo force)? Why can't we do it from the ground frame? why does the point about which we take the net torque and equate it to $I\alpha$ have to be non-accelerating?
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This page needs to be cleaned up. Please make this page better in any way that you can. Remove this box and the listing on the cleanup page after the article has been cleaned up. For tips on improving this article, read "How to edit a page" and "How to write Simple English articles".
Addition is putting two numbers together.
Arithmetic
In arithmetic, addition is putting with or more numbers together. The sign for addition is "+". The name for the sign "+" is "plus".
Counting examples
For example, are are objects in two groups. The objects are small circles: "o". One group has five of ase objects. The other group has 3 of ase objects. To find the number of objects in both groups, the objects can be counted. Another way to find the number of objects in both groups is to add the numbers in each group.
Another method is to add the numbers of objects in group A and group B, since ay are already counted. In symbols:
```5 + 3
```
There are rules for adding numbers that people learn. There are also rules for adding numbers that are built into machines. The rule says that:
``` 5+3=8
```
In another counting example, Sally and Bill have 2 children. Sally and Bill get 3 more children. Sally and Bill have added three children to air two children and now have five children.
A measurement example
Tom wants to know the distance between his house and Sally's house. Bob's house is 300 meters east of Tom's house. Sally's house is 120 meters east of Bob's house:
Tom's house<------------300 meters-------------->Bob's house<-----120 meters----->Sally's house
The distance from Tom's house to Sally's house can be found by adding the distances already measured. The distance from Tom's house to Bob's house added to the distance from Bob's house to Sally's house is the same as the distance from Tom's house to Sally's house. That is, three hundred meters plus 120 meters.
```300 + 120 = 420
```
Addition can also mean to make bigger.
${\displaystyle 40+20+3=(40+20)+3=60+3=63}$
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# Python vs. Perl: N Queens Problem
Posted in Computer Science
## Background
Revisiting the N queens problem, this time implementing the solution in Python.
Verb-oriented solution, functional, and based on Perl solution
More fair comparison - both are interpreted languages, not compiled languages
Compare Python and Perl, ease of implementation, speed, flexibility
## N Queens Problem
As a recap from the last post about the N queens problem, we're solving a chess puzzle that asks: how many different configurations are there for placing $$N$$ queens on an $$N \times N$$ chessboard such that no queen attacks any other queen?
This is a popular problem in computer science because solutions often implement recursive backtracking.
See the Wikipedia article for details.
## N Queens Solution
Here is the pseudocode of the N queens solution being implemented here:
explore(column):
if last column:
# base case
else:
# recursive case
for each row:
if this is a safe row:
place queen on this row
explore(column+1)
remove queen from this row
This solution implements recursive backtracking to explore choices of where to place each queen. It keeps solutions simple, and can be implemented using only primitive built-in data types. Solutions are stringified version of these arrays, consisting of 8 digits, so they are likewise very simple.
### Perl Solution
As a reminder, the Perl solution was originally from Rosetta Code. Here's another link to the Perl solution on Github.
Github gist: nqueens.pl
### Python Solution
The solution requires the use of one array of data that is fixed in size, which for a given column stores a list of rows already occupied by queens, and one array of data that is variable in size, which stores where each queen has been placed.
The Python solution makes use of lists by using a fixed-size list for the occupied rows and a variable size list for storing queen locations. It utilizes buit-in methods for the list data type to append and pop, or add and remove items from the end of the list.
Github gist: nqueens.py
## Head to Head: Walltime vs. Number of Queens
----------------------------------------------------------
| NQueens | Nsolutions | Java | Perl | Python |
|---------|------------|-----------|----------|----------|
| 8 | 92 | 0.003628 | 0.016 | 0.018 |
| 9 | 352 | 0.006709 | 0.067 | 0.077 |
| 10 | 724 | 0.017473 | 0.259 | 0.359 |
| 11 | 2680 | 0.061291 | 1.542 | 1.684 |
| 12 | 14200 | 0.240463 | 8.431 | 8.618 |
| 13 | 73712 | 1.113491 | 48.542 | 50.401 |
| 14 | 365596 | 6.557336 | 303.278 | 322.576 |
| 15 | 2279184 | 42.619426 | 2057.052 | 1979.343 |
----------------------------------------------------------
The results of this test show that Python and Perl are fairly closely matched.
## Perl Profiling
Results of profiling the Perl code with Devel::NYTProf were detailed in a prior post.
Here are those results once again, for the 11 queens problem:
# Profile data generated by Devel::NYTProf::Reader
# Version: v6.04
# Format: time,calls,time/call,code
0.000238,2,0.000119,use Time::HiRes qw(time);
0.000039,2,0.000019,use strict;
0.000491,2,0.000246,use warnings;
0.000021,1,0.000021,my $start = time; 0.010338,2680,0.000004,push @solutions, "@queens\n"; 0.009993,2680,0.000004,return; 0.186298,164246,0.000001,$#attacked = 2 * $board_size; 0.150338,164246,0.000001,for( 0 ..$#queens) {
0.675523,1.26035e+06,0.000001,$attacked[$ix2 ] = 1;
1.242624,164246,0.000008,for my $row (0 ..$board_size-1) {
0.267469,166925,0.000002,explore($depth+1); 0.125272,166925,0.000001,$occupied[$row] = 0; 0.000002,1,0.000002,explore(0); 0.000011,1,0.000011,my$duration = time - $start; 0.000075,1,0.000075,print "Found ", scalar(@solutions), " solutions\n"; 0.000050,1,0.000050,printf "Execution time: %0.3f s \n",$duration;
## Python Profiling
The Python N queens solution was profiled with two tools: cProfile and line_profiler.
The built-in profiling tool cProfile gives a summary of how much time was spent in each method, but nothing lower level than that. It is similar to Java profiling tools. (cProfile documentation)
The line_profiler tool is designed to profile Python code line-by-line, which gives a much more useful breakdown of where the code spent all of its time. It is also helpful because this can be compared one-to-one with the results from Perl, and we can get an equal basis for comparing the two languages.
## Python Profiling Results
### cProfile Results
Here are the results from running the N queens problem for N = 11 through cProfile:
**************************************
Profiling 11 queens problem with Python...
*******************************
cProfile:
*******************************
Found 2680 solutions
996197 function calls (829272 primitive calls) in 2.237 seconds
Ordered by: internal time
ncalls tottime percall cumtime percall filename:lineno(function)
166926/1 1.845 0.000 2.237 2.237 /Volumes/noospace/Users/charles/codes/hello-world/python/nqueens/nqueens.py:12(explore)
328492 0.275 0.000 0.275 0.000 {range}
166925 0.062 0.000 0.062 0.000 {method 'pop' of 'list' objects}
169605 0.029 0.000 0.029 0.000 {method 'append' of 'list' objects}
164247 0.026 0.000 0.026 0.000 {len}
1 0.000 0.000 2.237 2.237 /Volumes/noospace/Users/charles/codes/hello-world/python/nqueens/nqueens.py:4(<module>)
1 0.000 0.000 0.000 0.000 {method 'disable' of '_lsprof.Profiler' objects}
### line_profiler Results
The line_profiler tool gives a more detailed picture of the code and where it spends its time, breaking down profiling information line-by-line. This tool can be installed with pip: pip install line_profiler. The (INSERT LINK)(nqueens repository on github) has a file that demonstrates how to use this tool. See Python/Profiling
Here are the results from the line_profiler tool run on the same (11 queens) problem:
**************************************
Profiling 11 queens problem with Python...
Found 2680 solutions
Wrote profile results to nqueens.py.lprof
Timer unit: 1e-06 s
Total time: 14.2258 s
File: nqueens.py
Function: explore at line 11
Line # Hits Time Per Hit % Time Line Contents
==============================================================
11 @profile
12 def explore(depth):
13 # base case
14 166926 187573 1.1 1.3 if(depth==board_size):
15 # stringify/serialize the solution
16 2680 31497 11.8 0.2 solutions.append("%s"%(queens))
17 2680 3117 1.2 0.0 return
18
19 else:
20 164246 384688 2.3 2.7 attacked = 2*board_size*[0,]
21 1424595 1690693 1.2 11.9 for i in range(0,len(queens)):
22 1260349 1471304 1.2 10.3 ix1 = queens[i] + depth - i
23 1260349 1405141 1.1 9.9 attacked[ix1] = 1
24
25 1260349 1494095 1.2 10.5 ix2 = queens[i] - depth + i
26 1260349 1392563 1.1 9.8 attacked[ix2] = 1
27
28 1970952 2229922 1.1 15.7 for row in range(0,board_size):
29 1806706 2031139 1.1 14.3 if(occupied[row] or attacked[row]):
30 379432 374466 1.0 2.6 continue
31
32 # make a choice
33 166925 241114 1.4 1.7 queens.append(row)
34 166925 186833 1.1 1.3 occupied[row] = 1
35
36 # explore the consequences
37 166925 610396 3.7 4.3 explore(depth+1)
38
39 # unmake the choice
40 166925 288667 1.7 2.0 queens.pop()
41 166925 202555 1.2 1.4 occupied[row] = 0
## Python vs Perl: Walltime vs. Number of Solutions Tested
As with the prior post, I verified that both codes were testing the same number of solutions. Here is that table of the number of solutions for each value of N, together with the number of solutions tested:
-----------------------------------------------------------------------------
| NQueens | Nsolutions | NsolutionsTested | Java | Perl | Python |
|---------|------------|------------------|-----------|----------|----------|
| 8 | 92 | 1965 | 0.003628 | 0.016 | 0.018 |
| 9 | 352 | 8042 | 0.006709 | 0.067 | 0.077 |
| 10 | 724 | 34815 | 0.017473 | 0.259 | 0.359 |
| 11 | 2680 | 164246 | 0.061291 | 1.542 | 1.684 |
| 12 | 14200 | 841989 | 0.240463 | 8.431 | 8.618 |
| 13 | 73712 | 4601178 | 1.113491 | 48.542 | 50.401 |
| 14 | 365596 | 26992957 | 6.557336 | 303.278 | 322.576 |
| 15 | 2279184 | 168849888 | 42.619426 | 2057.052 | 1979.343 |
-----------------------------------------------------------------------------
## The Winner: Perl for Small Problems, Python for Big Ones
It was not a big surprise to see that Perl and Python were nearly identical in their performance, and it testament to the fact that interpreted scripting languages like Perl and Python operate at one speed, and compiled code in C++ or Java operates at a completely different speed that is an order of magnitude faster (see the comparison of Perl and Java in a prior blog post).
Perl and Python were close enough in performance that, although Perl came out ahead on smaller problems and Python came out ahead on the biggest, a different CPU or platform, micro-optimizations, and various butterfly effects could easily turn the tables.
## Sources
1. "N-Queens Problem". Rosetta Code, GNU Free Documentation License. Edited 6 March 2017. Accessed 21 March 2017. <https://web.archive.org/web/20170320081421/http://rosettacode.org/wiki/N-queens_problem>
2. "nqueens.py". Charles Reid. Github Gist, Github Inc. Edited 25 March 2017. Accessed 25 March 2017. <https://gist.github.com/charlesreid1/1a2ecb3a83284290d4a9daf747d0d7e4>
3. "nqueens.pl". Charles Reid. Github Gist, Github Inc. Edited 20 March 2017. Accessed 23 March 2017. <https://gist.github.com/charlesreid1/4ce97a5f896ff1c89855a5d038d51535>
4. "The Python Profilers". Python 2.7.13 Documentation, Python Software Foundation. Updated 20 February 2017. Accessed 23 March 2017. <https://docs.python.org/2/library/profile.html>
5. "line_profiler". Python Package Index, Python Software Foundation. Updated 20 October 2016. Accessed 23 March 2017. <https://pypi.python.org/pypi/line_profiler/>
6. "Github - rkern/line_profiler". rkern, Github Repository, Github Inc. Updated 20 October 2016. Accessed 23 March 2017. <https://github.com/rkern/line_profiler>
7. "Python/Profiling". Charlesreid1.com wiki, Charles Reid. Edited 23 March 2017. Accessed 23 March 2017. <https://web.archive.org/web/20170326031708/http://charlesreid1.com/wiki/Python/Profiling>
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# Subscriptions to newsmagazine x
Author Message
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Subscriptions to newsmagazine x [#permalink] 23 May 2017, 01:24
Expert's post
00:00
Question Stats:
84% (05:47) correct 15% (07:15) wrong based on 13 sessions
What was the total number of subscriptions for Newsmagazine x during the year in which Newsmagazine x accounted for 14.6 percent of nationwide news magazine subscriptions?
A) 1,020
B) 1,980
C) 6,300
D) 7,000
E) 7,200
[Reveal] Spoiler: OA
D
In which of the following years did subscriptions to Newsmagazine z account for approximately $$\frac{1}{6}$$ of the total nationwide magazine subscriptions?
A) 2009
B) 2006
C) 2003
D) 2000
E) 1997
[Reveal] Spoiler: OA
B
What was the approximate percent increase in nationwide subscriptions to newsmagazines between 1995 and 1996 ?
A) 4%
B) 11%
C) 26%
D) 51%
E) 73%
[Reveal] Spoiler: OA
D
In 1998, what was the approximate number of subscriptions to newsmagazines nationwide?
A) 3,000
B) 13,000
C) 16,000
D) 20,000
E) 67,000
[Reveal] Spoiler: OA
C
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Re: Subscriptions to newsmagazine x [#permalink] 26 Sep 2017, 08:21
2
KUDOS
Q1: we just have to check which is the year for wich magazine X's subscriptions were 14.6% from the right part of the first chart and we get that the total number of subscriptions is 7,000. Answer D.
Q2: we have to use the table at the bottom to compute for each year the ratio between subscriptionsof magazine z to the total and check the one most similar to 1/6=1.67. The answer is B.
Q3: To get the increase in nationwide subscriptions to any newsmagazine, we need to get the nationwide subscription for the two years of interest that can be computed using the percentage of magazine X subscriptions on the total. E.g. in 1995, the total nationwide number of subscriptions is 1.5/0.246 = 6.1 mln while in 1996, the number is equal to 2.6/0.283 = 9.2 mln. Then, the percent increase is computed as (9.2-6.1)/6.1*100 = 50.6%, rounded to 51%. Answer D.
Q4: same rationale as above. To find the nationwide number of subscription in 1998 we can use the percentage of magazine x subscriptions as 3.3/0.205 = 16.1, rounded to 16 mln. Answer C.
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Re: Subscriptions to newsmagazine x [#permalink] 05 Apr 2018, 11:26
IlCreatore wrote:
Q1: we just have to check which is the year for wich magazine X's subscriptions were 14.6% from the right part of the first chart and we get that the total number of subscriptions is 7,000. Answer D.
Q2: we have to use the table at the bottom to compute for each year the ratio between subscriptionsof magazine z to the total and check the one most similar to 1/6=1.67. The answer is B.
Q3: To get the increase in nationwide subscriptions to any newsmagazine, we need to get the nationwide subscription for the two years of interest that can be computed using the percentage of magazine X subscriptions on the total. E.g. in 1995, the total nationwide number of subscriptions is 1.5/0.246 = 6.1 mln while in 1996, the number is equal to 2.6/0.283 = 9.2 mln. Then, the percent increase is computed as (9.2-6.1)/6.1*100 = 50.6%, rounded to 51%. Answer D.
Q4: same rationale as above. To find the nationwide number of subscription in 1998 we can use the percentage of magazine x subscriptions as 3.3/0.205 = 16.1, rounded to 16 mln. Answer C.
All good ... but not 16 mln... noting is in millions, it 16,097 ~ 16,000.
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Re: Subscriptions to newsmagazine x [#permalink] 06 Apr 2018, 01:14
I can’t see the image. Is it only me?
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Re: Subscriptions to newsmagazine x [#permalink] 06 Apr 2018, 05:31
Expert's post
I do think, yes.
For me all is fine.
Regards
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Re: Subscriptions to newsmagazine x [#permalink] 01 Jul 2018, 13:02
IlCreatore wrote:
Q1: we just have to check which is the year for wich magazine X's subscriptions were 14.6% from the right part of the first chart and we get that the total number of subscriptions is 7,000. Answer D.
Q2: we have to use the table at the bottom to compute for each year the ratio between subscriptionsof magazine z to the total and check the one most similar to 1/6=1.67. The answer is B.
Q3: To get the increase in nationwide subscriptions to any newsmagazine, we need to get the nationwide subscription for the two years of interest that can be computed using the percentage of magazine X subscriptions on the total. E.g. in 1995, the total nationwide number of subscriptions is 1.5/0.246 = 6.1 mln while in 1996, the number is equal to 2.6/0.283 = 9.2 mln. Then, the percent increase is computed as (9.2-6.1)/6.1*100 = 50.6%, rounded to 51%. Answer D.
Q4: same rationale as above. To find the nationwide number of subscription in 1998 we can use the percentage of magazine x subscriptions as 3.3/0.205 = 16.1, rounded to 16 mln. Answer C.
Why we calculate it like this 1.5/0.246 = 6.1?
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Re: Subscriptions to newsmagazine x [#permalink] 02 Jul 2018, 15:38
Expert's post
Attachment:
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Actually, in 1985, 1.5 means 1,500,000 of people are 24.6 % of what number ?? roughly 6.1 millions.
Same calculation in 1986 and we do have 9.2 million. The difference is about 50%
Hope this helps
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Re: Subscriptions to newsmagazine x [#permalink] 07 Aug 2018, 00:28
Can someone decode the graph for me please? I dont understand 'Number of subscriptions as a percent of nationwide subscriptions to newsmaagzine'? Like what exactly is the graph saying? And how is everyone getting the answers in millions?
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Re: Subscriptions to newsmagazine x [#permalink] 07 Aug 2018, 08:25
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kruttikaaggarwal wrote:
Can someone decode the graph for me please? I dont understand 'Number of subscriptions as a percent of nationwide subscriptions to newsmaagzine'? Like what exactly is the graph saying? And how is everyone getting the answers in millions?
First of all leave out the million.
Look into the first graph and 1st column : it says "Number of subscription in Thousands" :- meaning the each year the value will be = no. given * 1000
For example in the year 1995:
Number of subscription = 1.5 * 1000 = 1500
Now look into the second column of the 1st graph : it says " Number of subscriptions as a percent of nationwide subscriptions to newsmaagzine" :
let us take for the year 1995:
Number of subscriptions as a percent of nationwide subscriptions to newsmaagzine : 1500 = 24.6% of Nationwide subscription ;
or Nationwide subscription for the year 1995 =$$\frac{1500}{0.25} = 6000$$ (taking 24.6% is nearly equal = 25% for easy calculation)
Hope this clears!!!!!!!!
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# Archived Find the vortices of a square after a transformation given by a tensor
1. Nov 21, 2013
### Jalo
1. The problem statement, all variables and given/known data
Given a square and the respective distension tensor, ε, find the position on his vortices after the transformation.
ε = 0.1.....0.25
.....0.25.....0.1
2. Relevant equations
3. The attempt at a solution
I got kind of lost in this question. I started thinking that maybe a vortic at the coordinates (a,b) would later be at the position (a',b') given by:
(a',b') = (a,b)ε
This got me some weird results tho, which led me to believe it was wrong.
I later tried to solve it using each component of the tensor alone. I know that the components of the diagonal give me the elongation, therefore I used them to find the positions of the vortices after the elongation. The problem came when I had to deal with the distortion. How can I find the position of the vortices there? I managed to find the position of the vortices that were at either the x axis or y axis, using trigonometry
sin(εxy)=b'/L , where L is the length of the square after the elongation.
My problem now rests with finding the position of the vortix at (L,L).
Using trigonometry I found that the size of the diagonal,D, after the distortion was:
D = Lcos(π/4-εxy)
Since the vortix will still have an y coordinate equal to the x coordinate after the distortion I can say that:
2A^2=D^2 , where A will be the position of the vortix after the distortion.
I think my way of solving the problem is correct, however I can't help but think there's a better way... If someone could throw me some light I'd appreciate.
2. Mar 15, 2016
### HallsofIvy
Staff Emeritus
I presume you mean "vertices". "Vortices" are completely different things!
Is it (a, b)ε or ε(a, b)? What convention are you using?
What were the vertices of your square to begin with? Is your square centered at (0, 0)? Are you taking into account rotation and dilation?
You say that one vertex is at (L, L) so are you assuming a square with vertices at (0, 0), (L, 0), (0, L), and (L, L)? If so applying $\epsilon$ to each vertex gives
$$\begin{bmatrix}0.1 & 0.25 \\ 0.25 & 0.1\end{bmatrix} \begin{bmatrix}0 \\ 0 \end{bmatrix}= \begin{bmatrix}0 \\ 0 \end{bmatrix}$$
$$\begin{bmatrix}0.1 & 0.25 \\ 0.25 & 0.1\end{bmatrix} \begin{bmatrix}L \\ 0 \end{bmatrix}= \begin{bmatrix}.25L \\ .1L \end{bmatrix}$$
$$\begin{bmatrix}0.1 & 0.25 \\ 0.25 & 0.1\end{bmatrix} \begin{bmatrix}0 \\ L \end{bmatrix}= \begin{bmatrix} .1L\\ .25L \end{bmatrix}$$
and
$$\begin{bmatrix}0.1 & 0.25 \\ 0.25 & 0.1\end{bmatrix} \begin{bmatrix}L \\ L \end{bmatrix}= \begin{bmatrix}.35L \\ .35L \end{bmatrix}$$
Last edited: Mar 17, 2016
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# zbMATH — the first resource for mathematics
##### Examples
Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.
##### Operators
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The explicit nonlinear wave solutions and their bifurcations of the generalized Camassa-Holm equation. (English) Zbl 1258.35050
Summary: We study explicit nonlinear wave solutions and their bifurcations of the generalized Camassa-Holm equation
${u}_{t}+2k{u}_{x}-{u}_{xxt}+3{u}^{2}{u}_{x}=2{u}_{x}{u}_{xx}+u{u}_{xxx}·$
Not only are the precise expressions of the explicit nonlinear wave solutions obtained, but some interesting bifurcation phenomena are revealed.
Firstly, it is verified that $k=3/8$ is a bifurcation parametric value for several types of explicit nonlinear wave solutions.
When $k<3/8$, there are five types of explicit nonlinear wave solutions, which are
(i) hyperbolic peakon wave solution,
(ii) fractional peakon wave solution,
(iii) fractional singular wave solution,
(iv) hyperbolic singular wave solution,
(v) hyperbolic smooth solitary wave solution.
When $k=3/8$, there are two types of explicit nonlinear wave solutions, which are fractional peakon wave solution and fractional singular wave solution.
When $k>3/8$, there is not any type of explicit nonlinear wave solutions.
Secondly, it is shown that there are some bifurcation wave speed values such that the peakon wave and the anti-peakon wave appear alternately.
Thirdly, it is displayed that there are other bifurcation wave speed values such that the hyperbolic peakon wave solution becomes the fractional peakon wave solution, and the hyperbolic singular wave solution becomes the fractional singular wave solution.
##### MSC:
35C08 Soliton solutions of PDE 35C07 Traveling wave solutions of PDE 35K55 Nonlinear parabolic equations 34A05 Methods of solution of ODE 34B15 Nonlinear boundary value problems for ODE 34C37 Homoclinic and heteroclinic solutions of ODE 34C23 Bifurcation (ODE)
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# Symmetric Matrix , Eigenvectors are not orthogonal to the same eigenvalue.
I know that the symmetric matrix has orthogonal eigenvectors corresponding eigenvalues. Also, the eigenvectors from the same eigenvalue are linearly independent.
I need an example of symmetric matrix such that the eigenvectors from the same eigenvalue are not orthogonal.
Also, I know that the eigenvalues of the symmetric matrix are real. But , this statement true iff the entries are real numbers.Right? Since the matrix 2x2 $$A=\begin{bmatrix} 1 &i \\ i &1 \end{bmatrix}$$
has non real eigenvalues
• If $x_1$ and $x_2$ are eigenvectors with the same eigenvalue $\lambda$, the vector $x_1+x_2$ will also be an eigenvector with the same eigenvalue $\lambda$, and it won't be orthogonal to either $x_1$ or $x_2$. – Paul Apr 19 '17 at 19:44
More generally, any combination of two eigenvectors with the same eigenvalue $\lambda$ is itself an eigenvector (with eigenvalue $\lambda$); even if your two original eigenvectors are orthogonal, a linear combinations thereof will not be orthogonal to either one.
For the second question, a complex-valued matrix has real eigenvalues iff the matrix is Hermitian, which is to say that it is equal to the conjugate of its transpose: $A^\dagger = (A^T)^* = A$. So while your $A$ is not Hermitian, the matrix $$B = \begin{bmatrix} 1 & i \\ -i & 1 \end{bmatrix}$$ is, and has two real eigenvalues (0 & 2).
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zbMATH — the first resource for mathematics
Weighted estimates for nonstationary Navier-Stokes equations. (English) Zbl 0910.35092
The Cauchy problem for the Navier-Stokes equations is considered in the whole 3-D space $\frac{\partial v}{\partial t}-\nu\Delta v+(u\cdot\nabla)v + \nabla p=0, \qquad\text{div }v =0,\quad x\in \mathbb{R}^3,\;t>0$ $v(x,0)=v_0(x)\quad x\in \mathbb{R}^3, \qquad v\to 0\;\text{as } | x| \to \infty$ Certain new results on the global solvability of the problem are obtained. Namely, if the given initial data $$v_0$$ is sufficiently small then a global strong solution exists. This solution satisfies $v,t^{\frac 12}v\in L^\infty((0,\infty);L^2(\mathbb{R}^3)),\quad \nabla v,\;t^{\frac 12}Av\in L^2((0,\infty);L^2(\mathbb{R}^3)),\quad (1+| x| ^2)v\in L^\infty(\mathbb{R}^3\times (0,\infty)).$ Here $$A$$ is the Stokes operator. Estimates of the solution are established in the corresponding functional spaces.
MSC:
35Q30 Navier-Stokes equations 35A05 General existence and uniqueness theorems (PDE) (MSC2000)
Full Text:
References:
[1] Sh. Agmon, Lectures on elliptic boundary value problems, New York, 1965 [2] Beirão da Veiga, H., On the suitable weak solutions to the navier – stokes equations in the whole space, J. math. pures appl., 64, 77-86, (1985) · Zbl 0615.35067 [3] Beirão da Veiga, H., Existence and asymptotic behavior for strong solution of the navier – stokes equations on the whole space, Indiana univ. math. J., 36, 149-166, (1987) · Zbl 0601.35093 [4] Fabes, B.E.; Jones, B.F.; Rivere, N.M., The initial value problem for the navier – stokes equations with data inL^p, Arch. rational mech. anal., 45, 222-240, (1972) · Zbl 0254.35097 [5] Galdi, G.P.; Simader, C.G., New estimates for the steady-state Stokes problem in exterior domains with applications tot he navier – stokes problem, Differential integral equations, 7, 847-861, (1994) · Zbl 0823.35142 [6] Giga, Y., Solutions for semilinear parabolic equations inL^pand regularity of weak solutions of the navier – stokes system, J. differential equations, 61, 186-212, (1986) · Zbl 0577.35058 [7] He, Cheng, The Cauchy problem for the navier – stokes equations, J. math. anal. appl., 209, 228-242, (1997) · Zbl 0880.35091 [8] He, Cheng, Existence and regularity of a class of weak solutions to the navier – stokes equations, J. math. anal. appl., 210, 512-530, (1997) · Zbl 0888.35079 [9] Heywood, J.G., Open problems in the theory of the navier – stokes equations for viscous incompressible flow, Lecture notes in math., (1990), Springer-Verlag New York/Berlin, p. 1-22 [10] Kato, T., Nonstantionary flows of viscous and ideal fluids inR3, J. funct. anal., 9, 296-305, (1972) [11] Kato, T., StrongL^p-solutions of the navier – stokes equations inrn, with application to weak solutions, Math. Z., 187, 471-480, (1984) · Zbl 0545.35073 [12] Kato, T., Liapunov functions and monotonolity in the navier – stokes equation, Lecture notes in math., (1989), Springer-Verlag New York/Berlin, p. 53-64 [13] Kato, T.; Ponce, G., Commutator estimates and the Euler and navier – stokes equations, Comm. pure. appl. math., 41, 891-907, (1988) · Zbl 0671.35066 [14] Kajikiya, R.; Miyakaya, T., OnL2rn, Math. Z., 192, 135-148, (1986) [15] Ladyzhenskaya, O.A., The mathematical theory of viscous incompressible flow, (1969), Gordon & Breach New York · Zbl 0184.52603 [16] Ladyzhenskaya, O.A.; Solonnikov, V.A.; Uralceva, N.N., Linear and quasilinear equations of parabolic type, Transl. math. monographs, 23, (1968) · Zbl 0174.15403 [17] Leray, J., Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta math., 63, 193-248, (1934) · JFM 60.0726.05 [18] Lions, J.L., Quelques méthodes de résolution des problèmes aux limites non linéaires, (1969), Dunod Paris · Zbl 0189.40603 [19] Schonbek, M.E., L2, Arch. rational mech. anal., 88, 209-222, (1985) [20] Stein, E.M., Singular integrals and differentiability properties of functions, (1970), Princeton Univ. Press Princeton · Zbl 0207.13501 [21] Stein, E.M., Note on singular integrals, Proc. amer. math. soc., 8, 250-254, (1957) · Zbl 0077.27301 [22] Temam, R., Navier – stokes equations, (1977), North-Holland Amsterdam · Zbl 0335.35077 [23] Wiegner, M., Decay results for weak solutions of the navier – stokes equations onR^n, J. London math. soc. (2), 35, 303-313, (1987) · Zbl 0652.35095
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.
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