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# Find the value of 'b' if sqrt7 is a root of the equation x^2
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Find the value of 'b' if sqrt7 is a root of the equation x^2 [#permalink]
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30 Jan 2004, 16:20
This topic is locked. If you want to discuss this question please re-post it in the respective forum.
Find the value of 'b' if sqrt7 is a root of the equation x^2 + ax - 14 = 0 and the equation x^2 + ax + b = 0 has real and equal roots.
7/4
-7/4
49/4
none of these
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30 Jan 2004, 16:29
from the second eqn: equal and real roots
a^2 = 4b
from the first eqn, let k be the other root
sqrt(7)+ k = - a
ksqrt(7) = -14
Thus b = (1/4)[sqrt(7) - [14/sqrt(7)]]^2
Last edited by Zhung Gazi on 30 Jan 2004, 17:12, edited 1 time in total.
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30 Jan 2004, 16:33
Rakesh, we did a similar problem yesterday, but I am having problems with this question. Can u help
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30 Jan 2004, 16:54
Yeah ur posting such wonderful avathars. We go step by step
from the first eq. let p & q be the roots then we know that p+q= - b/a (the eq is of the form ax^2 + bx +c ), so p+q = -a/1 = -a ( eq is x^2+ax-14) and pq = c/a = -14, we are given that one root is sqrt7 then let p=sqrt7 , we know that pq=-14, so (sqrt7* q) = -14 solve and we get q = -2sqrt7, we also know that p =sqrt7 and p+q = -a, substitute the values of p & q and we get a=sqrt7----------------(1)
from the second eq, we r given that it has real and equal roots so it means b^2- 4ac=0 ( for eq of the form ax^2 +bx +c) so as the given eq is of the form x^2 +ax+b=0 , we get a^2 - 4 * 1 * b=0, so a^2=4b, we know that a=sqrt7 ( from -----(1) above) substitute this value of a in a^2=4b and we get b=7/4.
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30 Jan 2004, 17:32
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30 Jan 2004, 17:32
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# Thread: more word problems for the ASVAB
1. ## more word problems for the ASVAB
35. Jimmy made a 15% profit on the sale of a custom designed boat, and the original cost of the boat was $15,000. The boat sold for how much? A.$17,250.00
B. $16,540.44 C.$16,230.34
D. $15,980.55 E.$15,870.8
33. Sue receives a base salary of $90 weekly plus a 12% commission on all sales. Sue had$3,000 in sales this week. How much did she make total?
A. $375 B.$450
C. $480 D.$510
E. $525 24. In his pocket, a boy has 3 red marbles, 4 blue marbles, and 4 green marbles. How many will he have to take out of his pocket to ensure that he has taken out at least one of each color? A. 3 B. 7 C. 8 D. 9 E. 11 22. The cost to ride on a ferry is$5.00 per vehicle and driver with an additional cost of 50 cents per passenger. If the charge to get on the ferry is $6.50, how many people were in the vehicle? A. 1 B. 2 C. 3 D. 4 E. 5 15. What will it cost to tile a kitchen floor that is 12 feet wide by 20 feet long if the tile cost$8.91 per square yard?
A. $224.51 B.$237.60
C. $246.55 D.$271.38
E. \$282.32
2. To learn the basics of how to set up and solve "percent of" word problems, try here.
Once you have studied the topic, please attempt at least one of the exercises. If you get stuck, you will then be able to reply with a clear listing of your steps and reasoning so far, at which point we'll be able to respond with intelligent assistance. Thank you!
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# Velocity of light
## Recommended Posts
How can the velocity of light be slower or faster depending on colour (n) and at the same time have a constant vacuum velocity of c? (Answers presupposing special or general relativity not of interest.)
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the velocity of light does not change with colour. i don't know where you heard it but it isn't true.
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Light's color varies depending on the frequency of the wave, the constant velocity doesn't change with color.
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Light does change speed within a medium, and that does generally vary with the wavelength. This is known as dispersion. v = c/n, where n is the index of refraction (n=1 for a vacuum as you might suspect from the previous answers)
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Light's color varies depending on the frequency of the wave, the constant velocity doesn't change with color.
Thought I would add this; a simple expression to help swallow the concept:
$f = \frac{c}{\lambda}$
Where "c" is the speed of light, $\lambda$ is the frequency of the light, and "f" is the frequency. This only works in a vacuum because light doesn't have a average velocity of "c" in different media as Swansont stated.
Edited by mississippichem
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How can the velocity of light be slower or faster depending on colour (n) and at the same time have a constant vacuum velocity of c? (Answers presupposing special or general relativity not of interest.)
Are you thinking about red shift and blue shift, if you are then it depends on the speed of the object emmiting or recieving the light not the speed of light itself (which is always constant). If the speed of different colours of light were different then you would see this during an eclipse
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the velocity of light does not change with colour. i don't know where you heard it but it isn't true.
Of course the velocity of light changes with colour. I don't know where you heard otherwise (since you haven't told me), but it isn't true.
If we start with the basic equation v = λn, then with a Doppler effect we get a change in n (fcy/colour), which, given constant λ (wavelength), implies a change in v. Or is v = λn no longer viable?
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ah, you're using the equation wrong. frequency and wavelength are variables the velocity is not. you can't have a constant wavelength and varying frequency.
pretty much all physics texts and experiments confirm the speed is constant with relation to colour. the only ones that don't are those unrelated to the topic of light.
Edited by insane_alien
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[/size]ost='576204']
Thought I would add this; a simple expression to help swallow the concept:
$f = \frac{c}{\lambda}$
Where "c" is the speed of light, $\lambda$ is the frequency of the light, and "f" is the frequency. This only works in a vacuum because light doesn't have a average velocity of "c" in different media as Swansont stated.
To start, let's not use c, but v, since c, being a constant, begs the question. Then we get f = v/λ, or, in the notation I'm familiar with, n = v/λ, or v = λn. Thus λ is not, as I'm sure you didn't mean it to be, fcy (colour); n (= f) is fcy (= colour). But then my comments to other contributors should apply.
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whether you call it c or v it's still a constant. as far as i can tell you are just saying its not.
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ah, you're using the equation wrong. frequency and wavelength are variables the velocity is not. you can't have a constant wavelength and varying frequency.
pretty much all physics texts and experiments confirm the speed is constant with relation to colour. the only ones that don't are those unrelated to the topic of light.
No, I should say, fcy and vel are variables; wavelength is not. At least this is what I think I've leared from my study of the subject. Cf e.g Otis (1963), p. 10. The motion of a spectroscope towards or away from a star, caused by the orbital and rotational motions of the earth, cannot in any way affect the ‘wavelength’ (λ) of the light coming from the star. This means that when the motion of the earth causes the spectroscope to approach a star, the shift of the spectrum of the star towards the violet clearly indicates an increase in the frequency (n) of the reception of the constituents of the starlight by the spectroscope. And a red-shift when the spectroscope recedes from the star clearly indicates a decrease in the frequency.
“Since velocity equals wavelength multiplied by frequency (v = λn), it follows that when the wavelength of the light coming from a star is unchanged [= the light coming from the star doesn’t change], and its frequency of reception by the spectroscope changes, as indicated by the shift of the spectrum of the light, the velocity (λn) of the light relative to the spectroscope changes.” And it follows further that, more particularly, it is the frequency of the radiation that changes, given that the wavelength of the light emitted is constant, as in the above example.
Otis (1963), p. 13. According to the light postulate there is no way by which an observer can detect any difference in the velocity of light relative to him. [so, given the special theory, the postulate that the speed of light is constant cannot be checked?] Nevertheless we see that the shift of the spectrum of light when the spectroscope is moved towards the source provides us with definite empirical evidence that the frequency of reception of the wavefronts by the spectroscope is increased, and hence (l remaining unchanged) the velocity of the light relative to the spectroscope (and relative to the laboratory and observer) is increased, thus contradicting the light postulate.
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Of course the velocity of light changes with colour. I don't know where you heard otherwise (since you haven't told me), but it isn't true.
If we start with the basic equation v = λn, then with a Doppler effect we get a change in n (fcy/colour), which, given constant λ (wavelength), implies a change in v. Or is v = λn no longer viable?
No. I'm not sure where you heard this, but it's not true. In a vacuum the speed of light is a constant, which is a ramification of Maxwell's equations. If it weren't your car radio wouldn't work. The Doppler effect changes both frequency and wavelength in inverse relation, so that the speed remains constant.
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whether you call it c or v it's still a constant. as far as i can tell you are just saying its not.
The question is whether it is or not. It's not a matter of what either of us say.
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it has been experimentally shown to be constant time and time again. the literature on the subject is overwhelmingly in support of a constant speed of light.
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To start, let's not use c, but v, since c, being a constant, begs the question. Then we get f = v/λ, or, in the notation I'm familiar with, n = v/λ, or v = λn. Thus λ is not, as I'm sure you didn't mean it to be, fcy (colour); n (= f) is fcy (= colour). But then my comments to other contributors should apply.
Color can be thought of as frequency or wavelength. They are really just different expressions of the same thing because light always travels at c in a vacuum. One just has units of length while the other has units of count/time.
If c isn't constant in a vacuum, the majority of modern physics would have to be overturned.
Edited by mississippichem
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The question is whether it is or not. It's not a matter of what either of us say.
Actually yes it is, when one is drawing on more than a hundred years' worth of research that depends on whether it's true or not. As mississippichem has stated, you would rewrite a lot of physics, and you'd be in the mystifying state of having things like GPS working, and yet the theoretical basis for its operation being wrong.
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With energy/photons there is a component that remains at C. We also have two finite components that we call wavelength and frequency. When light changes color, only these two finite aspects of the photon will change, but the C aspect will not be effected by the color change. Picture C as the root of a photon. Its two finite branches can move, but the root stays fixed. C is sort of an anchor state, with photons not able to move plus or minus C and still remain photons.
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it has been experimentally shown to be constant time and time again. the literature on the subject is overwhelmingly in support of a constant speed of light.
This reply begs the question. In the present context I would benefit most if you directed your comments to the reasoning involved in my Otis reference. Thanks!
With energy/photons there is a component that remains at C. We also have two finite components that we call wavelength and frequency. When light changes color, only these two finite aspects of the photon will change, but the C aspect will not be effected by the color change. Picture C as the root of a photon. Its two finite branches can move, but the root stays fixed. C is sort of an anchor state, with photons not able to move plus or minus C and still remain photons.
Could you please meet the line of thinking involved in my reference to Otis? Thanks! (What you present here just side-steps the issue.)
it has been experimentally shown to be constant time and time again. the literature on the subject is overwhelmingly in support of a constant speed of light.
Sorry, this isn't an argument, only hand-waving.
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Empirical demonstration is "begging the question?"
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Actually yes it is, when one is drawing on more than a hundred years' worth of research that depends on whether it's true or not. As mississippichem has stated, you would rewrite a lot of physics, and you'd be in the mystifying state of having things like GPS working, and yet the theoretical basis for its operation being wrong.
Yes, it presents a conundrum. But I'd appreciate your showing how Otis' (and and at least one so-far unmentioned other's) reasoning on this point is mistaken, rather than just referring to "a hundred years of research." Modern physics is a very tricky business, particularly since physicists now seem to accept such things as (Maxwellian, Lorentzian, Hertzian) electrodynamics (your GSP reference), which presupposes the existence of waves in a medium, and at the same time special relativity, which excludes the medium, as well as both the special and general theories of relativity, which are incompatible. By the way, by advocating the constancy of the speed of light you're denying the viability of the general theory of relativity, according to which the speed of light is variable. I think that much of this confusion stems from physicists' inclination to apply either wave or particle thinking whenever convenient (as QM has institutionalised).
Your suggestion of re-writing a lot of physics is interesting, since that's precisely what I'm engaged in!
As regards the GPS business, of course, logically, the results of Maxwellian electrodynamics could well be right while the theory itself is wrong.
Empirical demonstration is "begging the question?"
Could you please reply to the line of reasoning in my reference to Otis? Thanks!
Color can be thought of as frequency or wavelength. They are really just different expressions of the same thing because light always travels at c in a vacuum. One just has units of length while the other has units of count/time.
If c isn't constant in a vacuum, the majority of modern physics would have to be overturned.
You've got it!
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This reply begs the question. In the present context I would benefit most if you directed your comments to the reasoning involved in my Otis reference. Thanks!
This reference: Arthur Sinton Otis, Ph.D, "Light velocity and relativity: the problem of light velocity, disproof of the Einstein postulate" ?
Two words: Crack Pot. He wasn't even a physicist, and had no advanced training in physics. Otis was a psychiatrist or psychologist who apparently designed a school aptitude test. In his own field he may have been very good, but not in physics.
it has been experimentally shown to be constant time and time again. the literature on the subject is overwhelmingly in support of a constant speed of light.
Sorry' date=' this isn't an argument, only hand-waving.
[/quote']
No, claiming that experiment after experiment after experiment after experiment has shown that light does travel at c in vacuum regardless of the relative velocity between the source and the receiver and regardless of the frequency of the light is not hand-waving.
Edited by D H
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Are you thinking about red shift and blue shift, if you are then it depends on the speed of the object emmiting or recieving the light not the speed of light itself (which is always constant). If the speed of different colours of light were different then you would see this during an eclipse
When it comes to red- and blue-shift, you have to consider whether it is being understood in terms of a wave or particle theory. On a wave theory, such as Maxwell's, motion of the source is irrelevant. In the case of particle (electrodynamically relativistic) theories, the motion of the source is relevant, but only after the time it takes the radiation to reach the receiver.
As I mentioned to Swansont, that the vel. of light should be constant contravenes the general theory of relativity.
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Otis's work - some of which can be found on questia (I won't post link due to possible copyright problems) - directly contradicts maxwell, einstein etc. many seemingly consistent theories can be posited, however a constant speed of light which follows from maxwell's equations and is the basis of special relativity is the theory which matches experimental data. special and general relativity have been tested to an enormous extent, its predictions work, and real-world applications rely on the equations and physics it generates. From my very brief reading Otis claims that light does not have constant speed regardless of the motion of the observer/source - SR is based on the fact that it does. One is right, the other is wrong; I don't know if Otis's work can be mathematically self-consistent (he was/is a far greater mathematician than I) but even presuming that it is self-consistent it doesn't comply with known experimental results.
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This reference: Arthur Sinton Otis, Ph.D, "Light velocity and relativity: the problem of light velocity, disproof of the Einstein postulate" ?
Two words: Crack Pot. He wasn't even a physicist, and had no advanced training in physics. Otis was a psychiatrist or psychologist who apparently designed a school aptitude test. In his own field he may have been very good, but not in physics.
No, claiming that experiment after experiment after experiment after experiment has shown that light does travel at c in vacuum regardless of the relative velocity between the source and the receiver and regardless of the frequency of the light is not hand-waving.
Sorry, this is an ad hominem argument, which doesn't add to the discussion. Otis could be a mystic for all it matters. What you have to do is deal with what he says. And saying "experiment" four times doesn't strengthen your argument. But I'm curious. Which experiments do you have in mind, and do they presuppose a wave theory, or a particle theory, or neither?
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Math Calculators, Lessons and Formulas
It is time to solve your math problem
mathportal.org
« Solving Absolute Value Equations
Solving Equations: (lesson 3 of 4)
Definition:
A quadratic equation in the variable x is an equation that can be written in the form:
$a{x^2} + bx + c = 0$, $a \ne 0$
where a, b and c represent real number coefficients.
This form is sometimes called the standard form. The term quadratic is used for any equation where the highest power of the variable x is 2. The coefficient a cannot be zero, since otherwise it would be a linear equation.
${x_{1,2}} = \frac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$, $a \ne 0$
To solve quadratic equations, substitute the coefficients a, b and c into the quadratic formula.
The expression b2 - 4ac shown under the square root sign is called the discriminant, because it can "discriminate" between the all possible types of answer:
type 1: If b2 - 4ac ≥ 0 ⇒ equation has two real roots;
type 2: If b2 - 4ac = 0 ⇒ equation has two real roots but they are both the same.
type 3: If b2 - 4ac ≤ 0 ⇒ equation has two complex roots;
Example 1:
2x2 + 7x - 15 = 0
Solution:
In this case a = 2 b = 7 c= -15
The value of the discriminant is b2 - 4ac = 72 - 4(2)(-15) = 169 (Type 1)
${x_{1,2}} = \frac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}} = \frac{{ - 7 \pm \sqrt {169} }}{4} = \frac{{ - 7 \pm 13}}{4} \to {x_1} = - 5$ and ${x_2} = \frac{3}{2}$
Level 1
$$\color{blue}{{x^2} - 4x + 3 = 0}$$ ${x_1} = 3, {x_2} = - 1$ ${x_1} = - 3, {x_2} = 1$ ${x_1} = 3, {x_2} = 1$ ${x_1} = - 3, {x_2} = - 1$
Level 2
$$\color{blue}{3{x^2} - 4x - 4 = 0}$$ ${x_1} = 2,{x_2} = - \frac{2}{3}$ ${x_1} = - 2,{x_2} = - \frac{2}{3}$ ${x_1} = - 2,{x_2} = \frac{2}{3}$ ${x_1} = 2,{x_2} = \frac{2}{3}$
Example 2:
Solve the following equation using the quadratic formula.
4x2 - 20x + 25 = 0
Solution:
In this case a = 4 b = - 20 c = 25
The value of the discriminant is b2 - 4ac = 202 - 4(4)(25) = 0
${x_{1,2}} = \frac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}} = \frac{{20 \pm \sqrt 0 }}{8} = \frac{{20}}{8} = \frac{5}{2} \to x = - \frac{5}{2}$
That is, in this case since the value of the discriminant is zero, the two roots of the equation have the same of 2.5.
Level 1
$$\color{blue}{{x^2} - 2x + 1 = 0}$$ $x = 2$ $x = - 2$ $x = - 1$ $x = 1$
Level 2
$$\color{blue}{9{x^2} - 6x + 1 = 0}$$ $x = \frac{1}{3}$ $x = - \frac{1}{3}$ $x = \frac{2}{3}$ $x = - \frac{2}{3}$
Example 3:
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# True quantified Boolean formula
True quantified Boolean formula
The language TQBF is a formal language in computer science that contains True Quantified Boolean Formulas. A fully quantified boolean formula is a formula in first-order logic where every variable is quantified (or bound), using either existential or universal quantifiers, at the beginning of the sentence. Any such formula is always either true or false (since there are no free variables). If such a formula evaluates to true, then that formula is in the language TQBF. It is also known as QSAT (Quantified SAT).
Overview
In computational complexity theory, the quantified Boolean formula problem (QBF) is a generalization of the Boolean satisfiability problem in which both existential quantifiers and universal quantifiers can be applied to each variable. Put another way, it asks whether a first-order logic sentential form over a set of Boolean variables is true or false. For example, the following is an instance of QBF:
: $forall x exists y exists z \left(x lor y lor z\right) land \left(lnot x lor lnot y lor lnot z\right)$
QBF is the canonical complete problem for PSPACE, the class of problems solvable by a deterministic or nondeterministic Turing machine in polynomial space and unlimited time. [cite book | author = M. Garey and D. Johnson | title = Computers and intractability: a guide to the theory of NP-completeness | publisher = W. H. Freeman, San Francisco, California | year = 1979 | id = ISBN 0716710455] Given the formula in the form of an abstract syntax tree, the problem can be solved easily by a set of mutually recursive procedures which evaluate the formula. Such an algorithm uses space proportional to the height of the tree, which is linear in the worst case, but uses time exponential in the number of quantifiers.
Provided that MA ⊂ PSPACE, which is widely believed, QBF cannot be solved, nor can a given solution even be verified, in either deterministic or probabilistic polynomial time (in fact, unlike the satisfiability problem, there's no known way to specify a solution succinctly). It is trivial to solve using an alternating Turing machine in linear time, which is no surprise since in fact
cite journal | author = A. Chandra, D. Kozen, and L. Stockmeyer | url = http://portal.acm.org/citation.cfm?id=322243 | title = Alternation | journal = Journal of the ACM | volume = 28 | issue = 1 | pages = 114–133 | year = 1981 | doi = 10.1145/322234.322243]
When the seminal result IP = PSPACE was shown (see interactive proof system), it was done by exhibiting an interactive proof system that could solve QBF by solving a particular arithmetization of the problem. [cite journal | author = Adi Shamir| url = http://portal.acm.org/citation.cfm?doid=146585.146609 | title = Ip = Pspace | journal = Journal of the ACM | volume = 39 | issue = 4 | pages = 869–877 | year = 1992 | doi = 10.1145/146585.146609]
QBF formulas have a number of useful canonical forms. For example, it can be shown that there is a polynomial-time many-one reduction that will move all quantifiers to the front of the formula and make them alternate between universal and existential quantifiers. There is another reduction that proved useful in the IP = PSPACE proof where no more than one universal quantifier is placed between each variable's use and the quantifier binding that variable. This was critical in limiting the number of products in certain subexpressions of the arithmetization.
Prenex normal form
A fully quantified Boolean formula can be assumed to have a very specific form, called prenex normal form. It has two basic parts: a portion containing only quantifiers and a portion containing an unquantified Boolean formula usually denoted as $phi$. If there are "n" Boolean variables, the entire formula can be written as
:$exists x_1 forall x_2 exists x_3 cdots Q_n x_n phi\left(x_1, x_2, x_3, dots, x_n\right)$
where every variable falls within the scope of some quantifier. By introducing dummy variables, any formula in prenex normal form can be converted into a sentence where existential and universal quantifiers alternate. Using the dummy variable $y_1$,
:$exists x_1 exists x_2 phi\left(x_1, x_2\right) quad mapsto quadexists x_1 forall y_1 exists x_2 phi\left(x_1, x_2\right)$
The second sentence has the same truth value but follows the restricted syntax. Assuming fully quantified Boolean formulas to be in prenex normal form is a frequent feature of proofs.
Solving
There is a simple recursive algorithm for determining whether a TQBF is true. Given some QBF
:$Q_1 x_1 Q_2 x_2 cdots Q_n x_n phi\left(x_1, x_2, dots, x_n\right).$
If the formula contains no quantifiers, we can just return the formula. Otherwise, we take off the first quantifier and check both possible values for the first variable:
:$A = Q_2 x_2 cdots Q_n x_n phi\left(0, x_2, dots, x_n\right),$:$B = Q_2 x_2 cdots Q_n x_n phi\left(1, x_2, dots, x_n\right).$
If $Q_1 = exists$, then return $A lor B$. If $Q_1 = forall$, then return $A land B$.
How fast does this algorithm run?For every quantifier in the initial QBF, the algorithm makes two recursive calls on only a linearly smaller subproblem. This gives the algorithm an exponential runtime "O(2^n)".
How much space does this algorithm use?Within each invocation of the algorithm, it needs to store the intermediate results of computing A and B. Every recursive call takes off one quantifier, so the total recursive depth is linear in the number of quantifiers. Formulas that lack quantifiers can be evaluated in space logarithmic in the number of variables. The initial QBF was fully quantified, so there are at least as many quantifiers as variables. Thus, this algorithm uses "O"("n" + log "n") = "O"("n") space. This makes the TQBF language part of the PSPACE complexity class.
PSPACE-completeness
The TQBF language serves in complexity theory as the canonical PSPACE-complete problem. Being PSPACE-complete means that a language is in PSPACE and that the language is also PSPACE-hard. The algorithm above shows that TQBF is in PSPACE.Showing that TQBF is PSPACE-hard requires showing that any language in the complexity class PSPACE can be reduced to TQBF in polynomial time. I.e.,
:$forall Lin extrm\left\{PSPACE\right\}, Lleq_p extrm\left\{TQBF\right\}.$
This means that, for a PSPACE language L, whether an input $x$ is in L can be decided by checking whether $f\left(x\right)$ is in TQBF, for some function $f$ that is required to run in polynomial time (relative to the length of the input) Symbolically,
:$xin Liff f\left(x\right)in extrm\left\{TQBF\right\}.$
Proving that TQBF is PSPACE-hard, requires specification of $f$.
So, suppose that L is a PSPACE language. This means that L can be decided by a polynomial space deterministic Turing machine (DTM). This is very important for the reduction of L to TQBF, because the configurations of any such Turing Machine can be represented as Boolean formulas, with Boolean variables representing the state of the machine as well as the contents of each cell on the Turing Machine tape, with the position of the Turing Machine head encoded in the formula by the formula's ordering. In particular, our reduction will use the variables $c_1$ and $c_2$, which represent two possible configurations of the DTM for L, and a natural number t, in constructing a QBF $phi_\left\{c_1,c_2,t\right\}$ which is true if and only if the DTM for L can go from the configuration encoded in $c_1$ to the configuration encoded in $c_2$ in no more than t steps. The function $f$, then, will construct from the DTM for L a QBF $phi_\left\{c_\left\{start\right\},c_\left\{accept\right\},T\right\}$, where $c_\left\{start\right\}$ is the DTM's starting configuration, $c_\left\{accept\right\}$ is the DTM's accepting configuration, and T is the maximum number of steps the DTM could need to move from one configuration to the other. We know that "T" = "O"(exp("n")), where n is the length of the input, because this bounds the total number of possible configurations of the relevant DTM. Of course, it cannot take the DTM more steps than there are possible configurations to reach $c_mathrm\left\{accept\right\}$ unless it enters a loop, in which case it will never reach $c_mathrm\left\{accept\right\}$ anyway.
At this stage of the proof, we have already reduced the question of whether an input formula $w$ (encoded, of course, in $c_\left\{start\right\}$) is in L to the question of whether the QBF $phi_\left\{c_\left\{start\right\},c_\left\{accept\right\},T\right\}$, i.e., $f\left(w\right)$, is in TQBF. The remainder of this proof proves that $f$ can be computed in polynomial time.
For $t=1$, computation of $phi_\left\{c_1,c_2,t\right\}$ is straightforward--either one of the configurations changes to the other in one step or it does not. Since the Turing Machine that our formula represents is deterministic, this presents no problem.
For $t>1$, computation of $phi_\left\{c_1,c_2,t\right\}$ involves a recursive evaluation, looking for a so-called "middle point" $m_1$. In this case, we rewrite the formula as follows:
:$phi_\left\{c_1,c_2,t\right\}=exists m_1\left(phi_\left\{c_1,m_1,lceil t/2 ceil\right\}wedgephi_\left\{m_1,c_2,lceil t/2 ceil\right\}\right).$
This converts the question of whether $c_1$ can reach $c_2$ in t steps to the question of whether $c_1$ reaches a middle point $m_1$ in $t/2$ steps, which itself reaches $c_2$ in $t/2$ steps. The answer to the latter question of course gives the answer to the former.
Now, t is only bounded by T, which is exponential (and so not polynomial) in the length of the input. Additionally, each recursive layer virtually doubles the length of the formula. (The variable $m_1$ is only one midpoint--for greater t, there are more stops along the way, so to speak.) So the time required to recursively evaluate $phi_\left\{c_1,c_2,t\right\}$ in this manner could be exponential as well, simply because the formula could become exponentially large. This problem is solved by universally quantifying using variables $c_3$ and $c_4$ over the configuration pairs (e.g., $\left\{ \left(c_1,m_1\right),\left(m_1,c_2\right)\right\}$), which prevents the length of the formula from expanding due to recursive layers. This yields the following interpretation of $phi_\left\{c_1,c_2,t\right\}$:
:$phi_\left\{c_1,c_2,t\right\}=exists m_1forall \left(c_3,c_4\right)in \left\{ \left(c_1,m_1\right),\left(m_1,c_2\right)\right\}\left(phi_\left\{c_3,c_4,lceil t/2 ceil\right\}\right).$
This version of the formula can indeed be computed in polynomial time, since any one instance of it can be computed in polynomial time. The universally quantified ordered pair simply tells us that whichever choice of $\left(c_3,c_4\right)$ is made, $phi_\left\{c_1,c_2,t\right\}iffphi_\left\{c_3,c_4,lceil t/2 ceil\right\}$.
Thus, $forall Lin extrm\left\{PSPACE\right\}, Lleq_p extrm\left\{TQBF\right\}$, so TQBF is PSPACE-hard. Together with the above result that TQBF is in PSPACE, this completes the proof that TQBF is a PSPACE-complete language.
(This proof follows Sipser 2006 pp. 310-313 in all essentials. Papadimitriou 1994 also includes a proof.)
Miscellany
*One important subproblem in TQBF is the Boolean satisfiability problem. In this problem, you wish to know whether a given Boolean formula $phi$ can be made true with some assignment of variables. This is equivalent to the TQBF using only existential quantifiers:
:: $exists x_1 cdots exists x_n phi\left(x_1, ldots, x_n\right).$
:This is also an example of the larger result NP $subseteq$ PSPACE which follows directly from the observation that a polynomial time verifier for a proof of a language accepted by a NTM (Non-deterministic Turing Machine) requires polynomial space to store the proof.
*Any class in the polynomial hierarchy (PH) has TQBF as its complete problem. In other words, for the class comprised of all languages L for which there exists a poly-time TM V, a verifier, such that for all input x and some constant i,
:: $x in L Leftrightarrow exists y_1 forall y_2 cdots Q_i y_i V\left(x,y_1,y_2,dots,y_i\right) = 1$
: which has a specific QBF formulation that is given as
:: $exists phi$ such that $exists vec\left\{x_1\right\} forall vec\left\{x_2\right\} cdots Q_i vec\left\{x_i\right\} phi\left(vec\left\{x_1\right\},vec\left\{x_2\right\},dots,vec\left\{x_i\right\}\right) = 1$
:where the $vec\left\{x_i\right\}$'s are vectors of Boolean variables.
*It is important to note that while TQBF the language is defined as the collection of true quantified Boolean formulas, the abbreviation TQBF is often used (even in this article) to stand for a totally quantified Boolean formula, often simply called a QBF (quantified Boolean formula, understood as "fully" or "totally" quantified). It is important to distinguish contextually between the two uses of the abbreviation TQBF in reading the literature.
*A TQBF can be thought of as a game played between two players, with alternating moves. Existentially quantified variables are equivalent to the notion that a move is available to a player at a turn. Universally quantified variables mean that the outcome of the game does not depend on what move a player makes at that turn. Also, a TQBF whose first quantifier is existential corresponds to a formula game in which the first player has a winning strategy.
Notes and references
* Fortnow & Homer (2003) provides some historical background for PSPACE and TQBF.
* Zhang (2003) provides some historical background of Boolean formulas.
* Arora, Sanjeev. (2001). [http://www.cs.princeton.edu/~arora/pubs/aroracom.ps "COS 522: Computational Complexity"] . Lecture Notes, Princeton University. Retrieved October 10, 2005.
* Fortnow, Lance & Steve Homer. (2003, June). [http://people.cs.uchicago.edu/~fortnow/beatcs/column80.pdf A short history of computational complexity] . "The Computational Complexity Column," 80. Retrieved October 9, 2005.
* Papadimitriou, C. H. (1994). "Computational Complexity." Reading: Addison-Wesley.
* Sipser, Michael. (2006). "Introduction to the Theory of Computation." Boston: Thomson Course Technology.
* Zhang, Lintao. (2003). [http://research.microsoft.com/users/lintaoz/thesis_lintao_zhang.pdf "Searching for truth: Techniques for satisfiability of boolean formulas"] . Retrieved October 10, 2005.
See also
* Cook–Levin theorem, stating that SAT is NP-complete
* Generalized geography
External links
* The Quantified Boolean Formulas Library [http://www.qbflib.org (QBFLIB)]
Wikimedia Foundation. 2010.
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## Friday, May 16, 2014
### HaTeX-3.13: A summary of the latest developments
This week I have been coding for HaTeX, the LaTeX library of Haskell. If this is the first time you read about this library, take a look at it in Hackage or in GitHub.
I have closed really old tickets and made some important changes, and now I will let the library have a more stable time to check if these changes are worth in the long run. I think all these changes are positive, but I have to apologize for releasing two major versions in a single week. I don't want to give my users headaches, but I also want to provide them with a better library if that's in my hands.
## Property Tests (QuickCheck)
The first thing I want to mention is the addition of a test suite to HaTeX. It is rather small currently, but it is already giving us benefits. The greatest impact has been in the parser, when the following property has been added:
fmap render (parseLaTeX t) == Right t
Here t :: Text is a randomly generated syntactically correct LaTeX code. It is important to note that this property gives us two facts:
In other words, parseLaTeX is a partial function (if we consider Left values as errors) that is defined if and only if the input is a valid LaTeX file, and render is its left inverse. These are some properties that I would expect from parseLaTeX in order to do a reasonable job. The good thing is that now they are automatically checked and, thanks to that, I have discovered many small bugs I never noticed before (thanks QuickCheck!).
I want to say as well that having HaTeX added to Stackage is giving us good benefits. I have been quickly prompted when HaTeX did not build with the last version of transformers, or when a test suite was failing. Thank you Michael for your great work!
## Removal of TeXOp constructor
The LaTeX type has now one constructor less: TeXOp. This has simplified a little bit some other functions, mostly reducing code in case-by-case pattern matching. The reason to remove such constructor is that is was not providing anything that others constructors could not. Therefore, it didn't have much sense to have it there in the first place.
## Pretty-Printer
Some users have written me complaining that the output of the render function applied to LaTeX values is unreadable and hard to debug. It contains big lines of agglomerated code, making hard to distinguish - for example - where an environment starts and where it ends. This is on purpose. HaTeX won't add any line break that the user does not specify explicitly. If it were done that way it would, for instance, make a paragraph break where it should not be one. And worse, the user won't have any workaround to solve it. However, it is reasonable to ask for a prettier output. This is what the new Pretty module addresses. It has not been widely used yet, so it can probably be improved.
## The LaTeXC instance for LaTeXT
Back when the LaTeXC class was implemented, we needed to get values of type LaTeXT m a from LaTeX values for any type a, and the only value inhabiting every type is bottom, so we used that one. This has been done this way until now. HaTeX has been following an use-as-few-extensions-as-you-can policy, meaning that we stick with Haskell2010 as much as we can. But, since there was interest, I have added the TypeFamilies extension. The current LaTeXC instance has a ~ () in its context. This is also true for the IsString and Monoid instances, and for the numerical classes. Being honest, I still have to check what are the consequences of this, but I think time will tell us. At the moment, this change has simplified significantly the code of the Base.Writer module.
## Back to parsec
The first LaTeX parser was written in parsec, but was later rewritten by Tobias Schoofs using attoparsec. Since the new parser was better - in the sense that was closer to have the properties listed in the first section of this post - I accepted the patch gladly and we have been using it with some variations (some of them important) until today. With time, it became clear that the uninformative parsing error messages of attoparsec were unacceptable for this case, where many input files were written by hand or fixed manually, and most of them small enough to not be worth to have a faster parser. This is why today I decide to dedicate my evening to port the parser back to parsec, and so I did. A combination of the type checker and QuickCheck have made the work very amusing.
## Closure
If you are interested in a more detailed list of changes, it's probably worth a look at the commit list. If you think something in HaTeX has to be improved or fixed, do not hesitate in filling a ticket at the issue tracker. Thank you for reading to this point.
Happy hacking,
Daniel Díaz.
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# Underfull \hbox (badness 10000) in paragraph for table
I got a "bad box" warning said !h'
Underfull \hbox (badness 10000) in paragraph
Underfull \hbox (badness 10000) in paragraph
Underfull \hbox (badness 10000) in paragraph
Underfull \hbox (badness 2707) in paragraph
The source code is
\begin{table}[!ht]
\caption{Comparison of design requirements for adults and children}
\footnotesize
\begin{center}
\begin{tabular}{|p{0.2\linewidth}|p{0.35\linewidth}|p{0.35\linewidth}|}
\hline \centering{Message} & \centering{Adults} & \centering{Children} \tabularnewline
\hline *M1-constraint-requirement-requirement-summary & The target user is an \textbf{adult}. Constraints - Ergonomics can be found in standard \textbf{XX1}.''& The target user is a \textbf{child}. Constraints - Ergonomics can be found in standard \textbf{XX2}.'' \tabularnewline
\hline
\end{tabular}
\end{center}
\end{table}
and it looks like this
I don't know what's going wrong in my case.
• Welcome to TeX.SX! TeX is having problems in justification (that is in making lines of equal lengths) in those narrow columns. You probably want to use ragged right typesetting. – egreg Nov 19 '13 at 21:44
• what egreg said, also note \centering does not take an argument: \centering{Adults} should be \centering Adults (the extra group is in this case harmless but doing northing) – David Carlisle Nov 19 '13 at 23:34
The lines of the first cell of the second line doesn't fill the hbox-es (3 bad boxes). Same at the third cell for one line (1 bad box). Using \raggedright can make these bad boxes go away without the real change of output. This is because the box is small but the words are long: you can't really use justification when you have 1 word per line. (If you have a few words per line, you can try \sloppy.)
The last bad box is caused by the fact that your tabular is larger than the actual \hsize. That's because it is wider than the sum of the column widths. Using the showframe package, you can clearly see that in the example below (\raggedright already added, 1 bad box remaining). You can fix this by resizing a column or changing the column separation (or check this topic).
\documentclass{article}
\usepackage{showframe}
\begin{document}
\begin{table}[!ht]
\caption{Comparison of design requirements for adults and children}
\footnotesize
\begin{center}
\begin{tabular}{|p{0.2\textwidth}|p{0.35\textwidth}|p{0.35\textwidth}|}
\hline \centering{Message} & \centering{Adults} & \centering{Children} \tabularnewline
\hline \raggedright*M1-constraint-requirement-requirement-summary & \raggedrightThe target user is an \textbf{adult}. Constraints - Ergonomics can be found in standard \textbf{XX1}.''& \raggedrightThe target user is a \textbf{child}. Constraints - Ergonomics can be found in standard \textbf{XX2}.'' \tabularnewline
\hline
\end{tabular}
\end{center}
\end{table}
\end{document}
`
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While working on a problem, I derived an interesting result around sums of uniforms random variables. I wanted to record it here so I don't forget it (I haven't solved the more general problem yet!). Here's the summary of the result:
Let $$S_n = \sum_{i=1}^n U_i$$ be the sum of $$n$$ Uniform random variables. Let $$N$$ be the index of the first time the sum exceeds 1 (so $$S_{N-1} < 1$$ and $$S_{N} \ge 1$$). The distribution of $$U_N$$ is $$e - e^{1-u}$$.
This result is related to the really astounding result: $$N$$ (the same $$N$$ defined above) has expected value equal to $$e$$. Yes, that's right, the average number of uniform random variables needed to exceed 1 is exactely $$e$$. I just love this surprising result, but have still not found a good, intuitive reason as to why this is true. Of course it is true mathematically, and the proof is not too hard, but I just can't see why. Anyways, below is the proof of the first result.
### Break the problem into smaller pieces
We'll first use the total law of probability to break the distribution into smaller pieces: $$P(U_N \in du) = \sum_{n=2}^{\infty} P(U_N \in du \;|\; N=n)P(N=n)$$
### Numerical results first
For the below part, we'll be conditioning on a given $$N$$, so let's first look at the distibution of $$U_N \;|\; N=n$$. To do this, we'll just use Monte Carlo to sample from this distribution. Here are the results: So this is our target. If we can model these distributions, we can plug them into the above formula.
### Focusing on a particular $$N$$
Let's put our attention on the case $$N=2$$. We want to solve: \begin{align*} P(U_2 \in du \;|\; N=2) & = P(U_2 \in du \;|\; U_1 < 1, U_1 + U_2 \ge 1) \\ & = \frac{P(U_2 \in du, U_1 < 1, U_1 + U_2 \ge 1)}{P(U_1 < 1, U_1 + U_2 \ge 1)} \\ & = \frac{P(U_1 < 1, U_1 + U_2 \ge 1 \;|\; U_2 \in du)P(U_2 \in du)}{P(U_1 < 1, U_1 + U_2 \ge 1)} \\ \end{align*} Because $$U_2$$ is uniform, $$P(U_2 \in du) = du$$, so $$\frac{P(U_1 < 1, U_1 + U_2 \ge 1 \;|\; U_2 \in du) du}{P(U_1 < 1, U_1 + U_2 \ge 1)}$$ The denominator, while it can be worked out, is equal to the term $$P(N=n)$$ in the first equation, so those will cancel out later. It's equal to $$\frac{1}{n! + (n-1)!}$$, and I'll keep it in to compare our results to the distributions above. So we just have the numerator term to compute. I'll use a slight change in the problem statement here, so as to make the algebra more clear: \begin{align} P(U_1 < 1, U_1 + u \ge 1 \;|\; U_2 = u) & \\ & = P(U_1 < 1, U_1 \ge 1 - u \;|\; U_2 = u)\\ & = u \end{align} So we arrive at $$f_{U_2 \; | \; N=2}(u) = \frac{u}{1/2} = 2u$$, which nice aligns with our empirical distribution above.
### Slightly harder problem
Let's try $$N=3$$, and it gets more interesting here. We'll skip ahead to solving: \begin{align*} P(U_1 + U_2 < 1, U_1 + U_2 + u \ge 1 \;|\; U_3 = u) = \\ P(U_1 + U_2 < 1, U_1 + U_2 \ge 1 - u \;|\; U_3 = u) \end{align*} To solve this, we'll take geometric point of view: the two variables ($$U_1, U_2)$$ can be represented as the unit square in the first quadrant. The inequality $$U_1 + U_2 < 1$$ is an area in this square, bordered by the unit square and the line $$U_1 + U_2 = 1$$. Ditto with $$U_1 + U_2 \ge 1 - u$$. The probability is then just the area of the space between the lines. After computing triangles, the area comes out to $$\frac{1}{2}(1 - (1-u)^2)$$.
### In general
This geometric argument can be extended, with the help of the Irwin–Hall distribution, and in general we get: \begin{align*} f_{U_n \;|\; N = n}(u) &= \frac{1}{(n-1)! + (n-2)!} \int_{1-u}^{1} \frac{1}{(n-2)!}z^{n-2}dz \\ & =\frac{(n-1)! + (n-2)!}{(n-1)!}(1 - (1-u)^{n-1}) \end{align*} And the results are spot on:
### Near the finish line
Going back to our original equation: \begin{align*} f_{U_N}(u) &= \sum_{n=2}^{\infty} f_{U_N | N=n}(u)P(N=n)\\ &= \sum_{n=2}^{\infty} \frac{(n-1)! + (n-2)!}{(n-1)!}(1 - (1-u)^{n-1}) \frac{1}{(n-1)! + (n-2)!} \\ &= \sum_{n=2}^{\infty} \frac{1}{(n-1)!}(1 - (1-u)^{n-1})\\ &= \sum_{n=2}^{\infty} \frac{1}{(n-1)!} - \sum_{n=2}^{\infty} \frac{(1-u)^{n-1}}{(n-1)!}\\ &= \sum_{n=1}^{\infty} \frac{1}{n!} - \sum_{n=1}^{\infty} \frac{(1-u)^{n}}{n!}\\ &= e - e^{1-u}\\ \end{align*} Does this empirically line up? Again, I created samples via MC and plotted our derived curve over it: Terrific! This confirms that no mistake was made in the derivation.
### Conclusion
This worked out so nicely mostly because the probabilities were monomials with simple coefficients - then we could easily work out the solution to the infinite series. However, if I wanted a similar result for exceeding 2 (instead of 1), we get polynomials and much more complex coefficients. I haven't made much progress on this part. I think a solution is to condition the sum on some value, $$N'$$, such that the sum at $$N'$$ exceeds 1, and then I can subtract 1 (+ a smaller term) from 2, and I arrive back at a sum less than 1.
|
# Slope estimator for the regression line through the origin
For a regression line through the origin with the equation:
$$\tilde{y}=\tilde{\beta_1}x$$
How did we use OLS to get the below equation? I know it is by minimising the SSR but I can't seem to work it out by plugging in the values into the formula for SSR.
$$\sum (y_i -\tilde{\beta_1}x_i)^2$$
And furthermore, how do we use calculus to get the first order condition for equation directly above?
First order condition:
$$\sum x_i(y_i-\tilde{\beta_1}x_i) = 0$$
Is it a partial derivative? If so, where did the exponent (2) go?
This is just the derivative. For example, our loss function is the sum of squared residuals, or $$S(\beta) = \sum_{i=1}^n (y_i - \beta x_i)^2.$$ We want to minimize this, so we take the first derivative: $$\frac{dS}{d\beta} = -2 \sum_{i=1}^n (y_i - \beta x_i) x_i.$$
We set it equal to zero to find the minimum, so $$-2 \sum_{i=1}^n (y_i - \hat{\beta} x_i) x_i = 0 \\ \hat{\beta} =\frac{\sum_{i=1}^n y_i x_i}{\sum_{i=1}^n x_i^2}.$$
• Why is our loss function the sum of squared residuals? Is it because $\hat{u}=\tilde{y}-\tilde{beta_1}x$? – Munir Malik Nov 20 '19 at 19:47
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# Counting points in a box modulo a number
Consider the positive integers $\leq x$, then we know that there are $x/p + O(1)$ integers $\leq x$ that are $a \pmod{p}$ ($p$ prime).
Consider a similar problem, except this time, we are counting $(x, y) \in \mathbb{Z} \times \mathbb{Z}$ inside the box $|x| \leq B$ and $|y| \leq B$. I want to count the number of integers in this box with $x \equiv a \pmod{p}$ and $y \equiv b \pmod{p}$. Is the number of such pairs $4B^{2}/p^{2} + \text{error term}$. Is the error term $O(1)$ or $O(B)$? Can we have an error term of $O(1)$?
-
In fact, you don't need $p$ prime for this.
If we think about $p=3, a=1, b=1$ and let $B=3k$, the correct answer is $(2k-1)^2$ while our formula gives $4k^2$, with the error $4k \in O(B)$. Similarly if $B=3k+1$, the right answer is $(2k+1)^2=4k^2+4k+1$ while the formula gives $\frac {4(3k+1)^2}9 =\frac {36k^2+24k+1}9= 4k^2+ \frac {24}9k+\frac 19$ with error $\frac 43k \in O(B)$
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Molecular Disks in Early-type Galaxies
# Structure and Kinematics of Molecular Disks in Fast-rotator Early-type Galaxies
Lisa M. Young New Mexico Tech, 801 Leroy Place, Socorro, NM 87801 Martin Bureau and Michele Cappellari University of Oxford, Sub-department of Astrophysics, Denys Wilkinson Bldg., Keble Road, Oxford, OX1 3RH, UK
###### Abstract
We present interferometric observations resolving the CO emission in the four gas-rich lenticular galaxies NGC 3032, NGC 4150, NGC 4459, and NGC 4526, and we compare the CO distribution and kinematics to those of the stars and ionized gas. Counterrotation documents an external origin for the gas in at least one case (NGC 3032), and the comparisons to stellar and ionized gas substructures in all four galaxies offer insights into their formation histories. The molecular gas is found in kpc-scale disks with mostly regular kinematics and average surface densities of 100 to 200 pc. The disks are well aligned with the stellar photometric and kinematic axes. In the two more luminous Virgo Cluster members NGC 4459 and NGC 4526 the molecular gas shows excellent agreement with circular velocities derived independently from detailed modeling of stellar kinematic data. There are also two puzzling instances of disagreements between stellar kinematics and gas kinematics on sub-kpc scales. In the inner arcseconds of NGC 3032 the CO velocities are significantly lower than the inferred circular velocities, and the reasons may possibly be related to the external origin of the gas but are not well understood. In addition, the very young population of stars in the core of NGC 4150 appears to have the opposite sense of rotation from the molecular gas.
galaxies: elliptical and lenticular, cD — galaxies: ISM — galaxies: kinematics and dynamics — galaxies: structure — galaxies: evolution — galaxies: individual ( (catalog NGC 4459), (catalog NGC 4150), (catalog NGC 3032), (catalog NGC 4526))
slugcomment: ApJ, accepted 23 December 2007
## 1. Introduction
Early-type galaxies, the ellipticals and lenticulars, are generally poorer in cold gas than spirals (e.g. Lees et al., 1991) . This fact is, of course, ultimately responsible for the color difference between early- and late-type galaxies and their locations in the red sequence or the blue cloud (e.g. Baldry et al., 2004). However, cold atomic and molecular gas are not entirely absent from all early-type galaxies, and therein lie some important clues to both the past and the future of early-type galaxies. The origin of the cold gas in early-type galaxies can serve as a tracer of their assembly histories, and the properties of the cold gas (the raw material for star formation) offer insights into possible morphological and dynamical evolution through star formation.
Faber & Gallagher (1976) predicted that mass loss from evolved stars in early-type galaxies should produce detectable amounts of gas over a Hubble time. More recent work (e.g. Ciotti et al., 1991) makes predictions which are different quantitatively but not qualitatively. Of course, it is likely that the gas recycled into the interstellar medium will have a complex thermal history, including heating to X-ray temperatures and possible cooling all the way to the formation of molecules (Brighenti & Mathews, 1996, 1997). But since the stellar mass loss is unavoidable and Temi, Brightenti, & Mathews (2007) have shown that the dusty ejecta can cool to molecular temperatures in galaxy centers, it is very natural to expect that some early-type galaxies could contain cold gas. In fact, as Temi, Brightenti, & Mathews (2007) have alluded, the question of why there are early-type galaxies without cold gas may be just as interesting as why there are early-type galaxies with cold gas.
However, if the cold gas ultimately originated in the stars, one might expect a correlation between the luminosity of an early-type galaxy and its cold gas content, which has not been seen (Wardle & Knapp, 1986; Huchtmeier, Sage, & Henkel, 1995; Lees et al., 1991; Knapp & Rupen, 1996; Combes et al., 2007). The lack of such a correlation is then often used to suggest that whatever molecular gas does exist in early-type galaxies is completely unrelated to the internal stellar mass loss, having been acquired from external sources such as a satellite galaxy or perhaps even the intergalactic medium. In a slightly more sophisticated analysis of both the atomic and molecular gas contents of lenticular galaxies, Sage & Welch (2006) hypothesized a variation on this theme in which the atomic gas could be of external origin whereas the molecular gas could be of internal origin.
Due to the complex nature of the thermal evolution of the stellar mass loss, plus environmentally dependent interaction with an intracluster medium and potential time-dependent effects of AGN feedback on the ISM, we argue that the total gas masses by themselves do not offer compelling answers about the origin of the cold gas in early-type galaxies. However, a comparison of the gaseous and stellar angular momenta should provide much stronger constraints. The distribution and kinematics of the gas can reveal recent gravitational interactions, and gas which is counterrotating with respect to the stars almost certainly did not originate in those stars. Maps which resolve the molecular gas are thus crucial tools which help us to read the assembly histories of early-type galaxies.
Molecular gas in early-type galaxies is also particularly interesting because such gas is the raw material for star formation activity. In recent years there have been suggestions that current-day star formation is taking place in as many as 30% of early-type galaxies at (Yi et al., 2005; Kaviraj et al., 2006). Such inferences are at least superficially consistent with the CO detection rates found by Sage & Welch (2006), Sage, Welch, & Young (2007), and Combes et al. (2007). But the distribution of the molecular gas will determine whether the star formation takes place in a nuclear starburst or in an extended disk and, therefore, it will also determine the efficiency with which AGN feedback could disrupt the star formation. Testing our theoretical understanding of star formation in early-type galaxies also requires estimates of the molecular disk sizes and gas surface densities. Thus, resolved maps of molecular gas hold the keys to understanding the morphological evolution of early-type galaxies through star formation.
We begin to address these issues on the past and future of early-type galaxies through interferometric maps which resolve the CO emission in four nearby lenticular galaxies. Total molecular masses are commonly available for S0s, but resolved CO maps are rare (e.g. Okuda et al., 2005; Das et al., 2005) and the four presented here represent a significant addition to the literature on the subject. In all four cases the molecular gas is found in kpc-scale disks, and in one case the molecular gas counterrotates with respect to the stars. We also present comparisons between the CO kinematics and circular velocity curves derived independently from stellar kinematic data. We discuss the origin of the gas and its implications for mass modeling. A forthcoming paper will also present comparisons to the stellar populations in the galaxies and their ionized gas properties, with special emphasis on the distribution and kinematics of the young stellar population, showing clear evidence for disk growth through star formation.
## 2. Selection and properties of the galaxies
The four galaxies NGC 3032, NGC 4150, NGC 4459, and NGC 4526 were observed because of their membership in the SAURON integral-field survey of early-type galaxies (see de Zeeuw et al., 2002), which means that detailed maps of the stellar kinematics, ionized-gas kinematics, and stellar populations are available for the regions within an effective radius (e.g. Emsellem et al., 2004; Sarzi et al., 2005; Kuntschner et al., 2006). They also have CO detections (Thronson et al., 1989; Sage & Wrobel, 1989; Lees et al., 1991; Combes et al., 2007), making them suitable for high spatial resolution comparisons of the stellar and molecular disks. Basic parameters of the galaxies are summarized in Table 1.
NGC 3032 is a relatively faint () field lenticular with complex stellar kinematics. Having high angular momentum per unit mass, it belongs to the so-called fast rotator class (Emsellem et al., 2007), and has a small counter-rotating core (; McDermid et al. 2006a). The ionised gas is co-spatial with dusty spiral arms extending to about an effective radius and is also counter-rotating (Sarzi et al., 2005). H emission is the strongest in the SAURON sample while the ratio [O iii]/H is weakest (except in the very center). The absorption linestrength distributions are unusual, suggesting young stars everywhere with a lower metallicity and -enhancement in the center (Kuntschner et al. 2006; Kuntschner et al. 08, in prep.), probably indicating a recent starburst.
NGC 4150 belongs to the Coma I Cloud and is low-luminosity () lenticular in many respects similar to NGC 3032. It is a fast rotator (Emsellem et al., 2007) with moderate rotation and a dip in the central velocity dispersion (Emsellem et al., 2004), and also harbours a central counter-rotating core (McDermid et al., 2006b) where messy dust is found. Contrary to NGC 3032, however, the extended ionised gas is co-rotating with the bulk of the stars, although slightly misaligned (Sarzi et al., 2005). The linestrengths again indicate young stars everywhere, particularly in the center, but the metallicity and elements are rather flat (Kuntschner et al. 2006; Kuntschner et al. 08, in prep.).
NGC 4459 is an average () Virgo lenticular with rapid rotation and the pinch of the stellar isovelocity contours characteristic of decoupled central disks (Emsellem et al., 2004). This disk is co-spatial with a regular but flocculent dust disk and rapidly rotating ionised-gas with low [O iii]/H ratio (except in the very center; Sarzi et al. 2005). The linestrengths indicate much younger stars and higher metallicity in the central disk (Kuntschner et al. 2006; Kuntschner et al. 08, in prep.).
NGC 4526 is a Virgo lenticular similar to NGC 4459 but slightly brighter (). It harbours what is probably the strongest decoupled central stellar disk in the SAURON sample (observed nearly edge-on), embedded in a more slowly rotating bulge (Emsellem et al., 2004). The bulge has a slight triaxiality which suggests the presence of a bar. A rather large but regular dust and ionised-gas disk is co-spatial with the stellar disk, and has uniformly low [O iii]/H ratio (Sarzi et al., 2005). The linestrengths clearly show that the central disk is much younger and metal rich than the bulk of the galaxy (Kuntschner et al. 2006; Kuntschner et al. 08, in prep.).
The four galaxies studied here thus fall naturally into two classes. NGC 3032 and NGC 4150 are relatively faint and have low velocity dispersion, but harbour small counter-rotating cores and have extended marginally ordered gaseous disks with pervasive young stars. NGC 4459 and NGC 4526 have rapid rotation but even more rapidly co-rotating decoupled (and spatially well-constrained) central stellar disks, co-spatial with regular dust disk populated with young stars. The former thus point to a more troubled recent past, perhaps involving accretion or minor mergers, while the latter suggest a more peaceful immediate past dominated by secular evolution. The current molecular gas observations should thus allow to build on these data and test those ideas further.
## 3. CO data
NGC 3032, NGC 4150, NGC 4459, and NGC 4526 were observed in the CO line with the 10-element Berkeley-Illinois-Maryland Association (BIMA) millimeter interferometer at Hat Creek, CA (Welch et al., 1996). These observations were carried out in the C configuration (projected baselines 3 to 34 k) in the spring and fall of 2003. Additional data for NGC 3032 and NGC 4526 were obtained in the B configuration (projected baselines 4 to 83 k) in March 2003. Total observing times were 8 to 17 hours per galaxy in the C configuration and 10 to 16 hours in the B configuration. A single pointing centered on the optical position was used for all galaxies. Each observation covered a velocity range of 1300 km s, and these data have sensitivity to structures from point sources up to objects 60–80″ in diameter. System temperatures were mostly in the 300–500 K range. Table 2 summarizes important parameters of these observations.
Reduction of the BIMA data was carried out using standard tasks in the MIRIAD package (Sault, Teuben, & Wright, 1995). Electrical line length calibration was applied to all tracks. The C configuration data for NGC 3032 and 60% of the C configuration data for NGC 4526 were also explicitly corrected for amplitude decorrelation on longer baselines using data from an atmospheric phase monitor and the MIRIAD task uvdecor (Lay, 1999; Akeson, 1998; Regan et al., 2001; Wong, 2001). The atmospheric decorrelation is estimated using a small interferometer with a fixed 100 meter baseline which measures the rms path length difference in the signal from a commercial broadcast satellite. Data requiring the decorrelation correction had rms path length differences in the range of 300 to 700 microns, and the median amplitude correction factor was about 13%. However, the longest baselines in these datasets were later flagged because of their very poor phase stability. The remainder of the tracks were taken in stabler weather and were not explicitly corrected for decorrelation, because normal amplitude calibration can take out most of the effect (Wong, 2001).
Absolute flux calibration was based on observations of Mars and 3C273, and comparisons of flux measurements on all the observed calibrators suggest that the absolute flux uncertainties are in the range of 15%–20%. Phase drifts as a function of time were corrected by means of a nearby calibrator observed every 30 to 40 minutes. Gain variations as a function of frequency were corrected by the online passband calibration system. Inspection of data for 3C273 indicate that residual passband variations are on the order of 10% or less in amplitude and 2° in phase across the entire band.
The calibrated visibility data were weighted by the inverse square of the system temperature and the inverse square of the amplitude decorrelation correction factor (if used), then Fourier transformed. Dirty images were lightly deconvolved with the Clark clean algorithm (Clark, 1980), as appropriate for these compact, rather low signal-to-noise detections. No continuum subtraction was needed. Integrated intensity and mean velocity maps were produced by the masking method: the deconvolved image cube was smoothed along both spatial and velocity axes, and the smoothed cube was clipped at times the rms noise in a channel. The clipped version of the smoothed cube was then used as a mask to define a three-dimensional volume in the original, unsmoothed cube in which the emission was integrated over velocity (Wong, 2001; Regan et al., 2001). Velocity maps were also constructed from Gaussian fits to the line profile at each position.
Integrated spectra were constructed by first using the integrated intensity (moment 0) maps to define the spatial extent of the emission, then integrating over that fixed extent for every channel. Continuum images were also made by averaging all of the line-free channels in the final spectral line cubes, but no continuum emission was detected. Table 2 gives 3 limits for point source continuum emission at the centers of the galaxies, and Table 3 gives beam size and sensitivity information for the final spectral line cubes.
## 4. Results
### 4.1. Total CO fluxes
Comparisons of the total CO fluxes in the BIMA images and in previous single dish data show that the interferometer usually recovers all of the CO emission. In the case of NGC 4459, the best single dish flux is from a survey of the SAURON early-type galaxies made with the IRAM 30m telescope by Combes et al. (2007). That work finds 54.0 , with a statistical uncertainty of 2.4 and an absolute calibration uncertainty of 15% to 20%. The present BIMA data yield 56 11 . An earlier, much noisier detection by Thronson et al. (1989) found 120 55 . Thus, the single dish CO fluxes are entirely consistent with the flux measured from the interferometric map. We adopt here a CO-to-H conversion factor of 3.0 and distances from Tonry et al. (2001), scaled down by 3% following Mei et al. (2005). At 15.7 Mpc, the mass of NGC 4459 is (1.6 0.3) .
Our BIMA-derived CO flux of NGC 3032 (93 18 ) is also consistent with most of the previous single dish measurements. Thronson et al. (1989) measured 85 25 in the 45″ beam of the FCRAO 14m telescope, and Sage & Wrobel (1989) measured 92 10 in the 55″ beam of the NRAO 12m. However, the more recent measurements of Combes et al. (2007) give only 46.0 1.5 (again, with typically 15% to 20% absolute calibration uncertainty) in the 22″ beam of the IRAM 30m telescope. Since the BIMA data show emission over a region at least 25″ in diameter, it is plausible that some emission was missed by the 30m beam. At a distance of 21.4 Mpc, our flux corresponds to (5.0 1.0) of .
In contrast, the CO flux we find for NGC 4526 (180 36 ) is significantly larger than previous single dish measurements. Lees et al. (1991) quote a value of 11825 measured by Sage & Wrobel (1989) with the NRAO 12m telescope, and Combes et al. (2007) measured 118 4 (statistical) at the IRAM 30m telescope. Since the CO images show emission over a region nearly 30″ in diameter, the 30m telescope may have missed significant flux. Furthermore, the spectrum of Sage & Wrobel (1989) is quite asymmetric, which may indicate mispointing or an inaccurate spectral baseline (neither of which plague the interferometric fluxes). For a distance of 16.4 Mpc, our flux corresponds to a mass of (5.7 1.1) of . Thus, in NGC 3032, NGC 4459, and NGC 4526 the evidence suggests that the interferometric maps have recovered all of the CO emission.
There is some inconsistency in the single dish CO fluxes of NGC 4150, which makes it more difficult to judge whether the interferometric data are missing flux. The most recent 30m CO flux of NGC 4150 is 30.2 2.4 (statistical) 5 (calibration) from Combes et al. (2007), whereas the BIMA data yield 26 5 . These two measurements are in good agreement. Welch & Sage (2003) measured 45 2 7 , also with the IRAM 30m telescope, and this value may not be inconsistent with the previous two. However, a flux of 77 14 reported by Leroy et al. (2005) from the ARO 12m telescope (the old NRAO 12m telescope) is significantly higher, which might be due to an absolute calibration uncertainty, an underestimated baseline level, or even some extended molecular gas which is sampled by the 55″ beam of the 12m but not by the 22″ beam of the 30m. At a distance of 13.4 Mpc, our BIMA flux corresponds to an mass of (5.5 1.1) .
### 4.2. Ngc 3032
The CO in NGC 3032 is found in a centrally concentrated structure of rather narrow linewidth. Emission is detected over 145 km s centered on a systemic velocity of 1555 km s (Figure 1), so the CO systemic velocity is in good agreement with the stellar absorption line velocity measurements of 1555 41 km s from Falco et al. (1999) and 1559 10 km s from Emsellem et al. (2004). The diameter of the CO emission is approximately 30″, with a peak on the nucleus of the galaxy and a tail extending 15″ to the southeast (Figure 2). Based on a comparison with dust maps (see below), we infer that the molecular gas lies in a circular disk of radius 14″ (1.5 kpc), yielding an average surface density of pc including helium. NGC 3032 is also known to contain 0.9 (1.0 ) of HI emission (Duprie & Schneider, 1996), giving M()/M(HI) .
Individual CO channel maps (Figure 3) show good agreement between the distribution of molecular gas and dust in NGC 3032; dust and gas are both found in an inclined, rotating disk. An unsharp-masked WFPC2 image (Figure 3) shows a bright nucleus surrounded by a dark dusty ring (or two tightly wrapped arms) roughly 6″ in major axis diameter, and beyond that ring is a disk of flocculent spiral dust features interspersed with bright point sources. The dust features cover a region 28″ in diameter. CO channels have a butterfly-wing structure with compact emission in the end channels and central channels elongated in the direction of the kinematic minor axis. The extent of the CO emission matches that of the dust disk and the gas kinematic major axis matches the dust morphological major axis. The southeast “tail” of emission in Figure 2 is also visible in the channel maps at 1580 to 1616 km s as emission at about the 75 level, smoothly tracking to smaller radii with increasing velocity; it does not appear to be associated with a dust feature.
Kinematic analysis of the CO in NGC 3032 was made with several different techniques whose results are all in good agreement. To the velocity field shown in Figure 4 we fitted a model described by an inclined disk with the rotation curve , as encoded in the National Radio Astronomy Observatory’s AIPS task GAL. The details of the shape of this rotation curve are not critical, but it has the desired behavior of rising quickly at small radii and asymptotically flattening. We also employed the kinemetry analysis of Krajnović et al. (2006), and we made a tilted ring analysis with the rotcur task of the GIPSY package from the Kapteyn Institute, Rijksuniversiteit Groningen. In using the rotcur task we followed the procedure outlined by Swaters et al. (1999).
The fitted kinematic center positions are within an arcsecond of the optical nucleus measured from the SDSS image. The systemic velocity of the galaxy is also robustly fitted to be 1555 1 km s. The inclination of the disk is poorly constrained by the CO data, however, and is more accurately taken from the axis ratio of the dust disk which gives 44° 4°. The Jeans dynamical modeling we did in this work is consistent and suggests .
Measurements of the globally averaged CO kinematic position angle fall in the range 90° to 97°, measured to the receding major axis, while the tilted ring model and the kinemetry both suggest a gentle twist from 81° in the central resolution element to 97° at 8″ 14″. This twist is also suggested by the hint of an integral-sign shape in the zero velocity curve of the velocity field, although warping is not obvious in the images of the dust disk. A major axis position-velocity diagram sliced through the data cube at 95° is shown in Figure 5. This position-velocity diagram shows the typical pattern with a steeply rising inner portion followed by a flattening (especially noticeable on the eastern side of the galaxy) beyond radii 4″.
The kinematic position angle for the molecular disk is consistent with the photometric position angle, defined as that of the ellipse of inertia of the surface brightness (99.6°, Cappellari et al., 2007), but is 180° offset from the stellar kinematic angle of (Emsellem et al., 2004; Cappellari et al., 2007).111Cappellari et al. (2007) quote a stellar kinematic angle of , as they define it as it as the direction along which is maximum. The velocity fields shown in Emsellem et al. (2004) make it clear that this is to the stellar major axis on the approaching side rather than the receding side. Thus, the molecular gas in NGC 3032 is counterrotating with respect to the bulk of the stars in the galaxy. However, McDermid et al. (2006a, b) found a large radial age gradient in the galaxy and a counterrotating stellar core, which suggests recent star formation in the molecular disk. The sense of rotation of the molecular gas is consistent with that of the young counterrotating stars but inconsistent with that of the older, more extended population.
### 4.3. Ngc 4150
NGC 4150 shows a relatively small amount of molecular gas in a very compact structure at the center of the galaxy. We fit the integrated spectrum in Figure 6 with a Gaussian whose central velocity is 239 20 km s, which we take as the systemic velocity of the CO emission. This CO velocity is in good agreement with the stellar velocity measurements of 226 22 km s (Fisher et al., 1995), 208 30 km s (Falco et al., 1999), and 219 10 km s (Emsellem et al., 2004). CO emission is detected over a velocity range of 180 km s, and the bulk of this gas is within a few arcseconds of the nucleus of the galaxy (Figure 7).
Several dust structures are apparent in the center of NGC 4150 (Figure 8). A dark dust lane bisects the nucleus from northwest to southeast, and the galaxy has more dust clouds a few arcseconds northwest and east of the nucleus. There is an irregular dust ring, stronger in the north, of semimajor and semiminor axes 7.5″ 4″, with a fainter dust arm curving around the southeast side of the nucleus to radii 12″. The peaks in the channel maps suggest that most of the molecular gas is located in the bisecting dust lane, and the CO kinematic major axis roughly matches the major axis of the dust ring. Figure 7 shows a faint tail of emission stretching to 30″ south of the nucleus, also visible in the channel maps from 265 km s to 325 km s. Morganti et al. (2006) show that the 2.5 of HI in the galaxy (M()/M(HI) = 23) is elongated roughly along the optical major axis in a structure of 1′ diameter, with additional HI to the south and southwest, so the similarities between the CO and HI lend credence to the reality of the CO tail.
Due to the small angular size of the CO emission the velocity field (Figure 9) does not offer good constraints on the kinematic parameters of the galaxy. Cappellari et al. (2007) quote both the photometric and stellar kinematic position angles as , with uncertainties of a few degrees. The orientation of the dust ring (Figure 8) is , and although the CO velocity field is complex, it is not inconsistent with this value. A slice at position angle gives the major axis position-velocity diagram of Figure 10. It appears that very little of the molecular gas is located beyond the turnover point in the rotation curve. The inclination of the galaxy is measured from the axis ratio of the dust ring to be 54°5°, consistent with the value of 52° from the dynamical modelling (Cappellari et al., 2006).
There is a counterrotating stellar core in the central arcseconds of NGC 4150 (McDermid et al., 2006a, b, and it is also visible in the position-velocity diagram of Figure 10). This counterrotating core is believed to have formed recently as the galaxy shows a very strong radial age gradient. Curiously, though, the core does not appear to have a counterpart in the molecular gas at the current resolution and sensitivity. The counterrotating core is also not obvious in the [O iii] velocities of NGC 4150 (Figure 10), and indeed the CO velocities of this galaxy are a better match to the [O iii] velocities than they are to the stellar velocities. A good match between CO velocities and ionized gas velocities suggests the possibility that the ionized gas could trace star formation activity, but the relationship to the counterrotating stellar core is not yet clear.
### 4.4. Ngc 4459
The CO emission from NGC 4459 is also in a central disk, with hints of a double-horned structure in the integrated spectrum (Figure 11). A double-horned spectrum is typically produced by gas in the flat part of the rotation curve. Emission is detected over a velocity range of 400 km s centered near a systemic velocity of 1210 20 km s (but see below). The CO velocity is thus in good agreement with the stellar velocity of 1232 40 km s from Falco et al. (1999) and 1200 10 km s from Emsellem et al. (2004).
The integrated intensity map (Figure 12) shows a compact structure centered on the galaxy nucleus, no more than 20″ in diameter. Little structure is evident, and there is no strong evidence for arms or tails of molecular gas. The individual channel maps in Figure 13 again show a typical disk pattern with higher intensities at the extreme velocities than near the systemic velocity. The centroids of the emission in each channel show excellent agreement with the extent of the prominent, well-developed dust disk. The dust fills a flocculent disk with a rather sharp outer edge at a semimajor axis of 8.5″ and only a few faint spiral dust features beyond. From a radius of about 3.75″, two prominent dusty spiral arms can be traced inward to an inner ring of semimajor axis 2.0″. The larger disk has a major axis position angle of 102° but the inner ring appears to have a slightly different orientation, with position angle 90°. The bulk of the CO emission appears associated with the larger-scale, 8.5″ disk. For a circular disk of radius 8.5″ (670 pc), the average gas surface density is 170 pc, including helium. The north side of the disk must be the near side, suggesting that both the flocculent and inner spiral arms are trailing.
The velocity field (Figure 14) shows predominantly straight, parallel isovelocity contours, so again the inclination of the gas disk is better obtained from the dust disk than from the gas itself. The axial ratio of the dust disk gives an estimated inclination of , consistent with 47° from dynamical modelling (Cappellari et al., 2006). For the kinematic analysis we also constrain the kinematic center of the galaxy to be the position of the optical nucleus. Stellar isophotes have a position angle of 280° outside the dust disk, while a kinemetric analysis of the stellar velocities gives position angles of 280° outside the dust disk, decreasing to 274° inside it (Krajnović et al., 2006). A global kinematic position angle measurement for the molecular gas, measured both as in Krajnović et al. (2006) and by fitting a model velocity field, yields values in the range 269° to 273° to the receding major axis. The molecular gas kinematics are thus very well aligned with the stellar kinematics and the stellar isophotes. A tilted ring analysis suggests the systemic velocity to be 11962 km s. The major axis position-velocity diagram (Figure 15) is very strongly peaked at its maximum velocities, suggesting that the bulk of the molecular gas is located in the flat part of the galaxy’s rotation curve.
### 4.5. Ngc 4526
In NGC 4526 the CO lies in a well-developed, nearly edge-on disk. The integrated CO profile shows the familiar double-horned shape (Figure 16), with both horns approximately equally bright. The systemic velocity of the galaxy is 613 10 km s and emission is detected over a total width of 683 km s. A line width of this magnitude, while not unknown, is unusually large for lenticulars. In fact, we argue in section 5.3 that the galaxy’s maximum circular velocity is about 355 km s and in the compilation of 243 galaxies (54 lenticulars) made by Courteau et al. (2007), NGC 4526’s circular velocity is exceeded by only two other lenticulars, one Sa galaxy, and 10 ellipticals.
The observed CO systemic velocity is consistent with the stellar velocity of 592 48 km s given by Falco et al. (1999) and 626 10 km s given by Emsellem et al. (2004), but is inconsistent with the HI velocity of 448 8 km s (de Vaucouleurs et al., 1991; Davies & Lewis, 1973). In fact, a comparison of Figure 16 with the HI spectrum in Davies & Lewis (1973) shows that the latter authors only detected one “horn”, so their HI velocity is not a good estimate of the systemic velocity. Interferometric HI observations would undoubtedly be useful in revealing the relationships between atomic and molecular gas.
The CO distribution (Figure 17) is elongated in the direction of the optical major axis. Fitting a two-dimensional Gaussian to the integrated intensity image reveals that the disk is poorly resolved in the short dimension; the deconvolved minor axis FWHM is 4.3″, comparable to the beam size. The position angle of this gas disk is estimated as 1.8°.
Individual channel maps (Figure 18) show a close correspondence between the CO and dust disks. Unsharp-masked HST images show that the dust is confined to radii less than 14″ – 15″. Although the inclination is quite high, there appear to be at least two dominant dust rings of radii 14″ and 10″. (These radii are uncertain because the features are most prominent along the minor axis.) The centroids of CO emission in the extreme channels suggest that the bulk of the CO may be associated with the inner of these two rings. Assuming the gas to be in a circular disk of radius 14″ (1.1 kpc), the average surface density is 200 pc including helium.
As in the case of NGC 4459, the velocity field (Figure 19) shows straight, parallel isovelocity contours throughout; it constrains the kinematic position angle but not the inclination of the gas disk. However, the dynamical modelling of Cappellari et al. (2006) gives an inclination of 79° which is consistent with values of 75° 2° inferred from the axis ratio of the dust disk. Fits of a model exponential velocity field show that the kinematic center of the molecular gas is consistent with the position of the optical nucleus, within 0.5″. The global CO kinematic position angle, calculated from those fits and also via the method of Krajnović et al. (2006), is 3° to the receding major axis. More detailed kinemetric analysis suggests a gradual trend from PA in the inner resolution element to at radii 6″. In comparison, the stellar photometric major axis is and the stellar kinematic major axis is (Cappellari et al., 2007), so the molecular gas (especially at its outer radii) is aligned with the stellar body of the galaxy to within a few degrees. Figure 20, the major axis position-velocity diagram, clearly shows that the gas traces a turnover in the rotation curve with concentrations of gas near the turnover radius. There is a modest degree of asymmetry, with somewhat stronger CO emission on the receding (west) side of the major axis.
## 5. Discussion
### 5.1. Origins of the Molecular Gas
In NGC 4459 and NGC 4526 the molecular gas kinematics are consistent with both the stellar photometric axes and the stellar kinematic axes. The differences are on the order of a few degrees, which is the level of precision available from existing data. In NGC 4150 the molecular gas kinematics are more difficult to describe but the orientation of the dust ring is also consistent with the stellar photometric and kinematic axes to a degree or better. In NGC 3032 the molecular gas kinematic position angle is consistent with the photometric axis to within a few degrees, but it is nearly 180° away from the stellar kinematic axis. Despite the dramatic counterrotation of the gas in NGC 3032, these alignments imply that all four galaxies have nearly axisymmetric potentials and that the gas is well settled into the equatorial plane.
In NGC 4150, NGC 4459, and NGC 4526 the agreements between the sense of rotation of the molecular gas and stars also imply that the molecular gas could have originated in internal stellar mass loss. Such an internal origin is not required by the data, of course, as the gas could have been captured into prograde rotation from outside sources, and internal secular evolution over several dynamical timescales would gradually bring it to the galaxy’s equatorial plane. (Orbital periods are a few yr at the edges of the disks.)
In contrast to the other three galaxies, the conspicuous counterrotation of the molecular gas and stars in NGC 3032 provides clear evidence that this molecular gas could not have originated in internal stellar mass loss. The only scenario that could explain the counterrotation of internally produced gas is to invoke some gravitational interaction which strongly torques the gas (possibly at large radii) while leaving the remainder of the galaxy seemingly untouched. This scenario seems unnecessarily complex, however, so we suggest instead that the molecular gas was captured through “cold” accretion from the intergalactic medium or in an interaction or a minor merger with a gas-rich neighbor. Perhaps it is even a remnant of a major merger which formed the present galaxy. The high degree of regularity in the gas kinematics and stellar morphology suggests that this event did not occur recently, as the orbital period at the outer edge of the CO disk is on the order of yr. Furthermore, modeling the optical image (Section 5.2) shows that the isophotes are very regular to at least 70″ (7.3 kpc, or five times the radius of the molecular gas).
NGC 3032 is significant, then, because it is one of the few early-type galaxies in which the possibility of an internal origin for the molecular gas can be firmly excluded. It contradicts the picture outlined by Sage & Welch (2006), who suggested that most of the molecular gas in lenticulars should have an internal origin like that described by Temi, Brightenti, & Mathews (2007). The evidence for the suggestion of Sage & Welch (2006) came from contrasting the atomic and molecular properties of lenticulars: the atomic gas, which is usually more extended and shows different kinematics than the molecular gas, tends to dominate the gas content in gas-rich galaxies, whereas the molecular phase dominates in gas-poor galaxies. Thus Sage & Welch (2006) hypothesized that the atomic gas could be primarily attributed to external sources but the molecular gas to internal sources. While that picture may still be roughly correct, it cannot account for the molecular gas in NGC 3032.
On the other hand, it must still be true that the evolved stars in NGC 3032 expelled mass into the ISM. According to the models of Faber & Gallagher (1976) and Ciotti et al. (1991), given the galaxy’s present luminosity the stellar mass loss would amount to 4 to 6 over a Hubble time. We have not detected that gas yet, as it would show prograde rotation. The stellar mass loss must now be in the form of hot gas, or perhaps it has been removed from the galaxy entirely (e.g. Temi, Brightenti, & Mathews, 2007).
It is not unusual to find counterrotating ionized gas in lenticular galaxies. From their own data and a compilation of the literature, Bureau & Chung (2006) find that 15% 4% of all S0s contain counterrotating ionized gas while 23% 5% of the S0s with ionized gas have that gas in retrograde motion. Earlier work by Bertola et al. (1992) and Kuijken, Fisher, & Merrifield (1996) is in agreement with this result. A more detailed study of stellar - ionized gas misalignments was carried out by Sarzi et al. (2005) using integral-field observations. They find that among the lenticulars of the SAURON early-type sample, 9 of 20 galaxies have a star-gas kinematic misalignment greater than 30°. NGC 3032 is the only one of the 9 that has a misalignment 150°, though, so it is somewhat unusual in having ionized gas so nearly retrograde. We should be careful not to make assumptions about the behavior of the molecular gas based on the behavior of the ionized gas, however, since in general it is not obvious that these two phases of the interstellar medium are closely linked. For example, in NGC 4150 the ionized gas emission (Sarzi et al., 2005) is distributed over a much larger scale than the molecular gas; ionized gas fills the 30″ 40″ SAURON field of view whereas molecular gas is concentrated in the central 6″. Clearly, a larger sample of resolved molecular maps of lenticular galaxies will be required before we can make better judgments about the origin of their molecular gas.
### 5.2. Deriving circular velocities
Molecular gas is naturally cold and dissipational and therefore, in equilibrium in an axisymmetric potential, it should settle in circular orbits in the equatorial plane. To the extent that the velocity dispersion of the molecular gas is low (typically on the order of 10 to 20 km s) it should then provide an excellent tracer of the circular velocity of the galaxy. And the circular velocity is, ultimately, the dynamical indicator of the total matter content of a galaxy.
It is also possible to estimate the circular velocity from stellar kinematic data, with additional complications. For a given edge-on axisymmetric potential, dimensional arguments suggest that it may be possible to uniquely recover the three-dimensional (3D) orbital distribution of the stars in a galaxy with another 3D quantity, namely the knowledge of the stellar line-of-sight velocity-distribution at every position on the galaxy, as can be obtained with integral-field spectroscopy. It seems however unlikely that the axisymmetric potential itself, which is another two-dimensional function can also be uniquely recovered from the same 3D observations (see Section 3 of Valluri et al., 2004, for a discussion). In the stellar dynamical models either a constant mass-to-light ratio () or a parametric form for the potential are assumed. In this context it is worthwhile to compare the two different estimates of circular velocity in early-type galaxies coming from different assumptions.
Cappellari et al. (2006) have used the stellar kinematics presented in Emsellem et al. (2004) to derive high quality dynamical for a subset of the SAURON early-type galaxies, under the assumption of a constant . High resolution and wide-field images of the galaxies are modeled with a Multi-Gaussian Expansion (MGE; Emsellem et al., 1994; Cappellari, 2002) so that, if the inclination of the galaxy (assumed to be axisymmetric) is known, the fitted MGE model of the surface density can be deprojected and a three-dimensional stellar distribution can be estimated. Full two-dimensional maps of the stellar mean velocities, dispersions, and Gauss-Hermite parameters within about an effective radius are then used in combination with the stellar distribution to construct self-consistent two-integral Jeans and three-integral Schwarzschild models. The data are consistent with the assumption of a constant dynamical within an effective radius, and in galaxies with regular dust disks the inclination inferred from the Jeans models is consistent with the axial ratio of the dust disk.
Here we compute the circular velocities:
v2c(R)=R∂Φ(R,z)∂R,
where the gravitational potential is computed from the MGE models tabulated in Cappellari et al. (2006), deprojected at their best fitting inclination, and scaled by the constant best-fitting Schwarzschild’s . For NGC 3032 no previous dynamical model exist so the MGE model, inclination and were determined as described below. The circular velocity curves of the best-fit gravitational potentials for NGC 3032, NGC 4150, NGC 4459 and NGC 4526 are shown in Figures 5, 10, 15, and 20.
For NGC 3032 we constructed an MGE model for the photometry using the software of Cappellari (2002). To reduce the effect of dust extinction in the derivation of the stellar density distribution, in the MGE fit we combined an archival near-infrared HST/NICMOS/F160W (-band) image of the dusty central regions with a larger-field image taken with HST/WFPC2/F606W (-band). The -band image was adopted as photometric reference and used to converted the MGE model to the Johnson band using the calibration of Dolphin (2000). The distance-independent parameters of the PSF-deconvolved MGE models, corrected for galactic extinction following Schlegel et al. (1998), and adopting an absolute magnitude of the Sun mag (Table 2.1 of Binney & Merrifield, 1998), are given in Table 4.
We constructed a self-consistent two-integral axisymmetric MGE Jeans model for the second velocity moments of NGC 3032 and we determined the best-fitting and inclination , as in Cappellari et al. (2006). Using the color of Tonry et al. (2001) this translates into , which is among the lowest measured values for any SAURON galaxy (Cappellari et al., 2006). This is still consistent within the scatter with their relationship, considering the low value of km s of NGC 3032 from Emsellem et al. (2007). The low dynamical is also expected from their relation between and the fact that NGC 3032 has the lowest average H line-strength among all the galaxies in the SAURON sample of E/S0 galaxies (Kuntschner et al., 2006). This confirms that the variations in the stellar populations are the main driver for the observed differences in the dynamical . However the self-consistent model does not provide a good fit to the data, so the fitted represents an average value for the central regions of the galaxy which are sampled by the SAURON kinematics. In particular the self-consistent two-integral model shows a much steeper radial decrease of along the major axis than observed. This is the opposite of what is generally measured in flattened galaxies and disks (e.g. Cappellari et al., 2007) and suggests that anisotropy cannot explain the discrepancy. An improved non-self-consistent model will be discussed in Section 5.3.
### 5.3. Comparing circular velocities with the observed CO velocities
We make a comparison between the stellar and the molecular indicators of circular velocity with tilted ring models, projected and convolved to the same resolution and sampling as the CO data. The GALMOD routine in the GIPSY software package is used to generate the models. We specify the rotation velocities of the rings as computed in Section 5.2 and listed in Tables 5, 6, and 7. The inclinations of the gas disks are also known, as described above, and the velocity dispersion in the molecular gas is assumed to be 10 km s. The exact value of the velocity dispersion used is not critical, as long as it is smaller than the channel width and consistent with the sharp cutoffs in emission at the outlying velocities. Nominally the rotation velocities should also be corrected for this finite velocity dispersion (the asymmetric drift effect), but as the rotation velocities are 100 to 350 km s the correction is negligible. The optical dust images also suggest a very thin molecular disk, here assigned a scale height of 1″ (again the exact value is not critical as it is smaller than the spatial resolution). A gradual linear decrease in the gas surface density is assumed, with a sharp cutoff at the edge of the dust disk. The rotating disk model is then projected onto the plane of the sky at the specified inclination and the model emission is sampled at a pixel size and channel width matching the CO observations. The sampled cube is also spatially smoothed to have the same resolution as the CO beam. Finally the smoothed model cube is sliced along the galaxy major axis just as for the real data, and the slice is displayed with analogous contour levels (as percentages of the peak intensity, which is arbitrary).
Figure 21 compares the data and the model position-velocity diagrams for NGC 4526. The assumed gas surface densities drop linearly by a factor of 5 between 1″ and 14″ and are zero beyond that. Placing the gas in rings rotating at the circular velocity produced a model with a bit too large velocity amplitude; the peaks in the model position-velocity diagram, tracing gas along the line of nodes, were separated by 703 km s whereas in the data they are separated by 663 km s (the difference is two channels). Therefore, Figure 21 actually shows a model in which the rotation velocity of each ring is 5% smaller than the circular velocities derived with the of Cappellari et al. (2006), and it shows that this model produces a very close match to both the ridgeline and the envelope of the observed emission. Even the rounded shoulder of the circular velocity curve (specifically, the relatively shallow but still rising slope in ) is confirmed by the CO data. We have not made an exhaustive search in the rotation velocity or gas distribution parameter space, so we certainly cannot claim to have made the unique best match to the observed position-velocity diagram. However, the good agreement between the model and the data should be regarded as independent confirmation that the circular velocities derived with the of Cappellari et al. (2006) are accurate to about 5%. The is thus accurate to about 10%, roughly consistent with the quoted uncertainty.
Figure 22 presents a similar comparison for NGC 4459. In this case the circular velocities obtained with the of Cappellari et al. (2006) are used without correction and an excellent match is achieved, indicating again the high degree of accuracy of the dynamical . Figures 15 and 20 also show that in NGC 4459 and NGC 4526 the molecular gas rotation speeds are 1.6 to 2.0 times larger than the mean stellar velocities at the same radii. In these galaxies the stars have significant pressure support, but the molecular kinematics and the stellar kinematics are both consistent with the same gravitational potential.
A broader implication of this agreement is that the molecular and stellar kinematics for galaxies like NGC 4459 and NGC 4526 can be used with confidence in studies of the circular velocity and the bulge velocity dispersion (e.g. Courteau et al., 2007), where it is necessary to probe the relationships between the two dynamical indicators. For example, several authors (e.g. Ho, 2007; Shields et al., 2006) have studied the development of the relationship between a galaxy’s black hole mass and its bulge velocity dispersion. They have advocated use of the CO line width to trace the galaxy’s maximum circular velocity, hence its velocity dispersion (especially at high redshift where it is currently easier to measure the CO line width than other dynamical indicators of the galaxy mass). These studies use a CO Tully-Fisher relation to demonstrate that the CO line widths do indeed trace their host galaxies’ maximum circular velocity. Our detailed comparison of molecular kinematics and circular velocities supports this claim when the CO velocities and circular velocities at large radii are considered, even for early-type galaxies where the Tully-Fisher relation is not traditionally applied.
Figure 10 compares the CO position-velocity diagram in NGC 4150 to the inferred circular velocity. In this case the total velocity range covered by the molecular gas is nearly as large as what would be expected from the circular velocity curve. The CO velocities are also nearly equal to the mean stellar velocities at radii 10″, which is qualitatively consistent with the fact that the observed stellar velocity dispersions are much smaller than in NGC 4459 and NGC 4526 (Emsellem et al., 2004). It is worth noting that the adopted inclination for the galaxy, while consistent with the axis ratio of the dust ring, may not be appropriate for the molecular gas which is concentrated in the nuclear dust lane. Detailed models are probably not yet useful for these data, and the most that can be said is that the inferred circular velocity curve is not inconsistent with the observed CO and stellar velocities.
In contrast to the cases of NGC 4459 and NGC 4526, there are significant disagreements between the inferred circular velocity and the observed molecular gas rotation speeds of NGC 3032. Assuming a constant dynamical within an effective radius, the circular velocity rises quickly to a peak of 330 km s at 0.08″ and makes a sharp, nearly Keplerian decline through the inner arcseconds. It drops by a factor of two to 157 km s at 5″ and thereafter drops more slowly, passing through 132 km s at 10″ which is nearly the edge of the molecular disk. But as Figure 5 shows, these velocities are much larger than the observed CO velocities especially interior to 5″.
The strong radial gradient in the H absorption line strength index of NGC 3032 (Kuntschner et al., 2006), and the young inferred central age (McDermid et al., 2006a) suggest that the stellar population might be better described with a variable . Indeed, if the line strength indices are interpreted with the models of Thomas et al. (2003) and Maraston (2005), the SAURON data are consistent with a decline of a factor of two to three in the local stellar from 10″ to 1″, even in the band (from which the MGE models are derived). Moreover the inability of the self-consistent Jeans model of Section 5.2 to qualitatively reproduce the observed kinematics also indicates the need for a varying dynamical (total) . Thus a second MGE model was constructed, with a variable , such that the deprojected Gaussian components having widths (dispersions) 1″ have M/L values half as large as the components with dispersions 10″, and the variation is logarithmic between those extremes. Using this mass model a non-self-consistent Jeans model was computed and again fitted to the observed SAURON stellar kinematics to derive a new . Interestingly this revised Jeans model, which has an gradient dictated by the stellar population measurements, now also well reproduces the observed SAURON stellar kinematics. The inferred circular velocity curve has a somewhat smaller peak velocity, 285 km s at 0.08″, a steeper decline in the inner few arcseconds and a shallower slope at 10″, but its amplitude is the same at 10″ (132 km s) as for the model with the constant . Figure 23 shows the predicted CO velocities that would have been observed if the molecular gas in NGC 3032 were rotating at the inferred circular velocity (Table 7). Clearly the assumption of a variable M/L does not remove the discrepancy between observed and expected velocities.
A reasonably accurate measure of the discrepancy between observed and expected velocities can be made from a model which describes the CO velocities well. Table 7 and Figure 23 show the rotation velocities in this model; they have been altered by hand to reproduce the observed position-velocity diagram. As before, no formal fitting has been done so the model shown cannot be claimed unique or best and its gross properties should be trusted to a greater degree than its details. This “working” model has a rotation velocity of 98 km s at 6.5″ and 113 km s at 13.5″. At the outer edge of the CO disk its velocity is only 10% smaller than that of the inferred circular velocity curve. However, at 6.5″ (700 pc) its rotation velocity is only 70% of the circular velocity curve, which means that the enclosed mass is a factor of two smaller than that suggested by the circular velocity. The discrepancy is even worse at smaller radii, where the circular velocity is nearly a factor of three higher than the rotation velocities in the molecular gas and the implied mass difference is a factor of 10.
At present the source of the discrepancy between the observed CO velocities and the inferred circular velocity curve is not understood. The unusually strong peak in the circular velocity curve is driven by the very bright blue nucleus in optical images and by the high central stellar velocity dispersion. Therefore, one possibility is that the measured stellar mean velocity and/or velocity dispersion in the center of NGC 3032 are significantly overestimated. However, the velocity dispersion measurements in the central regions made by McDermid et al. (2006a) with the OASIS instrument are consistent with the SAURON data. Another striking possibility is that the molecular gas in NGC 3032 may not be rotating at the circular velocity. We argued above that the gas had to have been acquired from some external source, and perhaps it is still undergoing strong dynamical evolution as it settles into equilibrium. Yet if its velocities are 70% of the circular velocity its orbits would be highly noncircular, which seems inconsistent with the regular, symmetric appearance of the CO velocity field (Figure 4). Neither of these explanations is completely satisfactory.
In short, the CO velocities in NGC 4459 and NGC 4526 are in excellent agreement with the circular velocities inferred for those two galaxies. These matches indicate that the stellar and CO kinematics are well understood. In NGC 3032 and NGC 4150 at radii 10″, the observed CO velocities are nearly equal to the mean stellar velocities and 10% to 20% smaller than the inferred circular velocities. However, the CO velocities of NGC 3032 are conspicuously low when compared to its inferred circular velocity within 6″ of the nucleus, and it would be valuable to carry out this kind of a comparison in other galaxies with counterrotating molecular gas in order to ascertain whether NGC 3032 is unusual.
## 6. Molecular and Ionized Gas Velocities
In normal spiral galaxies, the bulk of the ionized gas emission traces HII regions recently formed out of the molecular gas and sharing the disky kinematics of the molecular gas. The effect is that in a major axis position-velocity diagram the ionized gas tends to trace the ridgeline of the molecular and atomic gas or to be displaced towards the high velocity envelope (due primarily to resolution effects). Examples can be seen in Kregel & van der Kruit (2004) and Young et al. (2006). Thus, comparisons between the molecular and ionized kinematics can help to elucidate the role of star formation activity (or the lack thereof) in producing the ionized gas.
Excluding the case of NGC 4150, which is discussed in SS4.3, there appears to be a systematic difference between ionized gas kinematics and molecular kinematics of these lenticulars. The rotation speeds of the ionized gas are 20% smaller than those of the CO. There is also a slight tendency for the [O iii] velocities to be smaller than H velocities; this latter feature is most obvious at radii 5″ in NGC 4526 (Figure 20). Lower velocities in the ionized gas than in the molecular gas is the opposite situation to what is usually observed in the disks of spirals and is also in the opposite sense to the effects of beam smearing on the CO velocities. These data suggest that there may be a kinematic component of the ionized gas which is not dynamically cold and not related to star formation. They also imply that if the ionized gas in early-type galaxies is to be used for a measurement of the galaxy’s circular velocity, an asymmetric drift correction may be significant (e.g. Cretton, Rix, & de Zeeuw, 2000). By way of a caveat, though, we note that these lenticulars are dusty and NGC 4526 in particular has a high inclination (79°), so the observed CO and ionized gas velocities may be biased by projection and optical depth effects. The details of the relationships between cold and warm ionized phases of the interstellar medium are not yet clear, and they deserve closer scrutiny for the insights they may give into the evolution of the ISM in early-type galaxies.
## 7. Summary
We present resolved images of the CO emission in the four lenticular galaxies NGC 3032, NGC 4150, NGC 4459, and NGC 4526. These are some of the most CO-rich galaxies in the SAURON survey of early type galaxies, so they are prime targets for investigations which use cold gas to trace the interaction/merger history of early type galaxies and also to document morphological change through star formation and disk growth. Their inferred masses are in the range 5 to 5 . The molecular gas is located in kpc-scale disks (in excellent agreement with the distribution of dust visible in broadband optical images), the smallest being NGC 4150 with a radius of 500 pc and the largest being NGC 3032 with a radius of 1.5 kpc. Average molecular surface densities (including helium) are 100 to 200 pc.
In three of the four galaxies (NGC 3032, NGC 4459, and NGC 4526) the molecular gas is distributed in disks which show regular rotation and little sign of recent disturbance. The kinematic major axes are well aligned with the host galaxies’ photometric and (stellar) kinematic major axes, suggesting that the gas has settled into the equatorial plane of nearly axisymmetric potentials. The velocity field of NGC 4150 shows a kinematic major axis in rough agreement with the galaxy’s optical major axis, but better spatial resolution will be necessary in order to assess the regularity of the CO kinematics. Furthermore, in NGC 3032 the molecular gas’s kinematic position angle is 180° offset from the stellar kinematic position angle; this dramatic counterrotation indicates that the molecular gas was acquired from an external source or perhaps is leftover from a major merger, but it cannot have been produced through internal stellar mass loss. The sense of the CO rotation in NGC 3032 is consistent with that of the young kinematically decoupled core, however, which suggests that this is an example of a stellar substructure forming through dissipational processes.
In two cases (NGC 4459 and NGC 4526) the CO kinematics provide powerful, independent confirmation of the dynamical mass-to-light ratios inferred by Cappellari et al. (2006). The mass-to-light ratios were derived from the SAURON stellar kinematic data via two-integral Jeans and three-integral Schwarzschild dynamical models and were used to infer circular velocity curves. Simple tilted ring models are presented in which the molecular gas rotates at the circular velocity (or, in NGC 4526, 95% of the circular velocity, a decrease which is roughly consistent with the uncertainty in the mass-to-light ratio). Comparison of the model and observed major axis position-velocity diagrams indicates that the circular velocity inferred from stellar kinematics alone is consistent with the behavior of the molecular gas. The agreement is significant because it is rare to be able to make this kind of an independent check of the stellar dynamical analysis.
Several puzzling results of this work deserve further study; these are disagreements between the gaseous and the stellar kinematics at sub-kpc scales. For example, in NGC 4150 the CO velocities appear to be inconsistent with those of the young (counterrotating) stellar core. In NGC 3032 the observed CO velocities are significantly lower than the circular velocity curve (amounting to an implied factor of two less mass interior to 700 pc and even more at smaller radii). At present it is not known whether the stellar kinematic data give an overestimated circular velocity for this galaxy or whether the recently accreted molecular gas is not following circular orbits. Higher resolution CO observations will also be necessary to check whether NGC 4150 shows an inconsistency between CO velocities and circular velocities. Finally, the relationships between molecular gas and ionized gas should contribute valuable insights into star formation and the evolution of the ISM in these lenticulars.
Davor Krajnović kindly provided his IDL routines for kinemetric analysis and assistance in using them. LMY thanks the University of Oxford sub-department of Astrophysics for hospitality during a sabbatical visit and acknowledges support from NSF AST-0507432. MC acknowledges support from a PPARC Advanced Fellowship (PP/D005574/1). This work is partially based on observations made with the NASA/ESA Hubble Space Telescope, obtained from the data archive at the Space Telescope Science Institute. STScI is operated by the Association of Universities for Research in Astronomy, Inc. under the NASA contract NAS 5-26555. Facilities: BIMA
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• Young (2002) Young, L. M. 2002, AJ124, 788
• Young et al. (2006) Young, L. M., Rosolowsky, E., van Gorkom, J. H., & Lamb, S. A. 2006, ApJ, 650, 166
• Yun, Reddy, & Condon (2001) Yun, M. S., Reddy, N. A., & Condon, J. J. 2001 ApJ 554, 803
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# Do a gauge transformation for a Chern-Simons theory?
Suppose we have the following Lagrangian density:
$$L=\epsilon^{\mu\nu\rho}\big(\sum_a A^a_{\mu}(x) \partial_\nu A^a_{\rho}(x)-\sum_{a,b,c}\frac{1}{3} f^{bca} A^a_{\mu}(x) A^b_{\nu}(x) A^c_{\rho}(x)\big)$$ under this infinitesimal transformation + $\dots$ $$A^a_\mu(x) \to {A^a_\mu}'(x)\equiv (A^a_\mu(x) + f^{abc} \alpha^b(x) A^c_\mu(x) + \partial_\mu \alpha^a(x) +\dots)$$ where $x\equiv(t,x_1,x_2)$, $\epsilon^{\mu\nu\rho}$ is anti-symmetric and cyclic, $f^{abc}$ is anti-symmetric, but may not be cyclic in general.
We end up with $L \to L'$ under $A^a_\mu \to {A^a_\mu}'$.
And my question is:
How to obtain a well-simplified $L'$ using Mathematica ?
The key point is: without knowing the detailed structure $f^{abc}$, but only implement $f^{abc}$'s property to simplify the answer $L'$.
ps. I just took a glimpse at this post -how-to-manipulate-gauge-theory-in-mathematica, but I wonder whether there is a simpler way, since I am only doing pure algebra?
-
Care to share a code snippet ? – Sektor Dec 19 '13 at 7:27
I don't have a code yet. I am looking for one, or writing one myself. :) – Idear Dec 19 '13 at 19:22
Of course, I shall say the purpose here is definitely NOT to teach me on gauge invariant: which is obvious from the perspective of a finite continuous gauge transformation: $U^\dagger (A-i d) U$ on the C-S term Tr[$A \wedge A +(2/3)A \wedge A \wedge A$]. The purpose here is to teach Mathematica how to simplify nontrivial algebra. This is what I need. :) – Idear Dec 19 '13 at 22:50
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<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
Difficulty Level: At Grade Created by: CK-12
This activity is intended to supplement Algebra I, Chapter 10, Lesson 1.
ID: 9406
Time required: 60 minutes
• Graph \begin{align*}a\end{align*} quadratic function \begin{align*}y = ax^2 + bx + c\end{align*} and display a table for integral values of the variable.
• Graph the equation \begin{align*}y = a(x - h)^2\end{align*} for various values of \begin{align*}a\end{align*} and describe its relationship to the graph of \begin{align*}y = (x - h)^2.\end{align*}
• Determine the vertex, zeros, and the equation of the axis of symmetry of the graph \begin{align*}y = x^2 + k\end{align*} and deduce the vertex, the zeros, and the equation of the axis of symmetry of the graph of \begin{align*}y = a(x - h)^2 + k\end{align*}
## Activity Overview
In this activity, students graph quadratic functions and study how the constants in the equations compare to the coordinates of the vertices and the axes of symmetry in the graphs. The first part of the activity focuses on the vertex form, while the second part focuses on the standard form. Both activities include opportunities for students to pair up and play a graphing game to test how well they really understand the equations of quadratic functions.
Teacher Preparation
This activity is designed to be used in an Algebra 1 classroom. It can also be used as review in an Algebra 2 classroom.
• Problem 1 introduces students to the vertex form of a quadratic equation, while Problem 2 introduces the standard form. You can modify the activity by working through only one of the problems.
• If you do not have a full hour to devote to the activity, work through Problem 1 on one day and then Problem 2 on the following day.
Classroom Management
• This activity is intended to be mainly teacher-led, with breaks for individual student work. Use the following pages to present the material to the class and encourage discussion. Students will follow along using their handhelds.
Associated Materials
## Solutions
Problem 1
1. Answers will vary. Sample answer: the curve appears symmetric, and becomes less steep as \begin{align*}x\end{align*} increases or decreases.
2. \begin{align*}2\end{align*}
3. \begin{align*}3\end{align*}
4. \begin{align*}1\end{align*}
5. The lowest point of the graph is to the right of the \begin{align*}y-\end{align*}axis.
6. The lowest point of the graph is to the left of the \begin{align*}y-\end{align*}axis.
7. The graph moves away from the \begin{align*}y-\end{align*}axis.
8. The graph moves closer to the \begin{align*}y-\end{align*}axis.
9. The graph will center on the \begin{align*}y-\end{align*}axis.
12. The \begin{align*}x-\end{align*}coordinates of the points are opposites of each other. The \begin{align*}y-\end{align*}coordinates of the points are the same.
13. The left and right points are equidistant from the \begin{align*}y-\end{align*}axis.
14. \begin{align*}x = 0\end{align*}
15. The vertex is \begin{align*}(h, \ k).\end{align*}
16. The parabola opens up.
17. The parabola opens down.
18. The parabola becomes “narrower.”
19. The parabola becomes “wider.”
20. \begin{align*}a\end{align*}
21. \begin{align*}(h, \ k)\end{align*}
22. \begin{align*}h\end{align*}
23. Check students graphs.
24. Check students graphs.
25. Check students graphs.
Problem 2
1. \begin{align*}-\frac{b}{2a}\end{align*}
2. \begin{align*}2\end{align*}
3. \begin{align*}-4\end{align*}
4. \begin{align*}(0, \ 4)\end{align*}
5. No
6. When \begin{align*}a\end{align*} is positive, the parabola opens up. When \begin{align*}a\end{align*} is negative the parabola opens down. The greater the absolute value of \begin{align*}a\end{align*}, the “narrower” the parabola. The smaller the absolute value of \begin{align*}a\end{align*}, the “wider” the parabola.
7. \begin{align*}c\end{align*} is the \begin{align*}y-\end{align*}intercept of the parabola
8. Check students graphs.
9. Check students graphs.
10. Check students graphs.
Solutions for the Summary
1. \begin{align*}y = a(x - h)^2 + k\end{align*}
2. \begin{align*}a; (h, \ k); h\end{align*}
3. \begin{align*}y = ax^2 + bx + c\end{align*}
4. \begin{align*}-\frac{b}{2a}; \ x=-\frac{b}{2a}; \ c\end{align*}
Sketch the graph of each function. Identify the vertex and the equation of the axis of symmetry. Then check your graphs with your calculator.
5. \begin{align*}y=x^2+4\end{align*}
vertex \begin{align*}(0, \ 4)\end{align*}
axis of symmetry \begin{align*}x = 0\end{align*}
6. \begin{align*}y=(x-3)^2+5\end{align*}
vertex \begin{align*}(3, \ 5)\end{align*}
axis of symmetry \begin{align*}x = 3\end{align*}
7. \begin{align*}y=-(x-2)^2\end{align*}
vertex \begin{align*}(2, \ 0)\end{align*}
axis of symmetry \begin{align*}x = 2\end{align*}
8. \begin{align*}y=x^2+6x+9\end{align*}
vertex \begin{align*}(-3, \ 0)\end{align*}
axis of symmetry \begin{align*}x = -3\end{align*}
9. \begin{align*}y=-3x^2+6x+1\end{align*}
vertex \begin{align*}(1, \ 4)\end{align*}
axis of symmetry \begin{align*}x = 1\end{align*}
10. \begin{align*}y=x^2+1\end{align*}
vertex \begin{align*}(0, \ 1)\end{align*}
axis of symmetry \begin{align*}x = 0\end{align*}
### Notes/Highlights Having trouble? Report an issue.
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# A polling company found that, of 300 households surveyed
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A polling company found that, of 300 households surveyed [#permalink]
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06 Oct 2010, 18:35
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Difficulty:
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Question Stats:
71% (02:07) correct 29% (02:31) wrong based on 350 sessions
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A polling company found that, of 300 households surveyed, 120 spent at least $100 per month on both gasoline and electricity, 60 spent at least$100 per month on gasoline but not on electricity, and for every household that did not spend at least $100 per month on gasoline or electricity, 4 spent at least$100 per month on electricity but not on gasoline. How many of the 300 households did not spend at least $100 per month on either gasoline or electricity? A. 24 B. 30 C. 36 D. 90 E. 96 ##### Most Helpful Community Reply Current Student Joined: 29 Mar 2012 Posts: 317 Location: India GMAT 1: 640 Q50 V26 GMAT 2: 660 Q50 V28 GMAT 3: 730 Q50 V38 Re: Gasoline and Electricity [#permalink] ### Show Tags Updated on: 09 Jul 2012, 01:08 5 Quote: A polling company found that, of 300 households surveyed, 120 spent at least$100 per month on both gasoline and electricity, 60 spent at least $100 per month on gasoline but not on electricity, and for every household that did not spend at least$100 per month on gasoline or electricity, 4 spent at least $100 per month on electricity but not on gasoline. How many of the 300 households did not spend at least$100 per month on either gasoline or electricity?
A. 24
B. 30
C. 36
D. 90
E. 96
Hi,
Please refer to the attached Venn diagram:
Here, x = Number of households neither using electricity nor gasoline.
Now, $$60+120+4x+x=300$$
or x = 24
Regards,
Attachments
Venn.jpg [ 22.47 KiB | Viewed 5068 times ]
Originally posted by cyberjadugar on 20 Jun 2012, 23:06.
Last edited by cyberjadugar on 09 Jul 2012, 01:08, edited 1 time in total.
##### General Discussion
Retired Moderator
Joined: 02 Sep 2010
Posts: 775
Location: London
### Show Tags
07 Oct 2010, 00:52
krazo wrote:
Here is one I thought was pretty good...
A polling company found that, of 300 households surveyed, 120 spent at least $100 per month on both gasoline and electricity, 60 spent at least$100 per month on gasoline but not on electricity, and for every household that did not spend at least $100 per month on gasoline or electricity, 4 spent at least$100 per month on electricity but not on gasoline. How many of the 300 households did not spend at least $100 per month on either gasoline or electricity? (A) 24 (B) 30 (C) 36 (D) 90 (E) 96 120 spend$100 on gas & electricity
60 spend $100 on gas but not on electricity For every one that spent <$100 on both there are 4 that spent $100 on electricity not gas Let the ones that spent on neither be x The ones that spent on just electricity will be 4x Total households = Those who spend >$100 on both + Those who spend >$100 on just one + Those who spend <$100 on both = 120 + 60 + 4x + x
300 = 180 + 5x implies x=24
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Posts: 47183
Re: A polling company found that, of 300 households surveyed [#permalink]
### Show Tags
21 Jun 2012, 01:36
1
1
A polling company found that, of 300 households surveyed, 120 spent at least $100 per month on both gasoline and electricity, 60 spent at least$100 per month on gasoline but not on electricity, and for every household that did not spend at least $100 per month on gasoline or electricity, 4 spent at least$100 per month on electricity but not on gasoline. How many of the 300 households did not spend at least $100 per month on either gasoline or electricity? A. 24 B. 30 C. 36 D. 90 E. 96 {Total} = {Only Gasoline} + {Only Electricity} + {Both} + {Neither}. Notice that this formula is different from {Total} = {Group #1} + {Group #2}} - {Both} + {Neither}; $$300 = 60 + 4x + 120 + x$$ --> $$x=24$$. Answer: A. Hope it's clear. _________________ Director Joined: 27 May 2012 Posts: 513 Re: A polling company found that, of 300 households surveyed [#permalink] ### Show Tags 21 Jun 2012, 04:20 Bunuel wrote: A polling company found that, of 300 households surveyed, 120 spent at least$100 per month on both gasoline and electricity, 60 spent at least $100 per month on gasoline but not on electricity, and for every household that did not spend at least$100 per month on gasoline or electricity, 4 spent at least $100 per month on electricity but not on gasoline. How many of the 300 households did not spend at least$100 per month on either gasoline or electricity?
A. 24
B. 30
C. 36
D. 90
E. 96
{Total} = {Only Gasoline} + {Only Electricity} + {Both} + {Neither}. Notice that this formula is different from {Total} = {Group #1} + {Group #2}} - {Both} + {Neither};
$$300 = 60 + 4x + 120 + x$$ --> $$x=24$$.
Hope it's clear.
I know this is very basic , but a reconfirmation will help
" My confusion was the term " OR "
A OR B means AUB ( Total ) right so in this case E OR G means EUG( Total )
so EUG = only E + Only G + Both + Neither ( This one we are using )
also EUG = Group E + group G - both + Neither ( This one we are not using here )
Please let me now if anything that I have written in Blue is wrong !!
I want to understand if A or B = AUB = Total ??
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Re: A polling company found that, of 300 households surveyed [#permalink]
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21 Jun 2012, 04:51
stne wrote:
Bunuel wrote:
A polling company found that, of 300 households surveyed, 120 spent at least $100 per month on both gasoline and electricity, 60 spent at least$100 per month on gasoline but not on electricity, and for every household that did not spend at least $100 per month on gasoline or electricity, 4 spent at least$100 per month on electricity but not on gasoline. How many of the 300 households did not spend at least $100 per month on either gasoline or electricity? A. 24 B. 30 C. 36 D. 90 E. 96 {Total} = {Only Gasoline} + {Only Electricity} + {Both} + {Neither}. Notice that this formula is different from {Total} = {Group #1} + {Group #2}} - {Both} + {Neither}; $$300 = 60 + 4x + 120 + x$$ --> $$x=24$$. Answer: A. Hope it's clear. I know this is very basic , but a reconfirmation will help " My confusion was the term " OR " A OR B means AUB ( Total ) right so in this case E OR G means EUG( Total ) so EUG = only E + Only G + Both + Neither ( This one we are using ) also EUG = Group E + group G - both + Neither ( This one we are not using here ) Please let me now if anything that I have written in Blue is wrong !! I want to understand if A or B = AUB = Total ?? Two formulas are: {Total} = {Group #1} + {Group #2} - {Both} + {Neither} {Total} = {Only Group #1} + {Only Group #2} + {Both} + {Neither} Now, if we are told that there are 10 people in {Group #1} and 15 people in {Group #2}, then the number of people in {Group #1} OR in {Group #2} can be expressed as {Group #1} + {Group #2} - {Both}. But to get the total number of people you should add to that the number of people who are in neither of groups. Hope it's clear. _________________ Manager Status: Never ever give up on yourself.Period. Joined: 23 Aug 2012 Posts: 147 Location: India Concentration: Finance, Human Resources GMAT 1: 570 Q47 V21 GMAT 2: 690 Q50 V33 GPA: 3.5 WE: Information Technology (Investment Banking) Re: A polling company found that, of 300 households surveyed [#permalink] ### Show Tags 17 Sep 2012, 06:25 2 Different approach: I think if there are two overlapping sets then quick way to solve the problem is to form a chart rather than to draw a Venn diagram..here's the solution for above problem in chart way.. --------------Electricity----Non-Electricity--Total Gasoline------120-----------60--------------180 Non-Gasln-----4x-------------x--------------120 Total -------------------------------------------300 now 4x + x =120, so 5x=120 and hence x=24. _________________ Don't give up on yourself ever. Period. Beat it, no one wants to be defeated (My journey from 570 to 690) : http://gmatclub.com/forum/beat-it-no-one-wants-to-be-defeated-journey-570-to-149968.html Manager Joined: 10 Mar 2014 Posts: 212 Re: A polling company found that, of 300 households surveyed [#permalink] ### Show Tags 07 Aug 2014, 12:06 Bunuel wrote: A polling company found that, of 300 households surveyed, 120 spent at least$100 per month on both gasoline and electricity, 60 spent at least $100 per month on gasoline but not on electricity, and for every household that did not spend at least$100 per month on gasoline or electricity, 4 spent at least $100 per month on electricity but not on gasoline. How many of the 300 households did not spend at least$100 per month on either gasoline or electricity?
A. 24
B. 30
C. 36
D. 90
E. 96
{Total} = {Only Gasoline} + {Only Electricity} + {Both} + {Neither}. Notice that this formula is different from {Total} = {Group #1} + {Group #2}} - {Both} + {Neither};
$$300 = 60 + 4x + 120 + x$$ --> $$x=24$$.
Hope it's clear.
HI Bunuel,
I know its a silly question but i didn't understand how you took 4x from question stem.
Thanks.
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A polling company found that, of 300 households surveyed [#permalink]
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08 Aug 2014, 00:19
PathFinder007 wrote:
Bunuel wrote:
A polling company found that, of 300 households surveyed, 120 spent at least $100 per month on both gasoline and electricity, 60 spent at least$100 per month on gasoline but not on electricity, and for every household that did not spend at least $100 per month on gasoline or electricity, 4 spent at least$100 per month on electricity but not on gasoline. How many of the 300 households did not spend at least $100 per month on either gasoline or electricity? A. 24 B. 30 C. 36 D. 90 E. 96 {Total} = {Only Gasoline} + {Only Electricity} + {Both} + {Neither}. Notice that this formula is different from {Total} = {Group #1} + {Group #2}} - {Both} + {Neither}; $$300 = 60 + 4x + 120 + x$$ --> $$x=24$$. Answer: A. Hope it's clear. HI Bunuel, I know its a silly question but i didn't understand how you took 4x from question stem. Please clarify. Thanks. Not Bunuel, but lets try The answer lies in the highlighted part of the question Only Electricity is 4 times of (No Gas No electricity) So, if "x" = No Gas No electricity, then 4x = Only electricity Please refer the Venn diagram above; it explains perfect In this problem, the term "atleast 100" is the confusion creator which is repeated 5 times; although "100" is not required for any calculation. May be GMAT trick _________________ Kindly press "+1 Kudos" to appreciate Target Test Prep Representative Status: Head GMAT Instructor Affiliations: Target Test Prep Joined: 04 Mar 2011 Posts: 2679 Re: A polling company found that, of 300 households surveyed [#permalink] ### Show Tags 14 Nov 2017, 07:20 1 krazo wrote: A polling company found that, of 300 households surveyed, 120 spent at least$100 per month on both gasoline and electricity, 60 spent at least $100 per month on gasoline but not on electricity, and for every household that did not spend at least$100 per month on gasoline or electricity, 4 spent at least $100 per month on electricity but not on gasoline. How many of the 300 households did not spend at least$100 per month on either gasoline or electricity?
A. 24
B. 30
C. 36
D. 90
E. 96
We can create the following equation:
Total = number who spent at least 100 on gasoline only + number who spent at least 100 on electricity only + number who spend at least 100 on both + number who spent at least 100 on neither
If we let x = number who spent at least 100 on neither, then 4x = number who spent at least 100 on electricity only, and we have:
300 = 60 + 4x + 120 + x
300 = 180 + 5x
120 = 5x
x = 24
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Re: A polling company found that, of 300 households surveyed [#permalink]
### Show Tags
23 Nov 2017, 16:32
krazo wrote:
A polling company found that, of 300 households surveyed, 120 spent at least $100 per month on both gasoline and electricity, 60 spent at least$100 per month on gasoline but not on electricity, and for every household that did not spend at least $100 per month on gasoline or electricity, 4 spent at least$100 per month on electricity but not on gasoline. How many of the 300 households did not spend at least \$100 per month on either gasoline or electricity?
A. 24
B. 30
C. 36
D. 90
E. 96
a ratio between "none and only electricity" is given
so, none and only electricity = (300 - 120 -60) = 120
now, according to the ratio
120 * 1/5 = 24
thanks
Re: A polling company found that, of 300 households surveyed &nbs [#permalink] 23 Nov 2017, 16:32
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# linear magnification formula for lens
What is the formula for calculating linear magnification of a specimen when using a hand lens . Keep in mind that subsequent lenses can continue to invert your image. The required linear magnification is the ratio of the desired image diameter to the diamond’s actual diameter (Equation \ref{eq15}). Answers . Zacharias Jansen and his father combined lenses from simple magnifying glasses to build microscopes and, from there, microscopes and telescopes changed the world. The required linear magnification is the ratio of the desired image diameter to the diamond’s actual diameter (). The magnification of a mirror is represented by the letter m. Thus m = Or m = where, h 2 = size of image h 1 = size of object. The linear magnification or magnification of a spherical mirror may be defined as the ratio of the size (height) of the image to the size (height) of the object. The linear or transverse magnification is defined as the ratio of the size of the image to that of the object. It is a pure ratio and has no units. The linear magnification produced by a spherical lens (convex or concave) is defined as the ratio of the height of the image (h′) to the height of the object (h). But how can I prove the equation mathematically? Subtended angles are related to the linear size by non-linear trigonometric functions and depend on the distance from image to eye. Using lens formula the equation for magnification can also be obtained as . Solution. Understanding the focal length of lenses was crucial to combining their powers. Before the 1590s, simple lenses dating back as far as the Romans and Vikings allowed limited magnification and simple eyeglasses. It is denoted by the letter ‘m’ and is given by, Linear magnification is the ratio of the size of object and image. For each lens, treat the image of the previous lens as its object and use the lens equation and magnification equation to find your answers. a) What is the formula for calculating linear magnification of a specimen when using a hand lens b) Give a reason why staining is necessary when preparing specimens for observation under the microscope. If the camera forms an image which is 6mm high, (a) What is the magnification; (b) How far must the camera film be behind the lens for the image to be formed. Answer: Q1. As the object is always placed above the principal axis so the magnitude of h 1 is always positive. Waves And Optics - Lenses Lens Formula: Magnification Linear Magnification (m) = height of image height of object OR = Image distance Object distance Example: A building is 6m high, and it is 80m from a converging camera lens. and the thin-lens equation. m = h 2 /h 1 = v//u = (f-v)/f = f/(f+u) This equation is valid for both convex and concave lenses and for real and virtual images. a) Magnification = $\frac{length of drawing}{length of object}$ b) Staining is … a. Angular magnification is the ratio of the angle subtended by object and image. Because the jeweler holds the magnifying lens close to his eye, we can use Equation \ref{eq13} to find the focal length of the magnifying lens.
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# Solve the system (notation submission issues)
Solve the system
$$x_1+x_2+2x_3=3\\ 6x_1+7x_2-3x_3=-3$$
$$\begin{bmatrix} 1 & 1 & 2 & 3\\ 6 & 7 & -3 & -3\\ \end{bmatrix}\text{~}\begin{bmatrix} 1 & 1 & 2 & 3\\ 0 & 1 & -15 & -21\\ \end{bmatrix}\text{~}\begin{bmatrix} 1 & 0 & 17 & 24\\ 0 & 1 & -15 & -21\\ \end{bmatrix}$$
And using this I found that: $$x_1+17x_3=24\\ x_2-15x_3=-21$$
As far as I know this is correct, but my issue is how to state my solution.
$$\begin{bmatrix} x_1\\ x_2\\ x_3 \end{bmatrix}=\begin{bmatrix} ?\\ ?\\ ? \end{bmatrix}+\begin{bmatrix} ?\\ ?\\ ? \end{bmatrix}s$$
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Just like we did here. – JohnD Jan 27 '13 at 19:44
## 1 Answer
Notice that all of your equations have a common factor, namely $x_3$.
Rewrite $x_1$ as a function of $x_3$ and then rewrite $x_2$ as a function of $x_3$. Now by letting $x_3$ vary over your the given field, you'll attain all solutions.
You'll get $x_1=24-17x_3$ and $x_2=-21+15x_3$, so your set of solutions should be $\{ \begin{bmatrix} x_1 & x_2 & x_3 \end{bmatrix}^T : x_1, x_2, x_3\in F \wedge x_1=24-17x_3 \wedge x_2=-21+15x_3\}$ or more succinctly $\{ \begin{bmatrix} 24-17x_3 \space -21+15x_3 \space x_3 \end{bmatrix}^T : x_3\in F\}$, where $F$ is the field you're working on.
In this answer I assumed that your calculations are correct.
For similar problems check this and this.
EDIT: Your calculations are correct.
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And then to put it in the form OP wants, $$\pmatrix{24-17x_3\cr-21+15x_3\cr x_3\cr}=\pmatrix{24\cr-21\cr0}+\pmatrix{-17\cr15\cr1\cr}x_3$$ – Gerry Myerson Jan 28 '13 at 5:40
yes, thank you. – ground.clouds1 Feb 11 '13 at 1:47
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# Definition of Chooses. Meaning of Chooses. Synonyms of Chooses
Here you will find one or more explanations in English for the word Chooses. Also in the bottom left of the page several parts of wikipedia pages related to the word Chooses and, of course, Chooses synonyms and on the right images related to the word Chooses.
## Definition of Chooses
Choose
Choose Choose, v. i. 1. To make a selection; to decide. They had only to choose between implicit obedience and open rebellion. --Prescott. 2. To do otherwise. Can I choose but smile?' --Pope. Can not choose but, must necessarily. Thou canst not choose but know who I am. --Shak.
## Meaning of Chooses from wikipedia
- options followed by the corresponding action. For example, a traveler might choose a route for a journey based on the preference of arriving at a given destination...
- ('seven choose two') Morra, a hand game sometimes referred to as Choose Choose (film), a crime horror film directed by Marcus Graves "Choose", song by...
- Chooser can refer to: Choosing, to select freely and after consideration. A user interface on a computer that allows the user to choose items from large...
- "We choose to go to the Moon", officially titled as the Address at Rice University on the Nation's Space Effort, is a speech delivered by United States...
- Choose life may refer to: In a Bible verse, Deuteronomy 30:19: "I call heaven and earth to record this day against you, that I have set before you life...
- eliminated: A will not choose A3 since either A1 or A2 will produce a better result, no matter what B chooses; B will not choose B3 since some mixtures...
- "A Time for Choosing", also known as "The Speech", was a speech presented during the 1964 U.S. presidential election campaign by ****ure president Ronald...
- minimizing player chooses pure strategy i and the maximizing player chooses pure strategy j (i.e. the player trying to minimize the payoff chooses the row and...
- {\displaystyle {\frac {z+k \choose j}{k \choose j}}\to \left(1-{\frac {j}{k}}\right)^{-z}\quad {\text{and}}\quad {\frac {j \choose j-k}{j-z \choose j-k}}\to \left({\frac...
- 2018. Furones, David (December 12, 2016). "Former Chaminade WR Brown chooses Oklahoma out of junior college". Sun Sentinel. Retrieved May 24, 2018....
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Über dieses Buch
This volume collects thirteen expository or survey articles on topics including Fractal Geometry, Analysis of Fractals, Multifractal Analysis, Ergodic Theory and Dynamical Systems, Probability and Stochastic Analysis, written by the leading experts in their respective fields. The articles are based on papers presented at the International Conference on Advances on Fractals and Related Topics, held on December 10-14, 2012 at the Chinese University of Hong Kong. The volume offers insights into a number of exciting, cutting-edge developments in the area of fractals, which has close ties to and applications in other areas such as analysis, geometry, number theory, probability and mathematical physics.
Inhaltsverzeichnis
Mandelbrot Cascades and Related Topics
This article is an extended version of the talk given by the author at the conference “Advances in fractals and related topics”, in December 2012 at the Chinese Hong-Kong University. It gathers recent advances in Mandelbrot cascades theory and related topics, namely branching random walks, directed polymers on disordered trees, multifractal analysis, and dynamical systems.
Julien Barral
Law of Pure Types and Some Exotic Spectra of Fractal Spectral Measures
Let
$$\mu$$
μ
be a Borel probability measure with compact support in
$${\mathbb R}^d$$
R
d
and let
$$E(\Lambda )=\{e^{-2\pi \lambda \cdot x}: \lambda \in \Lambda \}$$
E
(
Λ
)
=
{
e
-
2
π
λ
·
x
:
λ
Λ
}
. We make a review on some recent progress about spectral measures. We first show that the law of pure types holds for spectral measures, i.e. if
$$E(\Lambda )$$
E
(
Λ
)
is a frame for
$$L^2(\mu )$$
L
2
(
μ
)
, then
$$\mu$$
μ
is discrete or absolutely continuous or singular continuous with respect to Lebesgue measure (see [
HLL13
]). And we discuss the spectral properties of Cantor measures (see [
DaHL13
]), where we focus on some exotic properties of the spectra of some Cantor measures.
Xin-Rong Dai, Xing-Gang He, Chun-Kit Lai
The Role of Transfer Operators and Shifts in the Study of Fractals: Encoding-Models, Analysis and Geometry, Commutative and Non-commutative
We study a class of dynamical systems in
$$L^2$$
L
2
spaces of infinite products
$$X$$
X
. Fix a compact Hausdorff space
$$B$$
B
. Our setting encompasses such cases when the dynamics on
$$X = B^\mathbb {N}$$
X
=
B
N
is determined by the one-sided shift in
$$X$$
X
, and by a given transition-operator
$$R$$
R
. Our results apply to any positive operator
$$R$$
R
in
$$C(B)$$
C
(
B
)
such that
$$R1 = 1$$
R
1
=
1
. From this we obtain induced measures
$$\Sigma$$
Σ
on
$$X$$
X
, and we study spectral theory in the associated
$$L^2(X,\Sigma )$$
L
2
(
X
,
Σ
)
. For the second class of dynamics, we introduce a fixed endomorphism
$$r$$
r
in the base space
$$B$$
B
, and specialize to the induced solenoid
$$\mathrm{Sol }(r)$$
Sol
(
r
)
. The solenoid
$$\mathrm{Sol }(r)$$
Sol
(
r
)
is then naturally embedded in
$$X = B^\mathbb {N}$$
X
=
B
N
, and
$$r$$
r
induces an automorphism in
$$\mathrm{Sol }(r)$$
Sol
(
r
)
. The induced systems will then live in
$$L^2(\mathrm{Sol }(r), \Sigma )$$
L
2
(
Sol
(
r
)
,
Σ
)
. The applications include wavelet analysis, both in the classical setting of
$$\mathbb {R}^n$$
R
n
, and Cantor-wavelets in the setting of fractals induced by affine iterated function systems (IFS). But our solenoid analysis includes such hyperbolic systems as the Smale-Williams attractor, with the endomorphism
$$r$$
r
there prescribed to preserve a foliation by meridional disks. And our setting includes the study of Julia set-attractors in complex dynamics.
Dorin Ervin Dutkay, Palle E. T. Jorgensen
Generalized Energy Inequalities and Higher Multifractal Moments
We present a class of generalized energy inequalities which may be used to investigate higher multifractal moments, in particular
$$L^q$$
-dimensions of images of measures under Brownian-type processes,
$$L^q$$
-dimensions of almost self-affine measures, and moments of random cascade measures.
Kenneth Falconer
Some Aspects of Multifractal Analysis
The aim of this survey is to present some aspects of multifractal analysis around the recently developed subject of multiple ergodic averages. Related topics include dimensions of measures, oriented walks, Riesz products etc. The exposition on the multifractal analysis of multiple ergodic averages is mainly based on [
FLM12
,
KPS12
,
FSW00
].
Ai-Hua Fan
Heat Kernels on Metric Measure Spaces
In this section we shall discuss the notion of the heat kernel on a metric measure space
$$( M,d,\mu )$$
(
M
,
d
,
μ
)
.
Alexander Grigor’yan, Jiaxin Hu, Ka-Sing Lau
Stochastic Completeness of Jump Processes on Metric Measure Spaces
We give criteria for stochastic completeness of jump processes on metric measure spaces and on graphs in terms of volume growth.
Alexander Grigor’yan, Xueping Huang
Self Similar Sets, Entropy and Additive Combinatorics
Hoc12
], that self-similar sets whose dimension is smaller than the trivial upper bound have “almost overlaps” between cylinders. We give a heuristic derivation of the theorem using elementary arguments about covering numbers. We also give a short introduction to additive combinatorics, focusing on inverse theorems, which play a pivotal role in the proof. Our elementary approach avoids many of the technicalities in [
Hoc12
], but also falls short of a complete proof; in the last section we discuss how the heuristic argument is turned into a rigorous one.
Michael Hochman
Quasisymmetric Modification of Metrics on Self-Similar Sets
Using the notions of scales and their gauge functions associated with self-similar sets, we give a necessary and sufficient condition for two metrics on a self-similar set being quasisymmetric to each other. As an application, we construct metrics on the Sierpinski carpet which is quasisymmetric with respect to the Euclidean metrics and obtain an upper estimate of the conformal dimension of the Sierpinski carpet.
Jun Kigami
Recent Progress on Dimensions of Projections
This is a survey on recent progress on the question: how do projections effect dimensions generically? I shall also discuss briefly dimensions of plane sections.
Pertti Mattila
The Geometry of Fractal Percolation
A well studied family of random fractals called fractal percolation is discussed. We focus on the projections of fractal percolation on the plane. Our goal is to present stronger versions of the classical Marstrand theorem, valid for almost every realization of fractal percolation. The extensions go in three directions:
$$\bullet$$
the statements work for all directions, not almost all,
$$\bullet$$
the statements are true for more general projections, for example radial projections onto a circle,
$$\bullet$$
in the case
$$\dim _H >1$$
dim
H
>
1
, each projection has not only positive Lebesgue measure but also has nonempty interior.
Michał Rams, Károly Simon
Self-affine Sets and the Continuity of Subadditive Pressure
The affinity dimension is a number associated to an iterated function system of affine maps, which is fundamental in the study of the fractal dimensions of self-affine sets. De-Jun Feng and the author recently solved a folklore open problem, by proving that the affinity dimension is a continuous function of the defining maps. The proof also yields the continuity of a topological pressure arising in the study of random matrix products. I survey the definition, motivation and main properties of the affinity dimension and the associated SVF topological pressure, and give a proof of their continuity in the special case of ambient dimension two.
Pablo Shmerkin
Stability Properties of Fractal Curvatures
Lipschitz-Killing curvatures of singular sets are known from geometric measure theory. These are extensions of classical notions from convex and differential geometry. In recent years their fractal versions have been introduced via approximation by parallel sets of small distances. In the present paper stability properties of the corresponding limits under small perturbations of these neighborhoods are studied. The well-known Minkowski content may be considered as marginal case.
Martina Zähle
Backmatter
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# What does the government spend the least money onhealthcare defense social security or education
###### Question:
What does the government spend the least money on healthcare
defense
social security
or education
### Mike's movie rental charges $3 for each rental plus$0.50 for each additional day. reed's
Mike's movie rental charges $3 for each rental plus$0.50 for each additional day. reed's rentals charges $1.50 for each rental plus a dollar and$1.25 for each additional day. for what number of additional days will the bill at both stores be the same?...
### Read the passage from gulliver's travels. there was a most ingenious architect, who had contrived a
Read the passage from gulliver's travels. there was a most ingenious architect, who had contrived a new method for building houses, by beginning at the roof, and working downward to the foundation. which is an objective summary of the passage? a-in his travels, gulliver is fooled by a silly archite...
### Â/8.37 points scalcet8 12.2.037. ask your teacher my notes question part points submissions used a
Â/8.37 points scalcet8 12.2.037. ask your teacher my notes question part points submissions used a block-and-tackle pulley hoist is suspended in a warehouse by ropes of lengths 2 m and 3 m. the hoist weighs 350 n. the ropes, fastened at different heights, make angles of 50â° and 38â° with the h...
### A small bag has 8 black beans, 5 white beans, and 12 red beans.What is the probability of not drawing a red bean?A. 36%B.
A small bag has 8 black beans, 5 white beans, and 12 red beans. What is the probability of not drawing a red bean? A. 36% B. 48% C. 52% D. 71%...
### If Ax) = 5 x-2, what is f? (x)?O f(x) = 9x + 18or?(x) = 5 x + 2o ml(x) = 9x+2om (x) = -2x +.
If Ax) = 5 x-2, what is f? (x)? O f(x) = 9x + 18 or?(x) = 5 x + 2 o ml(x) = 9x+2 om (x) = -2x + ....
### Ralph sold brownies for $0.45 a piece in order to earn money to buy baseball cards. if the cards cost $0.81 per
Ralph sold brownies for $0.45 a piece in order to earn money to buy baseball cards. if the cards cost $0.81 per pack, and if ralph had no money left over after buying them, what is the least number of brownies he must have sold?...
Please help this is due today Given parallelogram ABCD determine the missing information Don’t answer with one answer, blank or ridiculous answer please this is serious $Please help this is due today Given parallelogram ABCD determine the missing information Do$...
### Fill in the blank with the correct item. _ un homme très généreux. O A. C'est O B. Il est O C. Elle est OD. Ils sont SUBMIT
Fill in the blank with the correct item. _ un homme très généreux. O A. C'est O B. Il est O C. Elle est OD. Ils sont SUBMIT...
### The owner of a candy store wants to mix some peanuts worth $3 per pound, some cashews worth$10 per pound, and some brazil
The owner of a candy store wants to mix some peanuts worth $3 per pound, some cashews worth$10 per pound, and some brazil nuts worth $9 per pound to get 50 pounds of a mixture that will sell for$6.80 per pound. she uses 10 fewer pounds of cashews than peanuts. how many pounds of each did she use?...
### When narrator 3 says in line 6 that 'coconuts rained down all around.'she means that
When narrator 3 says in line 6 that "coconuts rained down all around."she means that...
PLEASE HELP i need help on the HIGHLIGHTED question!! $PLEASE HELPPPPPPP i need help on the HIGHLIGHTED question!!$...
### How many orbits/shells/energy levels does Rubidium have?
How many orbits/shells/energy levels does Rubidium have?...
### What are the unknowns in this problem? a problem states: 'there are 9 more pencils than pens in a container.
What are the unknowns in this problem? a problem states: "there are 9 more pencils than pens in a container. there are 25 writing utensils in the container in all. how many pens are there in the container? " select each correct answer. the total number of pens in the container in all the total num...
### Which of greek pottery style featured figures in silhouette?
Which of greek pottery style featured figures in silhouette?...
### How long did World War II last? 1 year 10 years 8 years 6 years
How long did World War II last? 1 year 10 years 8 years 6 years...
### Ascientist exploring a deep sea trench finds a new soft bodied species with an unknown shape. she is
Ascientist exploring a deep sea trench finds a new soft bodied species with an unknown shape. she is able to determine it to be a mollusk because it has a. body segmentation. b. an internal skeleton. c. bilateral symmetry. d. an internal shell....
### What was an effect of the Missouri Compromise? A It temporarily ended the slavery debate. B It required Southern states to pay
What was an effect of the Missouri Compromise? A It temporarily ended the slavery debate. B It required Southern states to pay higher taxes. C It led to the South seceding from the Union. D It upset the balance of slaves and free states in the Union....
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# Optimizing an arbitrary function of 10 variables
Let us be given a function $$f(x_1,\dots,x_{10})$$ of multiple variables $$x_1,\dots,x_{10}$$ given that $$\sum_{i=1}^{10} x_i \leq 7$$. How do we solve the following problem?
\begin{aligned} \max_{\{x_i\}, i=1,\dots,10} \quad & f(x_1,\dots,x_{10})\\ \textrm{s.t.} \quad & \sum_{i=1}^{10} x_i \leq 7\\ & x_i \in X_i, X_i = [0,10] \\ \end{aligned}
I with my professor are trying to work on building a neural network in such a way that we have multiple output from each layer. For example there are $$n$$ neurons in the $$i^{th}$$ layer, there will be, let us say $$k$$ outputs of each size $$n$$. Similarly, the next layer will also have same number of output and we will look at all the possible ways to connect $$k$$ outputs of the $$i^{th}$$ layer and $$k$$ outputs of the $$(i+1)^{st}$$ layer. And we take the max of the output in each layer while updating the loss function. My professor told me that this boils down to finally become a DP problem.
• Any additional information, such as the contribution of each $x_i$ is independent of all $x_j, j\ne i$ or all $x_i$s are integral? May 10 at 18:42
• If the variables are integers, there are only 19448 options to try. May 10 at 18:52
• Please edit the question to indicate whether the variables must be integers, or if not, what domain they come from.
– D.W.
May 13 at 6:32
• "$k$ outputs of each size $n$": what ?? May 13 at 8:42
• If all $X_i$ are identical, why $i$ ? And due to the sum and positiveness constraints, $x_i\le 7$ must hold anyway. May 13 at 8:43
There are only 19448 combinations of $$x$$'s that meet the constraints. If $$f$$ is arbitrary, the best you can do is enumerate all 19448 combinations and see which leads to the largest value of $$f$$. There is no faster algorithm. Dynamic programming does not seem relevant.
If you know something about the structure of $$f$$ (e.g., it is a separable function), then it might be possible to do better.
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# GMAT Question of the Day (Oct 6)
- Oct 6, 02:00 AM Comments [0]
Math (DS) Gold depreciated at a rate of $X\\%$ per year between 2000 and 2005. If 1 kg of gold cost $S$ dollars in 2001 and $T$ dollars in 2003, how much did it cost in 2002...
# GMAT Question of the Day (Oct 3)
- Oct 3, 02:00 AM Comments [0]
Math (PS) What is the product of three consecutive integers? (1) At least one of the integers is positive. (2) The sum of the integers is less than 6. Question Discussion & Explanation Correct Answer - C - (click and drag your mouse to see the answer) GMAT Daily Deals e-GMAT’s...
# GMAT Question of the Day (Oct 2)
- Oct 2, 02:00 AM Comments [0]
Math (DS) Is the perimeter of triangle with the sides $a$, $b$ and $c$ greater than 30? (1) $a-b=15$ (2) The area of the triangle is 50This is a DS Question. As the result the answer...
# GMAT Question of the Day (Oct 1)
- Oct 1, 02:00 AM Comments [1]
Math (PS) 4 professors and 6 students are being considered for membership on a supervisory committee which must consist of 3 people. If the committee has to include at least 1 professor, how many ways can this committee be formed? A. 36 B. 60 C. 72 D. 80 E. 100 Question Discussion...
# GMAT Question of the Day (Sep 30): Arithmetic and Critical Reasoning
- Sep 30, 02:00 AM Comments [0]
Math (PS) How many integers are divisible by 3 between $10!$ and $10! + 20$ inclusive? A. 6 B. 7 C. 8 D. 9 E. 10 ____________________________________________________________________________________________________________________________________ Question Discussion & Explanation Correct Answer - B - (click and drag your mouse to see the answer) GMAT Daily Deals Manhattan GMAT:...
# GMAT Question of the Day (Sep 29)
- Sep 29, 02:00 AM Comments [0]
Math (PS) Is $|x - 6| \\gt 5$ ? (1) $x$ is an integer (2) $x^2 \\lt 1$ Question Discussion & Explanation Correct Answer - B - (click and drag your mouse to see...
# GMAT Question of the Day (Sep 26): Arithmetic and Sentence Correction
- Sep 26, 02:00 AM Comments [0]
Math (PS) How many five-digit numbers can be formed using the digits 0, 1, 2, 3, 4 and 5 which are divisible by 3, without repeating the digits? (A) 15 (B) 96 (C) 120 (D) 181 (E) 216 ------------------------------------------------------------------------------------------------------------------------------------ Question Discussion & Explanation Correct Answer - E - (click and drag your mouse to...
# GMAT Question of the Day (Sep 25): Word Problem and Sentence Correction
- Sep 25, 02:00 AM Comments [0]
Math (PS) One fisherman was telling his friends that he caught a fish that had a 60-foot long head. It also had a tail that was as long as the fish's head and a half of its body. Finally, the body was half the size of...
# GMAT Question of the Day (Sep 24): Statistics and Critical Reasoning
- Sep 24, 02:00 AM Comments [0]
Math (PS) Which set(s) has the greatest standard deviation? 1. Set 1 consisting of 10 digits 2. Set 2 consisting of 10 first positive consecutive even numbers 3. Set 3 consisting of 10 first primes (A) set 1 (B) set 2 (C) set 3 (D) set 1 and 2 (E) set 1, 2, and...
# GMAT Question of the Day (Sep 23): Arithmetic and Critical Reasoning
- Sep 23, 02:00 AM Comments [0]
Math (DS) Are integers , , and consecutive? 1. equals the arithmetic mean of and 2. For the correct answer, please drag your mouse over the hidden text. Explanation is linked below. Question Discussion & Explanation Correct Answer - E - (click and drag your mouse...
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# Uniform algebra
A subalgebra $A$, closed with respect to the topology of uniform convergence, of the algebra $C ( X)$ of continuous functions on a compactum $X$ that contains all constant functions and separates the points of $X$. The last condition means that for each pair $x, y$ of distinct points in $X$ there is a function $f$ in $A$ for which $f ( x) \neq f ( y)$. Uniform algebras are usually provided with the sup norm:
$$\| f \| = \sup _ { X } | f ( x) |.$$
Here $\| f ^ { 2 } \| = \| f \| ^ {2}$. Every Banach algebra with an identity (even without assuming commutativity) and with a norm satisfying the latter condition is isomorphic to a uniform algebra.
The uniform algebras form an important subclass of the class of commutative Banach algebras (cf. Commutative Banach algebra) over the field of complex numbers $\mathbf C$.
To each point $x \in X$ corresponds a homomorphism $\phi _ {x} : A \rightarrow \mathbf C$, defined by $\phi _ {x} ( f ) = f ( x)$. Therefore $X$ is in a natural way topologically imbedded in the space $\mathop{\rm MSpec} ( A)$ of maximal ideals of $A$, and under the corresponding identification $X$ contains the Shilov boundary (cf. Boundary (in the theory of uniform algebras)). In the study of uniform algebras a major role is played by peak points (that is, points of $X$ at which the strict maximum modulus of at least one element of $A$ is attained), by multiplicative probability measures on $X$( that is, representing measures of homomorphisms from $A$ to $\mathbf C$) and by measures on $X$ that are orthogonal to $A$. Many concrete results relating to uniform algebras touch on the relations between these notions.
A uniform algebra is called symmetric if with each function its complex conjugate belongs to the algebra. According to the Stone–Weierstrass theorem, each symmetric uniform algebra on a compactum $X$ coincides with $C ( X)$. The so-called anti-symmetric uniform algebras, containing no real-valued functions apart from the constants, form a kind of opposite class. A typical example is the algebra of all functions that are analytic in the open unit disc of the complex plane and continuous on its closure (the disc algebra). The Shilov–Bishop theorem: Each uniform algebra can be obtained from anti-symmetric uniform algebras by "glueing" in a certain way. Even more refined classification theorems are known. At the same time arbitrary uniform algebras do not reduce to algebras of analytic functions of the type of the disc algebra. For example, it is possible to construct a uniform algebra on a one-dimensional compactum, which coincides with its space of maximal ideals, such that all points of the compactum are peak points and at the same time only the zero element of the algebra can be zero on a non-empty open set.
#### References
[1] T.W. Gamelin, "Uniform algebras" , Prentice-Hall (1969)
Instead of "uniform algebra" the terminology "function algebra" is also used.
Let $\xi \in \mathop{\rm MSpec} ( A)$, the maximal ideal space of $A$. A representing measure for $\xi$ is a positive measure on $X$ such that
$$\xi ( f ) = \int\limits f d \mu ,\ f \in A .$$
They exist by the Riesz representation theorem (cf. (the editorial comments to the second article) Riesz theorem). Of course, then $\int d \mu = 1( f ) = 1$, so that $\mu$ is a probability measure. A Jensen measure for $\xi$ is a positive measure on $X$ such that the Jensen inequality
$$\mathop{\rm log} | \xi ( f ) | \leq \int\limits \mathop{\rm log} | f | d \mu ,\ \ f \in A ,$$
holds. A Jensen measure is a representing measure, and a Jensen measure for $\xi$ always exists.
A measure $\mu$ on $X$ is orthogonal to $A$ if $\int f d \mu = 0$ for all $f \in A$.
#### References
[a1] T.W. Gamelin, "Uniform algebras and Jensen measures" , Cambridge Univ. Press (1978) [a2] G.M. Leibowitz, "Lectures on complex functions algebras" , Foresman (1970) [a3] E.L. Stout, "The theory of uniform algebras" , Bogden & Quigley (1971) [a4] J. Wermer, "Banach algebras and several complex variables" , Springer (1975) [a5] I. Suciu, "Function algebras" , Ed. Acad. Romania (1973) [a6] A. Browder, "Introduction to function algebras" , Benjamin (1969)
How to Cite This Entry:
Uniform algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Uniform_algebra&oldid=49068
This article was adapted from an original article by E.A. Gorin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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The varieties are infinite. Factors which can influence organisational culture include: the organisation's structure, the system and processes by which work is carried out, the behaviour and attitudes 1 Defining Management and Organization 1 In this era of globalization accompanied by complexity, ambiguity, rapid change, and diversity, managing an organization is a difficult task. structure: To be effective, organization must adjust structure consistent with –The type of environment it works in –The technology it uses –Its size –Its strategy –Other contextual factors. We also start the process of building a better matrix upon which to base many such simulations and analyses in staffing research. Yet, good management is criti-cal for the survival of an organization. The structures diagrammed and described are functional, product, customer, geographic, divisional or M-form, matrix, amorphous, and hybrids. form of attractive filtered generalized singular value decomposition (GSVD) Chap 3 : Organizational Structure – Organization versus Structure – Theories of contingency approach • Internal and external factors • a typology of organizations (Mintzberg) – Types of organizational structures • functional organization (U-form) • divisional organization (M-form, H-form) expansions, where the filters are given explicitly. ), Kogan Page Limited, 1994.) %%EOF Copyright, A Science-Based Alternative to the World's Scriptures, Comments on “Improved Bounds for Linear Discrete-Time Systems with Structured Perturbations”. the values capture the underlying constructs and the appropriate population. ―David J. Hickson, Emeritus Professor of International Management & Organization… 1 Review . Organizational life today is often a complex social environment of confrontation, miscommunication, manipulation, hostility, and conflict. "Books and articles come and go, endlessly. We investigate how such concerns might influence conclusions concerning key issues such as prediction of job performance and adverse impact of selection procedures, as well as noting wider applications of these issues. Download Organisational Behaviour Notes, PDF, Books, Syllabus for MBA, BBA, BCOM 2021. 1 Review . structure and functions of an organization are independent of its environment. Figure 1 Structure of a traditional hierarchical organization:1 B. And any form can suffer from a variety of problems that develop because of the design itself. narrower spans of control, less delegation, and a more centralized structure. the discrepancy principle to determine $k^*$. The conceptual models of how to think about the structure and functioning of organizational culture, and … the general-form Tikhonov regularization problem: This implies the existence of a discrete maximum principle and thus monotonicity of the discretization. The Adequate Way . But a few do stick, and this book is such a one. even that new principles will be discovered and employed. appropriate, given that different businesses may face different environments. " �A�4H�&_��l���2ځH�~0� ��� �"e/�ٜ]��[ RDH��wbd������?��M? For the engineering part of the Facebook organizational structure, you can find software development, research and analysis and so on. Important Considerations Affecting Organization Structur. M. Armstrong (ed. W, assigned to the manufacturing function, sales perhaps in its own “arm,” and marketing perhaps as. •The organic structure works well for organizations pursuing meaningful and … s�j Our algorithm is different from the hybrid one proposed by Kilmer {\em et al. enough to be creative and highly coordinated. Chief Executive Officer (CEO) Mark Elliot Zuckerberg who born in 1980’s is an American computer programmer. II. among divisions of a functional or product form organization. Download Organisational Behaviour Notes, PDF, Books, Syllabus for MBA, BBA, BCOM 2021. organizational structure that will be used for the project. Organizational structure How individual and team work within an organization are coordinated. An M-matrix has a nonnegative inverse, i.e. Organizational structure is the framework of the relations on jobs, systems, operating process, people and groups making efforts to achieve the goals. Strategy and Structure –Changes in corporate strategy should lead to changes in an organization’s structure that support the strategy. But a few do stick, and this book is such a one. Functional organizational structure, also referred to as centralised structure is one of several reporting structures a company could implement. uncontrolled growth can be very expensive and even fatal. decentralize, for example, and how to structure the various functions in their firms. recruiting system must be refined to select only those who fit the needs of the organization. Lesson Structure: 1.1 Introduction 1.2 Definition of Management ... Drucker has stated in his famous book "The Practice of Management" that, ... organization under which the enterprise is to operate and the selection of the principal officers." There are only line departments-departments directly involved in accomplishing the primary goal of the organisation. Application 14-1 Healthways’ Process Structure 325 The Network Structure … the various systems operating within it, organizational effectiveness will be compromised. ResearchGate has not been able to resolve any references for this publication. Excel Books India, 2009 - Industrial management - 524 pages. This is because they reflect a natural and common technique for human beings to deal with complexity. (Source: Kelvin Hard, Developing the right organization, in Strategies for Human Resource Management. The Information Age is creating even more forms as we move ahead. creates leaders. When business problems emerge, signs often exist within the design or components of the organizational structure. The other is a line of project managers who are charged with the budget for a program. Basically the structure can be mechanistic or organic in nature or a combination of thereof. ... As this book pertains to management in criminal justice, a brief Create ... Portland International Conference on Management of Engineering and Technology. Executives are constantly. Business Organisation Book Pdf Download > DOWNLOAD c2ef32f23e Business Organisation And Management Book Pdf Free Download pdf download business organisation and management book pdf free pdf business organisation and management book pdf On this page you can read or download Business Organisation Book Pdf Download in scheduling and shorter production runs, which are expensive. decisions. boss” or the line-of-authority principle and are unable to function in a matrix structure. An organizational structure describes how a company, division, team, or other organization is built; how all of its various components fit together. Access scientific knowledge from anywhere. III. we propose and develop an iterative regularization algorithm for the large scale ��,b organizational structures that affect their behavior, but also the behavior of the people has an influence on the organizational structure. Organization Structure & Design : Applications And Challenges. In this type of organisational culture a dominant head sits in the centre surrounded by intimates and subordinates who are the dependants (Harrison, 1993). organizations tend to revolve around one or more of the seven previously outlined. }, This note introduces basic principles of organizational design and the advantages of several common organizational structures. The organizational structure also determines how information flows between levels within the company. The algorithm is simple and effective, and numerical experiments illustrate hThe organizational structure resembles a pyramid. functional array, with managers for engineering, manufacturing, sales, procurement, and so forth. In most cases, organizations evolve through structures when they progress through and enhance their processes and manpower. They argued that these factors imposed economic or other constrains on organizations that forced them to choose a certain structure over others. To provide easy access to my life's work through video clips and new text. ORGANIZATIONAL PROCESSES. We use the L-curve criterion or This is the definitive book on that practical science of organizational structure, the culmination of over 30 years of study and practical experiences implementing restructurings in dozens of diverse organizations. 906 0 obj <>stream 2 Perhaps the oldest and most common method of grouping related functions is by specialized function, such as … The first and second editions of this book attempted to show this connection, and I hope that I have been able to strengthen the connection even more in this third edition. 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# char_rnn¶
class hanlp.layers.embeddings.char_rnn.CharRNN(field, vocab_size, embed: Union[int, torch.nn.modules.sparse.Embedding], hidden_size)[source]
Character level RNN embedding module.
Parameters
• field – The field in samples this encoder will work on.
• vocab_size – The size of character vocab.
• embed – An Embedding object or the feature size to create an Embedding object.
• hidden_size – The hidden size of RNNs.
Defines the computation performed at every call.
Should be overridden by all subclasses.
Note
Although the recipe for forward pass needs to be defined within this function, one should call the Module instance afterwards instead of this since the former takes care of running the registered hooks while the latter silently ignores them.
class hanlp.layers.embeddings.char_rnn.CharRNNEmbedding(field, embed, hidden_size, max_word_length=None)[source]
Character level RNN embedding module builder.
Parameters
• field – The field in samples this encoder will work on.
• embed – An Embedding object or the feature size to create an Embedding object.
• hidden_size – The hidden size of RNNs.
• max_word_length – Character sequence longer than max_word_length will be truncated.
module(vocabs: hanlp.common.transform.VocabDict, **kwargs) Optional[torch.nn.modules.module.Module][source]
Build a module for this embedding.
Parameters
**kwargs – Containing vocabs, training etc. Not finalized for now.
Returns
A module.
transform(vocabs: hanlp.common.transform.VocabDict, **kwargs) [source]
Build a transform function for this embedding.
Parameters
**kwargs – Containing vocabs, training etc. Not finalized for now.
Returns
A transform function.
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<lift:loc locid="stock.discuss"></lift:loc>
Generic mean curvature flow I; generic singularities
2009.08.26
http://arxiv.org/abs/0908.3788
It has long been conjectured that starting at a generic smooth closed embedded surface in R^3, the mean curvature flow remains smooth until it arrives at a singularity in a neighborhood of which the flow looks like concentric spheres or cylinders. That is, the only singularities of a generic flow are spherical or cylindrical. We will address this conjecture here and in a sequel. The higher dimensional case will be addressed elsewhere. The key in showing this conjecture is to show that shrinking spheres, cylinders and planes are the only stable self-shrinkers under the mean curvature flow. We prove this here in all dimensions. An easy consequence of this is that every other singularity than spheres and cylinders can be perturbed away.
Discussions
• Pls. be polite and constructive.
• You can input La|TeX for math formulas. E.g. $$x = {-b \pm \sqrt{b^2-4ac} \over 2a}$$
• Any attachment files should still be uploaded to arXiv.org
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+1 vote
51 views
Let $P(x)$ be the statement “x spends more than five hours every weekday in class.” where the domain for x consists of all students. Express each of these qualifications in English.
1. $\exists x P(x)$
2. $\forall x P(x)$
3. $\exists x \sim p(x)$
4. $\forall x \sim P(x)$
| 51 views
+1 vote
P(x) is “spends more than five hours every weekday in class”
a) xP(x) "There exist a student who spends more than five hours every weekday in class"
b) xP(x) "All students spend more than five hours evert weekday in class"
c) ¬P(x) "There exist a student who does not spend more than five hours every weekday in class"
d) ¬P(x) "Every Student don't spend more than 5 hours every weekday in class"
by Boss (35.4k points)
0
I think option D can be written like this
d) "There is no student who spends more than five hours every weekday in class."
0
that would be ¬∃xP(x) which is correct when we apply de morgan's law
+1 vote
1. $\exists x P(x)$
There exists a student who spends more than five hours every weekday in class.
2. $\forall x P(x)$
All students spends more than five hours every weekday in class.
3. $\exists x \sim P(x)$
There exists a student who do not spend more than five hours every weekday in class.
4. $\forall x \sim P(x)$
All students do not spend more than five hours every weekday in class.
by Boss (35.4k points)
+1 vote
.......
by Boss (35.3k points)
a.∃xP(x)
There is a student who spends more than five hours every weekday in class.
b.∀xP(x)
All student spends more than five hours every weekday in class.
c.∃x∼P(x)
There is a student who does not spend more than five hours every weekday in class.
d.∀x∼P(x)
All students do not spend more than five hours every weekday in class.
Another way to write the same statement as "No student spends more than five hours every weekday in class"
by (235 points)
+1 vote
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Functional equation
There is given the functional equation: $f(2x)+f(1/2)=f(x)+f(x+1/2)$ for $x \in [0,1/2]$. We also know that $f(0)=-1$ and $f(1)=1$. Additionally, we assume the continuity and strict monotonicity of $f$. Is it possible to get any information on $f(1/2)$? In particular, can we prove that $f(1/2)=0$?
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# What are the various techniques for detecting Walls in the Building(Architectural) floor plan Images?
I have to process building floor plan images to extract Walls from the structure. It is trivial in case of binary images; but the images in our case are colored, and each have different color of walls. This restricts us from using any thresholding operation as we cannot assume the Walls will always be the darkest. How possibly can I come up with an algorithm that can work on all sort of Images?
• You can't come up with a universal algorithm, 'cause who knows what all the different types of plans will look like. Restrict the problem to the types of plans you know about. Write algorithms that work for most of them and exceptions for those that don't. As for colour, why not segment using both intensity AND colour? – geometrikal May 23 '14 at 13:07
• All Images are NOT different. The thing which is common is all images is that they all have walls, and these compose of major portion of images. What I'm asking is that Is there an algorithm that exploits this quality of Image? – shreelock May 23 '14 at 14:12
• LOL ok. By "all sort of Images" you mean "all THESE sorts of images", that is, plans with colored walls. Sorry. You need a line detector, or perhaps Canny edge detector. Post an example image. – geometrikal May 23 '14 at 14:44
• Depends what line/edge detector you use and what further processing is applied. For example, limit it to long straight lines. Do you have an example image? – geometrikal May 26 '14 at 8:42
• Could you upload a sample floor plan image? That'll give us a better understanding of your problem – Shravya Boggarapu Feb 28 '18 at 13:37
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# Giving randomly 5 red balls and 5 blue balls to 5 kids, each kid get 2 balls. What is the expected value of number of kids got 2 different balls?
Giving randomly 5 red balls and 5 blue balls to 5 kids, each kid get 2 balls. What is the expected value of number of kids got 2 different balls?
I solve it by:
Let $X_i$ be random variable that holds whether kid $i$ got different balls (i.e. red and blue), or same balls (i.e. red and red or blue and blue).
$X_i = 1 \iff$ got red and blue. $X_i = 0 \iff$ got blue and blue or red and red.
Get probabilities: $P(X_i = 0) = red\cdot red + blue\cdot blue = \frac{1}{2}\frac{1}{2} + \frac{1}{2}\frac{1}2{} = \frac{1}{4} + \frac{1}{4} = \frac{1}{2}.$
From here, $P(X_i = 1) = 1 - P(X_i = 0) = \frac{1}{2}$.
Find $E(X_i = 1) =\sum_1^5 X_i\cdot P(X_i = 1) = 5\cdot \frac{1}{2} = \frac{5}{2}$.
It's not the answer (got 0/9 on the exam sheet).
• Note that given your initial conditions it is impossible to get RR BB for all of the kids. At least one kid will end up with two different colored balls. – adam May 13 '15 at 16:56
Your approach with indicator variables using linearity of expectation went exactly in the right direction. You just used the wrong probabilities. Give the first child a ball; no matter which colour it is, $5$ of the remaining $9$ balls are of the other colour, so the probability that the second ball you give them is of the other colour is $\frac59$. Thus the expected number of kids with different colours is $5\cdot\frac59=\frac{25}9$.
I solved it with some brute force but I found it to be the easiest way. If there were a higher number of balls and kids It would be better to use some algebra.
You can have $5$, $3$ or $1$ kid(s) with different colored balls. You only need to count in how many ways each can happen.
For {{R,B},{R,B},{R,B},{R,B},{R,B}} there's only one possibility.
For {{R,B},{R,B},{R,B},{R,R},{B,B}} there's $5\times 4=20$ options (5 possible kids who have the red balls, then 4 possible kids who have the blue balls).
For {{R,B},{R,R},{R,R},{B,B},{B,B}} there's $5\times 6=30$ options (5 possible kids who have different balls and $4\choose 2$ to order the other 4).
All together you have $51$ possibilities. Now let's calculate the mean of kids with different boys:
$$\mu = 5\times\frac{1}{51}+3\times\frac{20}{51}+1\times\frac{30}{51}= \frac{5+3\times 20+30}{51}=\frac{95}{51}\\ \mu \approx 1.863$$
Hope it helped you.
• These $51$ outcomes are not equiprobable, so you can't calculate a probability as the fraction of favourable outcomes. There are $10!$ elementary outcomes (the permutations of the $10$ balls), so the denominator of the expected value of an integer can't contain a factor of $17$. – joriki Jun 19 '16 at 11:21
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# YOLOv4
YOLOv4 is an object detection model that is included in the TAO Toolkit. YOLOv4 supports the following tasks:
• kmeans
• train
• evaluate
• inference
• prune
• export
These tasks can be invoked from the TAO Toolkit Launcher using the following convention on the command line:
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tao yolo_v4 <sub_task> <args_per_subtask>
where args_per_subtask are the command line arguments required for a given subtask. Each subtask is explained in detail below.
## Creating a Configuration File
Below is a sample for the YOLOv4 spec file. It has 6 major components: yolov4_config, training_config, eval_config, nms_config, augmentation_config, and dataset_config. The format of the spec file is a protobuf text (prototxt) message, and each of its fields can be either a basic data type or a nested message. The top-level structure of the spec file is summarized in the table below.
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random_seed: 42
yolov4_config {
big_anchor_shape: "[(114.94, 60.67), (159.06, 114.59), (297.59, 176.38)]"
mid_anchor_shape: "[(42.99, 31.91), (79.57, 31.75), (56.80, 56.93)]"
small_anchor_shape: "[(15.60, 13.88), (30.25, 20.25), (20.67, 49.63)]"
box_matching_iou: 0.25
matching_neutral_box_iou: 0.5
arch: "resnet"
nlayers: 18
arch_conv_blocks: 2
loss_loc_weight: 1.0
loss_neg_obj_weights: 1.0
loss_class_weights: 1.0
label_smoothing: 0.0
big_grid_xy_extend: 0.05
mid_grid_xy_extend: 0.1
small_grid_xy_extend: 0.2
freeze_bn: false
freeze_blocks: 0
force_relu: false
}
training_config {
batch_size_per_gpu: 8
num_epochs: 80
enable_qat: false
checkpoint_interval: 10
learning_rate {
soft_start_cosine_annealing_schedule {
min_learning_rate: 1e-7
max_learning_rate: 1e-4
soft_start: 0.3
}
}
regularizer {
type: L1
weight: 3e-5
}
optimizer {
epsilon: 1e-7
beta1: 0.9
beta2: 0.999
}
}
pretrain_model_path: "EXPERIMENT_DIR/pretrained_resnet18/tlt_pretrained_object_detection_vresnet18/resnet_18.hdf5"
}
eval_config {
average_precision_mode: SAMPLE
batch_size: 8
matching_iou_threshold: 0.5
}
nms_config {
confidence_threshold: 0.001
clustering_iou_threshold: 0.5
top_k: 200
}
augmentation_config {
hue: 0.1
saturation: 1.5
exposure:1.5
vertical_flip:0
horizontal_flip: 0.5
jitter: 0.3
output_width: 1248
output_height: 384
output_channel: 3
randomize_input_shape_period: 100
mosaic_prob: 0.5
mosaic_min_ratio:0.2
image_mean {
key: 'b'
value: 103.9
}
image_mean {
key: 'g'
value: 116.8
}
image_mean {
key: 'r'
value: 123.7
}
}
dataset_config {
data_sources: {
tfrecords_path: "/workspace/tao-experiments/data/training/tfrecords/<tfrecords pattern>"
image_directory_path: "/workspace/tao-experiments/data/training"
}
include_difficult_in_training: true
image_extension: "png"
target_class_mapping {
key: "car"
value: "car"
}
target_class_mapping {
key: "pedestrian"
value: "pedestrian"
}
target_class_mapping {
key: "cyclist"
value: "cyclist"
}
target_class_mapping {
key: "van"
value: "car"
}
target_class_mapping {
key: "person_sitting"
value: "pedestrian"
}
validation_data_sources: {
tfrecords_path: "/workspace/tao-experiments/data/val/tfrecords/<tfrecords pattern>"
image_directory_path: "/workspace/tao-experiments/data/val"
}
}
### Training Config
The training configuration (training_config) defines the parameters needed for training, evaluation, and inference. Details are summarized in the table below.
Field Description Data Type and Constraints Recommended/Typical Value batch_size_per_gpu The batch size for each GPU; the effective batch size is batch_size_per_gpu * num_gpus Unsigned int, positive – checkpoint_interval The number of training epochs per one model checkpoint/validation Unsigned int, positive 10 num_epochs The number of epochs to train the network Unsigned int, positive. – enable_qat Whether to use quantization-aware training Boolean Note: YOLOv4 does not support loading a pruned QAT model and retraining it with QAT disabled, or vice versa. For example, to get a pruned QAT model, perform the initial training with QAT enabled or enable_qat=True. learning_rate One soft_start_annealing_schedule and soft_start_cosine_annealing_schedule with the following nested parameters are supported: min_learning_rate: The minimum learning late during the entire experiment max_learning_rate: The maximum learning rate during the entire experiment soft_start: The time to lapse before warm up (expressed as a percentage of progress between 0 and 1) annealing: (only for soft_start_annealing_schedule) The time to start annealing the learning rate Message type – regularizer This parameter configures the regularizer to use while training and contains the following nested parameters: type: The type of regularizer to use. NVIDIA supports NO_REG, L1, or L2. weight: The floating point value for regularizer weight Message type L1 (Note: NVIDIA suggests using the L1 regularizer when training a network before pruning, as L1 regularization makes the network weights more prunable.) optimizer The optimizer can be one of adam, sgd, and rmsprop. Each type has the following parameters: adam: epsilon, beta1, beta2, amsgrad sgd: momentum, nesterov rmsprop: rho, momentum, epsilon, centered The meanings of above parameters are same as those in Keras. Message type – pretrain_model_path The path to the pretrained model, if any At most, one pretrain_model_path, resume_model_path, and pruned_model_path may be present. String – resume_model_path The path to the TAO checkpoint model to resume training, if any At most, one pretrain_model_path, resume_model_path, and pruned_model_path may be present. String – pruned_model_path The path to the TAO pruned model for re-training, if any At most, one pretrain_model_path, resume_model_path, and pruned_model_path may be present. String – max_queue_size The number of prefetch batches in data loading Unsigned int, positive – n_workers The number of workers for data loading per GPU Unsigned int, positive – use_multiprocessing Whether to use multiprocessing mode of keras sequence data loader Boolean true (in case of deadlock, restart training and use False)
Note
The learning rate is automatically scaled with the number of GPUs used during training, or the effective learning rate is learning_rate * n_gpu.
### Evaluation Config
The evaluation configuration (eval_config) defines the parameters needed for the evaluation either during training or standalone evaluation. Details are summarized in the table below.
Field Description Data Type and Constraints Recommended/Typical Value average_precision_mode Average Precision (AP) calculation mode can be either SAMPLE or INTEGRATE. SAMPLE is used as VOC metrics for VOC 2009 or before. INTEGRATE is used for VOC 2010 or after. ENUM type ( SAMPLE or INTEGRATE) SAMPLE matching_iou_threshold The lowest IoU of the predicted box and ground truth box that can be considered a match float 0.5
### NMS Config
The NMS configuration (nms_config) defines the parameters needed for the NMS postprocessing. NMS config applies to the NMS layer of the model in training, validation, evaluation, inference, and export. Details are summarized in the table below.
Field Description Data Type and Constraints Recommended/Typical Value confidence_threshold Boxes with a confidence score less than confidence_threshold are discarded before applying NMS. float 0.01 cluster_iou_threshold The IoU threshold below which boxes will go through the NMS process. float 0.6 top_k top_k boxes will be output after the NMS keras layer. If the number of valid boxes is less than k, the returned array will be padded with boxes whose confidence score is 0. Unsigned int 200 infer_nms_score_bits The number of bits to represent the score values in NMS plugin in TensorRT OSS. The valid range is integers in [1, 10]. Setting it to any other values will make it fall back to ordinary NMS. Currently this optimized NMS plugin is only avaible in FP16 but it should also be selected by INT8 data type as there is no INT8 NMS in TensorRT OSS and hence this fastest implementation in FP16 will be selected. If falling back to ordinary NMS, the actual data type when building the engine will decide the exact precision(FP16 or FP32) to run at. int. In the interval [1, 10]. 0 force_on_cpu A flag to force NMS to run on CPU. Setting it to True will force NMS to run on CPU during training. This is useful when using TFRecord dataset for validation during training since there is a known issue with TensorFlow NMS on GPU when using TFRecord dataset for validation. Note Note that this flag does not have any impact on TAO export and TensorRT/DeepStream inference. Boolean False
### Augmentation Config
The augmentation configuration (augmentation_config) defines the parameters needed for online data augmentation. Details are summarized in the table below.
Field Description Data Type and Constraints Recommended/Typical Value hue Image hue to be changed within [-hue, hue] * 180.0 float of [0, 1] 0.1 saturation Image saturation to be changed within [1.0 / saturation, saturation] times float >= 1.0 1.5 exposure Image exposure to be changed within [1.0 / exposure, exposure] times float >= 1.0 1.5 vertical_flip The probability of images to be vertically flipped float of [0, 1] 0 horizontal_flip The probability of images to be horizontally flipped float of [0, 1] 0.5 jitter The maximum jitter allowed in augmentation; “jitter” here refers to jitter augmentation in YOLO networks float of [0, 1] 0.3 output_width The base output image width of augmentation pipeline integer, multiple of 32 – output_height The base output image height of augmentation pipeline integer, multiple of 32 – output_channel The number of output channels of augmentation pipeline 1 or 3 – randomize_input_shape_period The batch interval to randomly change the output width and height. For value K, the augmentation pipeline will adjust output shape per K batches, and the adjusted output width/height will be within 0.6 to 1.5 times of the base width/height. Note: If K=0, the output width/height will always be the exact base width/height as configured, and training will be much faster. But the accuracy of the trained network might not be as good. non-negative integer 10 mosaic_prob The probability of mosaic augmentation to be applied on one image float of [0, 1] 0.5 mosaic_min_ratio The minimum ratio of width/height one sub-image should occupy float of (0, 0.5) 0.2 image_mean A key/value pair to specify image mean values. If omitted, ImageNet mean will be used for image preprocessing. If set, depending on output_channel, either ‘r/g/b’ or ‘l’ key/value pair must be configured. dict –
### Dataset Config
YOLOv4 supports two data formats: the sequence format (KITTI images folder and raw labels folder) and the tfrecords format (KITTI images folder and TFRecords). From our experience, if mosaic augmentation is disabled (mosaic_prob=0), training with TFRecords format is faster. If mosaic augmentation is enabled (mosaic_prob>0), training with sequence format is faster. The train and evaluate command will determine the data format based on your dataset_config.
The YOLOv4 dataloader assumes the training/validation split is already done and the data is prepared in KITTI format: images and labels are in two separate folders, where each image in the image folder has a .txt label file with the same filename in the label folder, and the label file content follows KITTI format.
#### Sequence format
The following is an example dataset_config element if you want to use sequence format:
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dataset_config {
data_sources: {
label_directory_path: "/workspace/tao-experiments/data/training/label_2"
image_directory_path: "/workspace/tao-experiments/data/training/image_2"
}
data_sources: {
label_directory_path: "/workspace/tao-experiments/data/training/label_3"
image_directory_path: "/workspace/tao-experiments/data/training/image_3"
}
include_difficult_in_training: true
target_class_mapping {
key: "car"
value: "car"
}
target_class_mapping {
key: "pedestrian"
value: "pedestrian"
}
target_class_mapping {
key: "cyclist"
value: "cyclist"
}
target_class_mapping {
key: "van"
value: "car"
}
target_class_mapping {
key: "person_sitting"
value: "pedestrian"
}
validation_data_sources: {
label_directory_path: "/workspace/tao-experiments/data/val/label_1"
image_directory_path: "/workspace/tao-experiments/data/val/image_1"
}
}
The parameters in dataset_config are defined as follows:
• data_sources: The path to datasets to train on. If you have multiple data sources for training, you may use multiple data_sources. This field contains 2 parameters:
• label_directory_path: The path to the data source label folder.
• image_directory_path: The path to the data source image folder.
• include_difficult_in_training: A flag specifying whether to include difficult boxes in training. If set to false, difficult boxes will be ignored. Difficult boxes are those with non-zero occlusion levels in KITTI labels.
• target_class_mapping: This parameter maps the class names in the labels to the target class to be trained in the network. An element is defined for every source class to target class mapping. This field was included with the intention of grouping similar class objects under one umbrella. For example, “car”, “van”, “heavy_truck”, etc. may be grouped under “automobile”. The “key” field is the value of the class name in the tfrecords file, and the “value” field corresponds to the value that the network is expected to learn.
• validation_data_sources: Captures the path to datasets to validate on. If you have multiple data sources for validation, you may use multiple validation_data_sources. Like data_sources, this field contains two parameters.
Note
The class names key in the target_class_mapping must be identical to the one shown in the KITTI labels so that the correct classes are picked up for training.
#### TFRecords format
TFRecords format requires tfrecords for all labels. This requires running of tao yolo_v4 dataset-convert command. The command has same functionality and argument requirements as that of detectnet_v2 and for details of how to generate tfrecords, check Pre-processing the Dataset in detectnet_v2.
The following is an example dataset_config element if you want to use tfrecords format. Here, we assume your tfrecords are all generated under a folder called tfrecords, which is under same parent folder with images and labels:
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dataset_config {
data_sources: {
tfrecords_path: "/workspace/tao-experiments/data/training/tfrecords/<tfrecords pattern>"
image_directory_path: "/workspace/tao-experiments/data/training"
}
include_difficult_in_training: true
image_extension: "png"
target_class_mapping {
key: "car"
value: "car"
}
target_class_mapping {
key: "pedestrian"
value: "pedestrian"
}
target_class_mapping {
key: "cyclist"
value: "cyclist"
}
target_class_mapping {
key: "van"
value: "car"
}
target_class_mapping {
key: "person_sitting"
value: "pedestrian"
}
validation_data_sources: {
tfrecords_path: "/workspace/tao-experiments/data/val/tfrecords/<tfrecords pattern>"
image_directory_path: "/workspace/tao-experiments/data/val"
}
}
The parameters in dataset_config are defined as follows:
• data_sources: The path to datasets to train on. If you have multiple data sources for training, you may use multiple data_sources. This field contains 2 parameters:
• tfrecords_path: The path to the data source tfrecords.
• image_directory_path: The path to the root directory containing the image folder.
• image_extension: Image extensions of images contained in the image folder. Note, to use tfrecords format, all images must have same extensions and currently we support jpg and png
• validation_data_sources: Captures the path to datasets to validate on. This field contains two parameters same as data_sources. If you have multiple data sources for validation, you may use multiple validation_data_sources.
All other fields are same as those in sequence format dataset_config.
### YOLO4 Config
The YOLOv4 configuration (yolov4_config) defines the parameters needed for building the YOLOv4 model. Details are summarized in the table below.
Field Description Data Type and Constraints Recommended/Typical Value big_anchor_shape, mid_anchor_shape, and small_anchor_shape These settings should be 1-d arrays inside quotation marks. The elements of those arrays are tuples representing the pre-defined anchor shape in the order of “width, height”. The default YOLOv4 configuration has nine predefined anchor shapes. They are divided into three groups corresponding to big, medium, and small objects. The detection output corresponding to different groups are from different depths in the network. You should run the kmeans command (tao yolo_v4 kmeans) to determine the best anchor shapes for your dataset and put those anchor shapes in the spec file. It is worth noting that the number of anchor shapes for any field is not limited to three; you only need to specify one anchor shape in each of those three fields. string Use the tao yolo_v4 kmeans command to generate those shapes box_matching_iou This field should be a float number between 0 and 1. Any anchor with at least this IoU to any ground truth boxes will be matched to the ground truth box it has the largest IoU with. In contrast with YOLOv3, one ground truth box might match to multiple anchors in YOLOv4. float 0.5 matching_neutral_box_iou This field should be a float number between 0.25 and 1. Any inferred bounding box with at least this IoU to any ground truth boxes will not be treated as negative box and will be assigned 0 for its negative objectiveness loss (neutral box) float 0.5 arch_conv_blocks Supported values are 0, 1, and 2. This value controls how many convolutional blocks are present among detection output layers. Set this value to 2 if you want to reproduce the meta architecture of the original YOLOv4 model paired with DarkNet 53. Note that this config setting only controls the size of the YOLO meta-architecture–the size of the feature extractor has nothing to do with this config field. 0, 1 or 2 2 loss_loc_weight, loss_neg_obj_weights, and loss_class_weights These loss weights can be configured as float numbers. The YOLOv4 loss is a summation of localization loss, negative objectiveness loss, positive objectiveness loss, and classification loss. The weight of positive objectiveness loss is set to 1, while the weights of other losses are read from the config file. float loss_loc_weight: 5.0 loss_neg_obj_weights: 50.0 loss_class_weights: 1.0 label_smoothing Label smoothing applied to classification loss. float of [0, 0.3] 0, 0.1, 0.2 big_grid_xy_extend, mid_grid_xy_extend, and small_grid_xy_extend These settings should be small positive floats. The calculated box center relative to the anchor box will be re-calibrated according to following: center_xy = calculated_xy * (grid_xy_extend + 1.0) - grid_xy_extend / 2.0 The default YOLOv4 has nine predefined anchor shapes. They are divided into three groups corresponding to big, medium, and small objects. The detection output corresponding to different groups are from different depths in the network. The three different grid_xy_extend configs allow users to define different grid_xy_extend values for different anchor-shape groups. The grid_xy_extend settings make it easier for the network to propose an inferenced box with a center that is close to or on the anchor border. float of [0, 0.3] 0.05, 0.1, 0.2 arch The backbone for feature extraction. Currently, “resnet”, “vgg”, “darknet”, “googlenet”, “mobilenet_v1”, “mobilenet_v2”, “cspdarknet”, and “squeezenet” are supported. string resnet activation The activation type used in YOLOv4 CSPDarkNet backbone. Only “relu”, “leaky_relu” and “mish” are supported. For other backbones, this parameter is not useful. string “mish” nlayers The number of conv layers in a specific architecture. For “resnet”, 10, 18, 34, 50 and 101 are supported. For “vgg”, 16 and 19 are supported. For “darknet” or “cspdarknet”, 19 and 53 are supported. All other networks don’t have this configuration, in which case you should just delete this config from the config file. Unsigned int – freeze_bn A flag specifying whether to freeze all batch normalization layers during training. Boolean False freeze_blocks The list of block IDs to be frozen in the model during training. You can choose to freeze some of the CNN blocks in the model to make the training more stable and/or easier to converge. The definition of a block is heuristic for a specific architecture (for example, by stride or by logical blocks in the model). However, the block ID numbers identify the blocks in the model in a sequential order so you don’t have to know the exact locations of the blocks when you do training. A general principle to keep in mind is that the smaller the block ID, the closer it is to the model input; the larger the block ID, the closer it is to the model output. You can divide the whole model into several blocks and optionally freeze a subset of it. Note that for FasterRCNN, you can only freeze the blocks that are before the ROI pooling layer. Any layer after the ROI pooling layer will not be frozen anyway. For different backbones, the number of blocks and the block ID for each block are different. It deserves some detailed explanations on how to specify the block IDs for each backbone. list(repeated integers) ResNet series. For the ResNet series, the block IDs valid for freezing is any subset of [0, 1, 2, 3] (inclusive) VGG series. For the VGG series, the block IDs valid for freezing is any subset of[1, 2, 3, 4, 5] (inclusive) GoogLeNet. For the GoogLeNet, the block IDs valid for freezing is any subset of[0, 1, 2, 3, 4, 5, 6, 7] (inclusive) MobileNet V1. For the MobileNet V1, the block IDs valid for freezing is any subset of [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11] (inclusive) MobileNet V2. For the MobileNet V2, the block IDs valid for freezing is any subset of [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13] (inclusive) DarkNet. For the DarkNet 19 and DarkNet 53, the block IDs valid for freezing is any subset of [0, 1, 2, 3, 4, 5] (inclusive) – force_relu A flag specifying whether to replace all activation functions with ReLU. This is useful for training models for NVDLA. Boolean False
## Generate anchor shape
The anchor shape should match most ground truth boxes in the dataset to help the network learn bounding boxes. The YOLOv4 paper proposes using the kmeans algorithm to get the anchor shapes, and the tao yolo_v4 kmeans command is implemented in the TAO algorithm. You should use the output as your anchor shape in the yolov4_config spec file.
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tao yolo_v4 kmeans [-h] -l <label_folders>
-i <image_folders>
-x <network base input width>
-y <network base input height>
[-n <num_clusters>]
[--max_steps <kmeans max steps>]
[--min_x <ignore boxes with width less than this value>]
[--min_y <ignore boxes with height less than this value>]
### Required Arguments
• -l: Paths to the training label folders. Multiple folder paths should be separated with spaces.
• -i: Paths to corresponding training image folders. Folder counts and orders must match label folders.
• -x: The base network input width. This should be the output_width in the augmentation config part of your spec file.
• -y: The base network input height. This should be the output_height in the augmentation config part of your spec file.
### Optional Arguments
• -n: The number of shape clusters. This defines how many shape centers the command will output. The default is 9 (3 per group and 3 groups).
• --max_steps: The max number of steps the kmeans algorithm should run. If the algorithm does not converge at this step, a suboptimal result will be returned. The default value is 10000.
• --min_x: Ignore ground truth boxes with width less than this value in a reshaped image (images are first reshaped to network base shape as -x, -y).
• --min_y: Ignore ground truth boxes with height less than this value in a reshaped image (images are first reshaped to network base shape as -x, -y).
• -h, --help: Show this help message and exit.
## Training the Model
Train the YOLOv4 model using this command:
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tao yolo_v4 train [-h] -e <experiment_spec>
-r <output_dir>
-k <key>
[--gpus <num_gpus>]
[--gpu_index <gpu_index>]
[--use_amp]
[--log_file <log_file_path>]
### Required Arguments
• -r, --results_dir: The path to the folder where the experiment output is written.
• -k, --key: The encryption key to decrypt the model.
• -e, --experiment_spec_file: The experiment specification file to set up the evaluation experiment. This should be the same as the training-specification file.
### Optional Arguments
• --gpus: The number of GPUs to use for training in a multi-GPU scenario (default: 1).
• --gpu_index: The GPU indices used to run the training. You can specify the indices of GPUs to use for training when the machine has multiple GPUs installed.
• --use_amp: A flag to enable AMP training.
• --log_file: THe path to the log file. The default path is stdout.
• -h, --help: Show this help message and exit.
### Input Requirement
• Input size: C * W * H (where C = 1 or 3, W >= 128, H >= 128, W, H are multiples of 32)
• Image format: JPG, JPEG, PNG
• Label format: KITTI detection
### Sample Usage
Here’s an example of using the train command on a YOLOv4 model:
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## Re-training the Pruned Model
Once the model has been pruned, there might be a slight decrease in accuracy because some previously useful weights may have been removed. To regain accuracy, we recommend retraining the pruned model over the same dataset. To do this, use the tao yolo_v4 train command as documented in Training the model, with an updated spec file that points to the newly pruned model as the pruned_model_path.
We recommend turning off the regularizer in the training_config for detectnet to recover the accuracy when retraining a pruned model. You may do this by setting the regularizer type to NO_REG as mentioned in the Training Config section. All other parameters in the spec file can be carried over from the previous training.
## Exporting the Model
Exporting the model decouples the training process from inference and allows conversion to TensorRT engines outside the TAO environment. TensorRT engines are specific to each hardware configuration and should be generated for each unique inference environment. The exported model may be used universally across training and deployment hardware. The exported model format is referred to as .etlt. Like .tlt, the .etlt model format is also an encrypted model format with the same key of the .tlt model that it is exported from. This key is required when deploying this model.
### INT8 Mode Overview
TensorRT engines can be generated in INT8 mode to improve performance, but require a calibration cache at engine creation-time. If tao yolo_v4 export is run with the --data_type flag set to int8, the calibration cache is generated using a calibration tensor file. Pre-generating the calibration information and caching it removes the need for calibrating the model on the inference machine. Moving the calibration cache is usually much more convenient than moving the calibration tensorfile since it is a much smaller file and can be moved with the exported model. Using the calibration cache also speeds up engine creation, as building the cache can take several minutes depending on the size of the Tensorfile and the model itself.
The export tool can generate an INT8 calibration cache by ingesting training data using either of these options:
• Option 1: Using the training data loader to load the training images for INT8 calibration. This option is now the recommended approach to support multiple image directories by leveraging the training dataset loader. This also ensures two important aspects of data during calibration:
• Data pre-processing in the INT8 calibration step is the same as in the training process.
• The data batches are sampled randomly across the entire training dataset, thereby improving the accuracy of the INT8 model.
• Option 2: Pointing the tool to a directory of images that you want to use to calibrate the model. For this option, make sure to create a sub-sampled directory of random images that best represent your training dataset.
### FP16/FP32 Model
The calibration.bin is only required if you need to run inference at INT8 precision. For FP16/FP32 based inference, the export step is much simpler. All that is required is to provide a .tlt model from the training/retraining step to be converted into .etlt.
### Exporting the Model
Here’s an example of the command line arguments of the tao yolo_v4 export command:
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tao yolo_v4 export [-h]
-m <path to the .tlt model file generated by tao train>
-k <key>
[-o <path to output file>]
[--cal_data_file <path to tensor file>]
[--cal_image_dir <path to the directory images to calibrate the model]
[--cal_cache_file <path to output calibration file>]
[--data_type <Data type for the TensorRT backend during export>]
[--batches <Number of batches to calibrate over>]
[--max_batch_size <maximum trt batch size>]
[--max_workspace_size <maximum workspace size]
[--batch_size <batch size to TensorRT engine>]
[--experiment_spec <path to experiment spec file>]
[--engine_file <path to the TensorRT engine file>]
[--gen_ds_config]
[--verbose]
[--strict_type_constraints]
[--force_ptq]
[--gpu_index <gpu_index>]
[--log_file <log_file_path>]
#### Required Arguments
• -m, --model: The path to the .tlt model file to be exported.
• -k, --key: The key used to save the .tlt model file.
• -e, --experiment_spec: The path to the spec file.
#### Optional Arguments
• -h, --help: Show this help message and exit.
• -o, --output_file: The path to save the exported model to. The default path is ./<input_file>.etlt.
• --data_type: The desired engine data type. The options are {fp32, fp16, int8} The default value is fp32. A calibration cache is generated in INT8 mode. If using INT8 mode, the following INT8 arguments are required.
• --gen_ds_config: A Boolean flag indicating whether to generate the template DeepStream related configuration (“nvinfer_config.txt”) as well as a label file (“labels.txt”) in the same directory as the output_file. Note that the config file is NOT a complete configuration file and requires the user to update the sample config files in DeepStream with the parameters generated.
• -s, --strict_type_constraints: A Boolean flag indicating whether to apply the TensorRT strict type constraints when building the TensorRT engine.
• --gpu_index: The index of the (discrete) GPU for exporting the model if the machine has multiple GPUs installed. Note that export can only run on a single GPU.
• --log_file: The path to the log file. The default path is stdout.
### INT8 Export Mode Required Arguments
• --cal_data_file: The tensorfile generated for calibrating the engine. This can also be an output file if used with --cal_image_dir.
• --cal_image_dir: A directory of images to use for calibration.
Note
The --cal_image_dir parameter for images applies the necessary preprocessing to generate a tensorfile at the path mentioned in the --cal_data_file parameter, which is in turn used for calibration. The number of generated batches in the tensorfile is obtained from the --batches parameter value, and the batch_size is obtained from the --batch_size parameter value. Ensure that the directory mentioned in --cal_image_dir has at least batch_size * batches number of images in it. The valid image extensions are .jpg, .jpeg, and .png. In this case, the input_dimensions of the calibration tensors are derived from the input layer of the .tlt model.
### INT8 Export Optional Arguments
• --cal_cache_file: The path to save the calibration cache file to. The default value is ./cal.bin.
• --batches: The number of batches to use for calibration and inference testing. The default value is 10.
• --batch_size: The batch size to use for calibration. The default value is 8.
• --max_batch_size: The maximum batch size of the TensorRT engine. The default value is 16.
• --max_workspace_size: The maximum workspace size of TensorRT engine. The default value is 1073741824 = 1<<30
• --engine_file: The path to the serialized TensorRT engine file. Note that this file is hardware specific, and cannot be generalized across GPUs. It is useful to quickly test your model accuracy using TensorRT on the host. As TensorRT engine file is hardware specific, you cannot use this engine file for deployment unless the deployment GPU is identical to the training GPU.
• --force_ptq: A boolean flag to force post training quantization on the exported etlt model.
Note
When exporting a model trained with QAT enabled, the tensor scale factors to calibrate the activations are peeled out of the model and serialized to a TensorRT readable cache file defined by the cal_cache_file argument. However, note that the current version of QAT doesn’t natively support DLA INT8 deployment on Jetson. To deploy this model on a Jetson with DLA int8, use the --force_ptq flag for TensorRT post-training quantization to generate the calibration cache file.
### Sample usage
The following is a sample command to export a YOLOv4 model in INT8 mode:
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tao yolo_v4 export -m /workspace/yolov4_resnet18_epoch_100.tlt \
-o /workspace/yolov4_resnet18_epoch_100_int8.etlt \
-e /workspace/yolov4_retrain_resnet18_kitti.txt \
-k $KEY \ --cal_image_dir /workspace/data/training/image_2 \ --data_type int8 \ --batch_size 8 \ --batches 10 \ --cal_cache_file /export/cal.bin \ --cal_data_file /export/cal.tensorfile ## Deploying to DeepStream The deep learning and computer vision models that you’ve trained can be deployed on edge devices, such as a Jetson Xavier or Jetson Nano, a discrete GPU, or in the cloud with NVIDIA GPUs. TAO Toolkit has been designed to integrate with DeepStream SDK, so models trained with TAO Toolkit will work out of the box with DeepStream SDK. DeepStream SDK is a streaming analytic toolkit to accelerate building AI-based video analytic applications. This section will describe how to deploy your trained model to DeepStream SDK. To deploy a model trained by TAO Toolkit to DeepStream we have two options: • Option 1: Integrate the .etlt model directly in the DeepStream app. The model file is generated by export. • Option 2: Generate a device specific optimized TensorRT engine using tao-converter. The generated TensorRT engine file can also be ingested by DeepStream. Machine-specific optimizations are done as part of the engine creation process, so a distinct engine should be generated for each environment and hardware configuration. If the TensorRT or CUDA libraries of the inference environment are updated (including minor version updates), or if a new model is generated, new engines need to be generated. Running an engine that was generated with a different version of TensorRT and CUDA is not supported and will cause unknown behavior that affects inference speed, accuracy, and stability, or it may fail to run altogether. Option 1 is very straightforward. The .etlt file and calibration cache are directly used by DeepStream. DeepStream will automatically generate the TensorRT engine file and then run inference. TensorRT engine generation can take some time depending on size of the model and type of hardware. Engine generation can be done ahead of time with Option 2. With option 2, the tao-converter is used to convert the .etlt file to TensorRT; this file is then provided directly to DeepStream. See the Exporting the Model section for more details on how to export a TAO model. ### TensorRT Open Source Software (OSS) The TensorRT OSS build is required for YOLOv4 models. This is required because several TensorRT plugins that are required by these models are only available in TensorRT open source repo and not in the general TensorRT release. Specifically, for YOLOv4, we need the batchTilePlugin and batchedNMSPlugin. If the deployment platform is x86 with an NVIDIA GPU, follow the TensorRT OSS on x86 instructions; if your deployment is on an NVIDIA Jetson platform, follow the TensorRT OSS on Jetson (ARM64) instructions. #### TensorRT OSS on x86 Building TensorRT OSS on x86: 1. Install Cmake (>=3.13). Note TensorRT OSS requires cmake >= v3.13, so install cmake 3.13 if your cmake version is lower than 3.13c Copy Copied! sudo apt remove --purge --auto-remove cmake wget https://github.com/Kitware/CMake/releases/download/v3.13.5/cmake-3.13.5.tar.gz tar xvf cmake-3.13.5.tar.gz cd cmake-3.13.5/ ./configure make -j$(nproc)
sudo make install
sudo ln -s /usr/local/bin/cmake /usr/bin/cmake
2. Get GPU architecture. The GPU_ARCHS value can be retrieved by the deviceQuery CUDA sample:
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cd /usr/local/cuda/samples/1_Utilities/deviceQuery
sudo make
./deviceQuery
If the /usr/local/cuda/samples doesn’t exist in your system, you could download deviceQuery.cpp from this GitHub repo. Compile and run deviceQuery.
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nvcc deviceQuery.cpp -o deviceQuery
./deviceQuery
This command will output something like this, which indicates the GPU_ARCHS is 75 based on CUDA Capability major/minor version.
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Detected 2 CUDA Capable device(s)
Device 0: "Tesla T4"
CUDA Driver Version / Runtime Version 10.2 / 10.2
CUDA Capability Major/Minor version number: 7.5
3. Build TensorRT OSS:
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git clone -b 21.08 https://github.com/nvidia/TensorRT
cd TensorRT/
git submodule update --init --recursive
export TRT_SOURCE=pwd
cd $TRT_SOURCE mkdir -p build && cd build Note Make sure your GPU_ARCHS from step 2 is in TensorRT OSS CMakeLists.txt. If GPU_ARCHS is not in TensorRT OSS CMakeLists.txt, add -DGPU_ARCHS=<VER> as below, where <VER> represents GPU_ARCHS from step 2. Copy Copied! /usr/local/bin/cmake .. -DGPU_ARCHS=xy -DTRT_LIB_DIR=/usr/lib/x86_64-linux-gnu/ -DCMAKE_C_COMPILER=/usr/bin/gcc -DTRT_BIN_DIR=pwd/out make nvinfer_plugin -j$(nproc)
After building ends successfully, libnvinfer_plugin.so* will be generated under \pwd\/out/.
4. Replace the original libnvinfer_plugin.so*:
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sudo mv /usr/lib/x86_64-linux-gnu/libnvinfer_plugin.so.8.x.y ${HOME}/libnvinfer_plugin.so.8.x.y.bak // backup original libnvinfer_plugin.so.x.y sudo cp$TRT_SOURCE/pwd/out/libnvinfer_plugin.so.8.m.n /usr/lib/x86_64-linux-gnu/libnvinfer_plugin.so.8.x.y
sudo ldconfig
#### TensorRT OSS on Jetson (ARM64)
1. Install Cmake (>=3.13)
Note
TensorRT OSS requires cmake >= v3.13, while the default cmake on Jetson/Ubuntu 18.04 is cmake 3.10.2.
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sudo apt remove --purge --auto-remove cmake
tar xvf cmake-3.13.5.tar.gz
cd cmake-3.13.5/
./configure
make -j$(nproc) sudo make install sudo ln -s /usr/local/bin/cmake /usr/bin/cmake 2. Get GPU architecture based on your platform. The GPU_ARCHS for different Jetson platform are given in the following table. Jetson Platform GPU_ARCHS Nano/Tx1 53 Tx2 62 AGX Xavier/Xavier NX 72 3. Build TensorRT OSS: Copy Copied! git clone -b 21.03 https://github.com/nvidia/TensorRT cd TensorRT/ git submodule update --init --recursive export TRT_SOURCE=pwd cd$TRT_SOURCE
mkdir -p build && cd build
Note
The -DGPU_ARCHS=72 below is for Xavier or NX, for other Jetson platform, change 72 referring to GPU_ARCHS from step 2.
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/usr/local/bin/cmake .. -DGPU_ARCHS=72 -DTRT_LIB_DIR=/usr/lib/aarch64-linux-gnu/ -DCMAKE_C_COMPILER=/usr/bin/gcc -DTRT_BIN_DIR=pwd/out
make nvinfer_plugin -j$(nproc) After building ends successfully, libnvinfer_plugin.so* will be generated under ‘pwd’/out/. 4. Replace "libnvinfer_plugin.so*" with the newly generated. Copy Copied! sudo mv /usr/lib/aarch64-linux-gnu/libnvinfer_plugin.so.8.x.y${HOME}/libnvinfer_plugin.so.8.x.y.bak // backup original libnvinfer_plugin.so.x.y
sudo cp pwd/out/libnvinfer_plugin.so.8.m.n /usr/lib/aarch64-linux-gnu/libnvinfer_plugin.so.8.x.y
sudo ldconfig
### Generating an Engine Using tao-converter
The tao-converter tool is provided with the TAO Toolkit to facilitate the deployment of TAO trained models on TensorRT and/or Deepstream. This section elaborates on how to generate a TensorRT engine using tao-converter.
For deployment platforms with an x86-based CPU and discrete GPUs, the tao-converter is distributed within the TAO docker. Therefore, we suggest using the docker to generate the engine. However, this requires that the user adhere to the same minor version of TensorRT as distributed with the docker. The TAO docker includes TensorRT version 8.0.
#### Instructions for x86
For an x86 platform with discrete GPUs, the default TAO package includes the tao-converter built for TensorRT 8.0 with CUDA 11.3 and CUDNN 8.2. However, for any other version of CUDA and TensorRT, please refer to the overview section for download. Once the tao-converter is downloaded, follow the instructions below to generate a TensorRT engine.
1. Unzip the zip file on the target machine.
2. Install the OpenSSL package using the command:
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sudo apt-get install libssl-dev
3. Export the following environment variables:
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$export TRT_LIB_PATH=”/usr/lib/x86_64-linux-gnu”$ export TRT_INC_PATH=”/usr/include/x86_64-linux-gnu”
1. Run the tao-converter using the sample command below and generate the engine.
2. Instructions to build TensorRT OSS on Jetson can be found in the TensorRT OSS on x86 section above or in this GitHub repo.
Note
Make sure to follow the output node names as mentioned in Exporting the Model section of the respective model.
#### Instructions for Jetson
For the Jetson platform, the tao-converter is available to download in the NVIDIA developer zone. You may choose the version you wish to download as listed in the overview section. Once the tao-converter is downloaded, please follow the instructions below to generate a TensorRT engine.
1. Unzip the zip file on the target machine.
2. Install the OpenSSL package using the command:
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sudo apt-get install libssl-dev
3. Export the following environment variables:
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$export TRT_LIB_PATH=”/usr/lib/aarch64-linux-gnu”$ export TRT_INC_PATH=”/usr/include/aarch64-linux-gnu”
1. For Jetson devices, TensorRT comes pre-installed with Jetpack. If you are using older JetPack, upgrade to JetPack 4.5 or 4.6.
2. Instructions to build TensorRT OSS on Jetson can be found in the TensorRT OSS on Jetson (ARM64) section above or in this GitHub repo.
3. Run the tao-converter using the sample command below and generate the engine.
Note
Make sure to follow the output node names as mentioned in Exporting the Model section of the respective model.
#### Using the tao-converter
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tao-converter [-h] -k <encryption_key>
-d <input_dimensions>
-o <comma separated output nodes>
[-c <path to calibration cache file>]
[-e <path to output engine>]
[-b <calibration batch size>]
[-m <maximum batch size of the TRT engine>]
[-t <engine datatype>]
[-w <maximum workspace size of the TRT Engine>]
[-i <input dimension ordering>]
[-p <optimization_profiles>]
[-s]
[-u <DLA_core>]
input_file
##### Required Arguments
• input_file: The path to the .etlt model exported using tao yolo_v4 export.
• -k: The key used to encode the .tlt model when training.
• -d: A comma-separated list of input dimensions that should match the dimensions used for tao yolo_v4 export.
• -o: A comma-separated list of output blob names that should match the output configuration used for tao yolo_v4 export. For YOLOv4, set this argument to BatchedNMS.
• -p: Optimization profiles for .etlt models with dynamic shape. Use a comma-separated list of optimization profile shapes in the format <input_name>,<min_shape>,<opt_shape>,<max_shape>, where each shape has the format: <n>x<c>x<h>x<w>. The input name for YOLOv4 is Input
##### Optional Arguments
• -e: The path to save the engine to. The default path is ./saved.engine.
• -t: The desired engine data type. The options are {fp32, fp16, int8}. The default value is fp32. A calibration cache is generated in INT8 mode.
• -w: The maximum workspace size for the TensorRT engine. The default value is 1073741824(1<<30).
• -i: The input dimension ordering; all other TAO commands use NCHW. The options are {nchw, nhwc, nc}.The default value is nchw, so you can omit this argument for YOLOv4.
• -s: TensorRT strict type constraints. A Boolean to apply TensorRT strict type constraints when building the TensorRT engine.
• -u: Specifies the DLA core index when building the TensorRT engine on Jetson devices (only needed if using DLA core).
##### INT8 Mode Arguments
• -c: The path to the calibration cache file (only used in INT8 mode). The default value is ./cal.bin.
• -b: The batch size used during the export step for INT8-calibration cache generation (default: 8).
• -m: The maximum batch size for the TensorRT engine (default: 16). If you encounter an out-of-memory issue, decrease the batch size accordingly. This parameter is not required for .etlt models generated with dynamic shape (which is only possible for new models introduced since version 3.0).
##### Sample Output Log
Here is a sample log for exporting a YOLOv4 model:
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tao-converter -k \$KEY \
-p Input,1x3x384x1248,8x3x384x1248,16x3x384x1248 \
-e /export/trt.fp16.engine \
-t fp16 \
/ws/yolov4_resnet18_epoch_100.etlt
### Integrating the model with DeepStream
To integrate a model trained by TAO Toolkit with DeepStream, you shoud generate a device-specific optimized TensorRT engine using tao-converter. The generated TensorRT engine file can then be ingested by DeepStream (Currently, YOLOv4 etlt files are not supported by DeepStream).
For YOLOv4, you will need to build the TensorRT open-source plugins and custom bounding-box parser. The instructions to build TensorRT open-source plugins are provided in the TensorRT Open Source Software (OSS) section above. The instructions to build a custom bounding-box parser are provided in the Prerequisites for YOLOv4 Model section below, and the required code can be found in this GitHub repo.
To integrate the models with DeepStream, you need the following:
1. Download and install DeepStream SDK. The installation instructions for DeepStream are provided in the DeepStream Development Guide.
2. An exported .etlt model file and optional calibration cache for INT8 precision.
3. A labels.txt file containing the labels for classes in the order in which the networks produces outputs.
4. A sample config_infer_*.txt file to configure the nvinfer element in DeepStream. The nvinfer element handles everything related to TensorRT optimization and engine creation in DeepStream.
DeepStream SDK ships with an end-to-end reference application which is fully configurable. Users can configure input sources, inference model, and output sinks. The app requires a primary object detection model, followed by an optional secondary classification model. The reference application is installed as deepstream-app. The graphic below shows the architecture of the reference application.
There are typically 2 or more configuration files that are used with this app. In the install directory, the config files are located in samples/configs/deepstream-app or sample/configs/tlt_pretrained_models. The main config file configures all the high level parameters in the pipeline above. This would set input source and resolution, number of inferences, tracker and output sinks. The other supporting config files are for each individual inference engine. The inference specific config files are used to specify models, inference resolution, batch size, number of classes and other customization. The main config file will call all the supporting config files. Here are some config files in samples/configs/deepstream-app for your reference.
• source4_1080p_dec_infer-resnet_tracker_sgie_tiled_display_int8.txt: Main config file
• config_infer_primary.txt: Supporting config file for primary detector in the pipeline above
• config_infer_secondary_*.txt: Supporting config file for secondary classifier in the pipeline above
The deepstream-app will only work with the main config file. This file will most likely remain the same for all models and can be used directly from the DeepStream SDK will little to no change. User will only have to modify or create config_infer_primary.txt and config_infer_secondary_*.txt.
#### Integrating a YOLOv4 Model
To run a YOLOv4 model in DeepStream, you need a label file and a DeepStream configuration file. In addition, you need to compile the TensorRT 7+ Open source software and YOLOv4 bounding box parser for DeepStream.
A DeepStream sample with documentation on how to run inference using the trained YOLOv4 models from TAO Toolkit is provided on GitHub repo..
##### Prerequisites for YOLOv4 Model
1. YOLOv4 requires batchTilePlugin, resizeNearestPlugin, and batchedNMSPlugin. These plugins are available in the TensorRT open source repo, but not in TensorRT 7.0. Detailed instructions to build TensorRT OSS can be found in TensorRT Open Source Software (OSS).
2. YOLOv4 requires YOLOv3 custom bounding box parsers that are not built-in inside the DeepStream SDK. The source code to build YOLOv3 custom bounding box parsers is available in GitHub repo. The following instructions can be used to build bounding box parser:
Step1: Install git-lfs (git >= 1.8.2)
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curl -s https://packagecloud.io/install/repositories/github/git-lfs/
script.deb.sh | sudo bash
sudo apt-get install git-lfs
git lfs install
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git clone -b release/tlt3.0 https://github.com/NVIDIA-AI-IOT/deepstream_tlt_apps
Step 3: Build
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// or Path for DS installation
export CUDA_VER=10.2 // CUDA version, e.g. 10.2
make
This generates libnvds_infercustomparser_tlt.so in the directory post_processor.
### Label File
The label file is a text file containing the names of the classes that the YOLOv4 model is trained to detect. The order in which the classes are listed here must match the order in which the model predicts the output. During the training, TAO YOLOv4 will specify all class names in lower case and sort them in alphabetical order. For example, if the dataset_config is:
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dataset_config {
data_sources: {
label_directory_path: "/workspace/tao-experiments/data/training/label_2"
image_directory_path: "/workspace/tao-experiments/data/training/image_2"
}
target_class_mapping {
key: "car"
value: "car"
}
target_class_mapping {
key: "person"
value: "person"
}
target_class_mapping {
key: "bicycle"
value: "bicycle"
}
validation_data_sources: {
label_directory_path: "/workspace/tao-experiments/data/val/label"
image_directory_path: "/workspace/tao-experiments/data/val/image"
}
}
Then the corresponding yolov4_labels.txt file would be:
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bicycle
car
person
### DeepStream Configuration File
The detection model is typically used as a primary inference engine. It can also be used as a secondary inference engine. To run this model in the sample deepstream-app, you must modify the existing config_infer_primary.txt file to point to this model.
Integrate the TensorRT engine file with the DeepStream app
Step 1: Generate TensorRT engine using tao-converter. Detailed instructions are provided in the Generating an engine using tao-converter section above.
Step 2: Once the engine file is generated successfully, modify the following parameters to use this engine with DeepStream.
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model-engine-file=<PATH to generated TensorRT engine>
All other parameters are common between the two approaches. To use the custom bounding box parser instead of the default parsers in DeepStream, modify the following parameters in [property] section of primary infer configuration file:
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parse-bbox-func-name=NvDsInferParseCustomBatchedNMSTLT
custom-lib-path=<PATH to libnvds_infercustomparser_tlt.so>
Add the label file generated above using:
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labelfile-path=<YOLOv4 labels>
For all the options, see the configuration file below. To learn about what all the parameters are used for, refer to the DeepStream Development Guide.
Here’s a sample config file, pgie_yolov4_config.txt:
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[property]
gpu-id=0
net-scale-factor=1.0
offsets=103.939;116.779;123.68
model-color-format=1
labelfile-path=<Path to yolov4_labels.txt>
model-engine-file=<PATH to generated TensorRT engine>
tlt-model-key=<Key to decrypt model>
infer-dims=3;384;1248
maintain-aspect-ratio=1
uff-input-order=0
uff-input-blob-name=Input
batch-size=1
## 0=FP32, 1=INT8, 2=FP16 mode
network-mode=0
num-detected-classes=3
interval=0
gie-unique-id=1
is-classifier=0
#network-type=0
#no cluster
cluster-mode=3
output-blob-names=BatchedNMS
parse-bbox-func-name=NvDsInferParseCustomBatchedNMSTLT
custom-lib-path=<Path to libnvds_infercustomparser_tlt.so>
[class-attrs-all]
pre-cluster-threshold=0.3
roi-top-offset=0
roi-bottom-offset=0
detected-min-w=0
detected-min-h=0
detected-max-w=0
detected-max-h=0
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# Okumura Model
The Okumura model for Urban Areas is a Radio propagation model that was built using the data collected in the city of Tokyo, Japan. The model is ideal for using in cities with many urban structures but not many tall blocking structures. The model served as a base for the Hata Model.
Okumura model was built into three modes. The ones for urban, suburban and open areas. The model for urban areas was built first and used as the base for others.
## Coverage
Frequency = 150 MHz to 1920 MHz
Mobile station antenna height: between 1 m and 10 m
Base station antenna height: between 30 m and 1000 m
Link distance: between 1 km and 100 km
## Mathematical formulation
The Okumura model is formally expressed as:
$L\;=\;L_{FSL}\;+\;A_{MU}\;-\;H_{MG}\;-\;H_{BG}\;-\;\sum{K_{correction}}\;$
where,
L = The median path loss. Unit: Decibel (dB)
LFSL = The Free Space Loss. Unit: Decibel (dB)
AMU = Median attenuation. Unit: Decibel (dB)
HMG = Mobile station antenna height gain factor.
HBG = Base station antenna height gain factor.
Kcorrection = Correction factor gain (such as type of environment, water surfaces, isolated obstacle etc.)
## Points to note
Okumura's model is one of the most widely used models for signal prediction in urban areas. This model is applicable for frequencies in the range 150 MHz to 1920 MHz (although it is typically extrapolated up to 3000 MHz) and distances of 1 km to 100 km. It can be used for base station antenna heights ranging from 30 m to 1000 m.
Okumura developed a set of curves giving the median attenuation relative to free space (Arnu), in an urban area over a quasi-smooth terrain with a base station effective antenna height (hte) of 200 m and a mobile antenna height (hre) of 3 m. These curves were developed from extensive measurements using vertical omni-directional antennas at both the base and mobile, and are plotted as a function of frequency in the range 100 MHz to 1920 MHz and as a function of distance from the base station in the range 1 km to 100 km. To determine path loss using Okumura's model, the free space path loss between the points of interest is first determined, and then the value of Amu(f, d) (as read from the curves) is added to it along with correction factors to account for the type of terrain. The model can be expressed as
L50(dB) = LF + Amu(f, d)- G(hte) — G(hre) — Garea
where L50 is the 50th percentile (i.e., median) value of propagation path loss, LF is the free space propagation loss, Amu is the median attenuation relative to free space, G(hte) is the base station antenna height gain factor, G(hre) is the mobile antenna height gain factor, and GAREA is the gain due to the type of environment. Note that the antenna height gains are strictly a function of height and have nothing to do with antenna patterns.
Plots of Amu(f, d) and GAREA for a wide range of frequencies are shown in Figure 3,23 and Figure 3.24. Furthermore, Okumura found that G(hte) varies at a rate of 20 dB/decade and G(hre) varies at a rate of 10 dB/decade for heights less than 3 m. G(hte) = 20 log(hte/200) 1000 m > hte > 30 m G(hre) = 10 log(hre/3) hre <= 3 m G(hre) = 20 log (hre/3) 10 m > hre > 3 m Other corrections may also be applied to Okumura's model. Some of the important terrain related parameters are the terrain undulation height (A/i), isolated ridge height, average slope of the terrain and the mixed land-sea parameter. Once the terrain related parameters are calculated, the necessary correction factors can be added or subtracted as required. All these correction factors are also available as Okumura curves [0ku68].
Okumura's model is wholly based on measured data and does not provide any analytical explanation. For many situations, extrapolations of the derived curves can be made to obtain values outside the measurement range, although the validity of such extrapolations depends on the circumstances and the smoothness of the curve in question.
Okumura's model is considered to be among the simplest and best in terms of accuracy in path loss prediction for mature cellular and land mobile radio systems in cluttered environments. It is very practical and has become a standard for system planning in modern land mobile radio systems in Japan. The major disadvantage with the model is its slow response to rapid changes in terrain, therefore the model is fairly good in urban and suburban areas, but not as good in rural areas. Common standard deviations between predicted and measured path loss values are around 10 dB to 14 dB.
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## Absolute Galois group of the field of Puiseux series over $\overline{\mathbb{F}}_p$
Let $K$ be the field of Puiseux series with coefficients in $\overline{\mathbb{F}}_p$ (the algebraic closure of the field with $p$ elements). What is the absolute Galois group of $K$?
Thank you to anyone who could help!
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One can compute the set of continuous homomorphisms from the absolute Galois group to $\mathbb F_p$ using the Artin-Schreier exact sequence. A basis, as a vector space over $\bar{\mathbb F}_p$ consists of functions of the form $T^{-a/b}$ with $a$, $b$, and $p$ pairwise relatively prime. This is a weird-looking countable-dimensional vector space, which makes it seem unlikely that one can find a nice description for the whole group. In particular, it cannot be finitely topologically generated. – Will Sawin Nov 16 at 6:11
Is this known if we replace $\bar{\mathbb{F}}_p$ with $\mathbb{C}$? Can one reason by the analogy? – Spice the Bird Nov 16 at 7:58
@Spice: if you replace $\overline{\mathbb{F}}_p$ iwth $\mathbb{C}$, then the field you get is algebraically closed. – Laurent Berger Nov 16 at 9:25
On a webpage entitled "Questions I'm thinking about", Kiran Kedlaya wrote "I have a method for computing in the algebraic closure of the rational function field over a finite field, using finite automata and generalized power series. Does it actually work in practice? I can't tell. (There has been a tiny bit of experimental work on this; contact me for details.)" math.ucsd.edu/~kedlaya/questions.shtml , last updated Dec. 2009. – David Speyer Nov 16 at 14:28
I think there is a 2001 paper by Kedlaya where he described the algebraic closure of $\overline{\mathbb{F}}_p((x))$ building on other's work, it is like the Puiseaux series together with towers of Artin-Schreier extensions. – 36min Nov 20 at 2:15
Let $E$ be the field $\overline{\mathbb{F}}_p((X))$. The field of Puiseux series whose exponents have denominators prime to $p$ is a subfield of $E^{sep}$, so the group you're asking about would then be the wild inertia subgroup of $Gal(E^{sep}/E)$. The group $Gal(E^{sep}/E)$ is quite complicated, and it comes up in arithmetic geometry, for example when studying the $\pi_1$ of curves. It also occurs as a closed subgroup of $Gal(\overline{\mathbb{Q}}_p / \mathbb{Q}_p)$ by the theory of the field of norms of Fontaine and Wintenberger. Its representations on $\mathbb{Z}_p$-modules are described by $\varphi$-modules'' (like $(\varphi,\Gamma)$-modules without the $\Gamma$). If you want to include Puiseux series whose exponents have denominators divisible by $p$, then you're looking at the perfection of $E$. The group does not change, as $E^{sep}$ is dense in $E^{alg}$ by a theorem of Ax.
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Thank you for the answers ! Would it be possible to explain in a few words what $(\phi, \Gamma)$-modules are ? Thanks again !
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Dear beginner, I leave to Laurent a better discussion on $(\phi,\Gamma)$-modules (although "in a few words" might be difficult even for him...), but observe that there are small boxes below answers labeled add comment which are normally used to insert these further questions or remarks. And welcome to MO, of course! – Filippo Alberto Edoardo Nov 20 at 1:05
I think that beginner has lost access to his account and his 70 points of reputation; with a new account and reputation as 1, you can neither comment nor edit your own question (since the system don't know it is yours). I remember when I was a beginner myself on this site that similar problems happened to me several times. Not sure that will help in this specific case, but here is a simple way to retrieve one's account: click to "Users" (between "Tags" and "Badge") on the top of the page, then use the search function to find your user's account, and then there should be a button saying: – Joël Nov 20 at 1:36
"is that your account?" and you're done. Anyway, welcome to MO. – Joël Nov 20 at 1:36
Thank you very much for your welcome and your explanation on how to use MO ! Joël is right: I had lost access to my account. – beginner Nov 20 at 2:38
@beginner If I can be allowed to advertize my own stuff: I wrote a survey on p-adic representations a few years ago, and one chapter concerns $(\varphi,\Gamma)$-modules. See [05] of perso.ens-lyon.fr/laurent.berger/publications.php – Laurent Berger Nov 20 at 12:27
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This post covers the basics of Approximate Agreement. We define the problem, explain in what way its an interesting variation of classic Agreement, and describe the idea behind the robust midpoint protocol.
In classic consensus, the space of possible decision values is just a set and the goal is to agree on a decision value that must be the input value if all non-faulty have the same input value. Approximate Agreement is a variation, first suggested by Dolev, Lynch, Pinter, Stark, Weihl, 1985-86, where the goal is to approximately agree (up to some $<\epsilon$ difference) on a value that is in the convex hull of the non-faulty input values. To make this well defined, in approximate agreement the space of possible decision values is a convex set, for this post just assume the rational numbers $\mathcal{R}$. This naturally induces a notion of two values being close to each other and that a value is in between two other values.
There are many cases where (approximate) agreement on a rational value makes sense. For example, if parties want to reach approximate agreement on the current exchange rate of dollar vs euro, we can define the distance between two exchange rates as say the distance in US cents. Similarly, if parties want to reach approximate agreement on the current temperature, we can define the distance between two measurements as say the distance in degrees of Celsius.
As in classic Agreement:
• Validity: if all non-faulty parties have the same value then this is the decision value.
• Termination: all non-faulty decide.
Unlike classic agreement, Approximate Agreement strengthens the validity property and weakens (relaxes) the agreement property:
• Convex validity: if a non-faulty party outputs $v$ then there exists some non-faulty parties with inputs $c_1$ and $c_2$ such that $c_1 \leq v \leq c_2$. More generally, we can say that $v$ must belong to the convex hull of the non-faulty input values.
• $\epsilon$-Approximate consensus: if two non-faulty parties output $v_i$ and $v_j$ then $|v_j-v_i|<\epsilon$ (or more generally, $d(v_i,v_j)<\epsilon$). In other words, the decisions of all non-faulty are $\epsilon$-close to each other.
## Why is Approximate Agreement interesting?
From a foundations perspective, relaxing to approximate consensus allows to circumvent fundamental limitations of classic (exact) consensus and obtain qualitatively better round complexity. We will see in a later post that this relaxation circumvents the $f+1$ round lower bound in synchrony and the infinite execution lower bound in asnychrony. On the other hand, the strengthening to convex validity seems to require more resources and hence is an interesting trade-off.
From a practical perspective, there are use cases (many in the blockchain world) where we would like an oracle service to (approximately) agree on some external values and often these values have a natural notion of convexity (for example, the exchange rate of two assets or the interest rate, etc). In many of these cases the input values of non-faulty parties are close but not exactly the same. The classic validity property allows any value to be output in this case, which clearly does not seem like the desired outcome. Convex validity captures a very natural requirement of outputting a value in the convex hull of these inputs (for example, if all non-faulty inputs are in the range [3.99,4.01], we would want the output value to be in that range as well!).
## Convex validity using a broadcast channel
Let’s jump right in and assume we have $n=3f+1$ parties and a malicious adversary that can control at most $f$ parties. Party $i$ has input $v_i \in \mathcal{R}$.
In this section we assume parties have access to a powerful a broadcast channel (a blockchain). This makes everything much easier :-). The first step of the protocol is trivial:
Each party i broadcasts its input v_i
Assuming synchrony for now, each party gathers a multi-set $V$ of between $n-f$ and $n$ values from the broadcast values. The only question is how does a party extract a single decision value from this multi-set $V$?
What can the adversary do? Intuitively, if the adversary chooses to send values inside the non-faulty convex hull, it wouldn’t do us much harm. These values are roughly where they should be. However, it can cause the $f$ parties it controls to post very high or very low inputs. So it is very natural to trim (or remove) the $f$ smallest and $f$ largest values from the multi-set $V$. Given a representation of $V$ as an ordered sequence $V=\{v_1,v_2,\dots, v_k\}$, define the trim of $V$ by $f$ values as the sub-multi-set formed by removing the lowest $f$ values and the highest $f$ values from the sequence.
$T=trim(V)=\{v_{f+1},\dots,v_{k-f}\}$
Observe that $|T| = |V|-2f$ so $T$ is only well defined when $|V| \geq 2f+1$ (indeed this is why we required $n=3f+1$ and hence assured to see at least $n-f \geq 2f+1$ values). BTW the idea of trimming the outliers is deeply connected to robust statistics.
Let $G$ be the multi-set of input values of all the non-faulty parties.
We are now ready to state a simple but important fact about $\max(T)$ and $\min(T)$ relative to $\max(G), \min(G)$ and $median(G)$.
Claim 1: $median(G) \leq \max(T) \leq \max(G)$ and similarly $\min(G) \leq \min(T) \leq median(G)$.
Proof: To see that $\max(T) \leq \max(G)$ observe that the worst the adversary can do is post $f$ values that are higher than $\max(G)$ but those will be removed by $T=trim(V)$.
Recall that $T=trim(V)$ and $V$ is the set of all values that are broadcast. So in particular we have $G \subseteq V$.
To see that $median(G) \leq \max(T)$ observe that $2f+1 \leq |G|$ hence even if the top $f$ values will be removed by $T=trim(V)$, the $median(G)$ value will remain.
The proof for $\min(G) \leq \min(T) \leq median(G)$ is identical.
We can finally complete the protocol, return the robust midpoint:
Given a multi-set V let T=trim(V)
output (max(T)+min(T))/2
Proof: Trivially we have termination and (exact) agreement from using the powerful the broadcast channel assumption. For convex validity, the claim above shows that $\min(G) \leq \min(T) \leq \max(T) \leq \max(G)$.
Note that any function that outputs a value in the convex hull of $T$ would be fine. For example some oracle services use the median (see Chainlink).
### In synchrony without a broadcast channel
Let’s look at the natural robust midpoint protocol:
Each party i sends its input v_i to all
Each party waits for 2 Delta to gather a multi-set V
let T=trim(V)
output (min(T)+max(T))/2
Proof: Termination is immediate. Convex validity follows from Claim 1. What about (approximate) agreement?
For example with $n=4$, party A may see $V_1=\{0,0,1,1\}$ from parties A,B,C,D (so it outputs 1/2) and party D may see $V_2 = \{0,1,1,1\}$ from parties $A,B,C,D$ (so it outputs 1). We don’t have agreement, did we make progress?
Yes! It’s time to introduce an important measure for a multi-set of non-faulty parties:
$span(G) = \max(G)-\min(G)$
Let $G$ be the multi-set of input values of the non-faulty parties and $G_1$ be the multi-set of output values of the non-faulty from the protocol (so $G_1=\{\frac{\min(T_i)+\max(T_i)}{2}\}_{i \in G}$):
Claim 2: $span(G_1) \leq span(G)/2$.
Proof: From Claim 1 we have
$median(G) \leq \max(T) \leq \max(G)$
and
$\min(G) \leq \min(T) \leq median(G)$
Recall that
$\min(G_1) = \min_{i \in G} \frac{\min(T_i)+\max(T_i)}{2}$
So we have
$\min(G_1) \geq \frac{\min(G) + median(G)}{2}$
and similarly
$\max(G_1) \leq \frac{\max(G) + median(G)}{2}$
Hence
$span(G_1) = \max(G_1) - \min(G_1) \\ \leq \frac{\max(G) + median(G)}{2} - \frac{\min(G) + median(G)}{2}\\ = \frac{\max(G) -\min(G)}{2} = span(G)/2$
Every time we do this calculation we are re-surprised how everything cleans up so nicely :-)
So what did we achieve? The robust mid-point protocol allows parties to output values in the convex hull of their inputs, while cutting their span in half!
In the next posts we will see how repeating this one round protocol can eventually cut the span to any desired $<\epsilon$ and how to terminate with approximate agreement.
Cant wait for the follow up post? Let us know if this protocol can also work in asynchrony?
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# Minimal bivariate diophantine equation solution space
I am facing the following type of diophantine equations:
$$axy + bx + cy + d = 0$$
Where $a$, $b$, $c$, $d$ are integers and solutions for $x$, $y$ in the integers are seeked. If $a=0$ one can apply the extended euclidean algorithm. $x$ and $y$ can then be viewed as generated by a new parameter $t$ and linear forms:
$$x = et + f \\ y = gt + h$$
Can something similar be said when $a\ne0$?
-
$axy+bx+cy+d=0$; $a^2xy+abx+acy+ad=0$; $(ax+c)(ay+b)+ad-bc=0$; $(ax+c)(ay+b)=bc-ad$. So what you have to do is factor $bc-ad$; then for every factorization $bc-ad=rs$, you have to see whether the simultaneous equations $ax+c=r$, $ay+b=s$ have a solution.
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## Chemistry and Chemical Reactivity (9th Edition)
Initially, all hydroxide added is neutralized: Number of moles: $20/1000\ L\times 0.10\ M=0.002\ mol$ $OH^-+NH_4^+\rightarrow NH_3 +H_2O$ The new concentration of ammonium and ammonia are: $NH_4^+: (0.183\ M\times 80.0/1000\ L-0.002\ mol)\div 100/1000\ L=0.126\ M$ $NH_3: (0.169\ M\times 80.0/1000\ L+0.002\ mol)\div 100/1000\ L=0.155\ M$ New equilibrium: $K_a=x(0.155+x)/(0.126-x)$ $x=4.55\cdot10^{-10}\ M=[H_3O^+]$ $pH=9.34$ From Henderson-Hasselbach, the original pH was 9.21, So it goes from 9.21 to 9.34
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# Back-propagation for computing derivative of certain line integral
Consider a function F (think of neural networks) with two sets of parameters: (1) model parameters $$\mathbf{w}$$, and (2) input data $${\bf x} \in {\mathbb R}^d$$. Fix $$i \in [d]$$, consider the following path integral:
$$C_i({\bf x}, {\bf w}) = {\bf x}_i \int_0^1\frac{\partial F}{\partial {\bf x}_i}(\alpha {\bf x})d\alpha$$
Basically, one can think of $$C_i({\bf x}, {\bf w})$$ as the "contribution" of the $$i$$-th dimension to the final output -- by integrating along a line from $$\bf 0$$ to $$\bf x$$.
My question is whether there are known methods (references, etc.) that give "backpropagation-like" algorithms for computing:
$$\frac{\partial C_i({\bf x}, {\bf w})}{\partial {\bf w}_i}$$
That is, basically I want to understand the first order information of the contribution of the $$i$$-th dimension w.r.t. $${\bf w}_i$$.
Thanks in advance for any insight.
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# A solvable nonic polynomial
Continuing from our demonstration that a certain sextic polynomial, which are not in general solvable, has an explicit factorization, we go on to describe how a class of degree 9 polynomials is solvable. Consider $a,b,c,d$ to be rational integers, not all zero, and the nonic polynomial
$F(x) = x^9 + a x^8 + b x^7 + c x^6 + d x^5 - (126 - 56 a + 21 b - 6 c + d)x^4$
$- (84 - 28 a + 7 b - c)x^3 - (36 - 8a + c)x^2 - (9 - a)x - 1.$
The claim is that all such polynomials are in fact solvable!
I will reveal the argument a little later, but it’ll be interesting to see what kind of arguments readers can come up with.
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# Latex thesis class file
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How to write a thesis in latex pt 1 - basic structure we need to choose is a document class in chapters/ before the file name so that latex knows where to. Preparing your master's thesis using latex the latex file using the old ucthesis document class, then you should update your thesis to use ucbthesis. Abstract a university of california davis dissertation/thesis latex class file the abstract submitted as part of your dissertation, in the introductory pages, does not. This latex template is used by many universities as the basis for template for a masters / doctoral thesis % the class file specifying the document.
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# Object synchronization.
This topic is 4039 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.
## Recommended Posts
Hi all! I'm looking for some hints on how to implement some sort of object synchronization over an existing network connection. Another way of putting it is that I need a shared object amongst two or more computers. I.e if one client changes the object the change needs to be propagated to the other local copies, something like this: With object I really doesn't mean any CPP class instance, I can live with a few hardcoded types. If we for instance assume that the shared object above is a std::map<std::string, float>, we can have any number of floating point "variables" referenced by a string. For instance if the client inserts a few strings like: "A", 10.0 "B", 2.0 "C", 12.0 The "system" should send these new additions to the server. If then the server changes some of the above the change would be sent to the client, i.e: "B" = 7.0 Am I making any sense at all? ;) What I'm after is not source or tools that let me do that, also I'm not looking for the answer "Use a database". All data communcation must be handled by my connection. I'm looking for a nice clean way to handle these sort of things, addressing problems like these: * What if the server and client changes the same variable at about the same time? * Handles huge objects but just sends the updates necessary? * Explicit synchronization points, i.e doesn't send any data before needed? * What if we have more than two parties involved? (Not really needed right now, but if there is a known methodology that is commonly used that can handle this, it's a bonus). If thinking about using a time stamp system and keeping a local copy of what the other computer(s) has, something like this (not working code):
struct SharedObject
{
T m_localObject;
T m_remoteObject;
std::map<int, T> m_sentObjects;
int m_remoteTimeStamp;
int m_localTimeStamp;
};
doSync()
{
++ m_localTimeStamp;
m_sentObjects.insert(std::make_pair(m_localTimeStamp, m_localObject));
sendToRemote(m_localTimeStamp, Diff(m_remoteObject, m_localObject));
}
recieveFromRemote()
{
while (m_remoteTimeStamp < Packet.m_timeStamp){
m_sentObjects.erase(m_remoteTimeStamp);
++ m_remoteTimeStamp;
}
if (!Packet.m_diff)
return;
sendToRemote(m_remoteTimeStamp, 0);
}
Edit: I could simplify the requirements slighly by disallowing different computer to modify the same variable. I.e all computers can read all variables, but only one computer may change it (the one that created it). I'm still looking for some good links to papers and or people who haved worked with this sort of things.
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Hi eq!
So you're basically looking for some kind of "Property Replication" mechanism, right? If so, you may have a look at the UnrealEngine which does this sort of thing. I don't have any links here, sorry but googling for "Unreal Replication" may help you.
>>
* What if the server and client changes the same variable at about the same time?
>
* Handles huge objects but just sends the updates necessary?
>
* Explicit synchronization points, i.e doesn't send any data before needed?
>
* What if we have more than two parties involved?
<
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EDIT: Sorry for the post above! There was some problem with quoting, so here again.
Hi eq!
So you're basically looking for some kind of "Property Replication" mechanism, right? If so, you may have a look at the UnrealEngine which does this sort of thing. I don't have any links here, sorry but googling for "Unreal Replication" may help you.
* What if the server and client changes the same variable at about the same time?
That must not happen! Exactly one host either one of the clients or the server must have authoritative access to a variable. And only that host is responsible for sending changes of that variable to other hosts. If another host wants to change a variable it has to send a "ChangeRequest" or something like that to the authoritative connection side.
* Handles huge objects but just sends the updates necessary?
That's a must! One easy way to do this, is that the authoritative host caches all his *owning* variables/objects with their initial values. You then check periodically (once per frame or so) whether a variable has changed and send it.
You may attach to each variable/object some kind of "UpdateFrequency" and/or "UpdatePriority", whether to send this variable reliably or unreliably, ordered/unordered etc.
* Explicit synchronization points, i.e doesn't send any data before needed?
You can attach to each variable a *condition*! Think of it as a script-function or so. And only if this condition evaluates to true, you send it.
* What if we have more than two parties involved?
I assume that party is a synonym for client!
Well, as I sad before only one host is *authoritative*!
cu,
Chris
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Two things pop into my mind, remembering courses and the Quake series (although, as the AP mentioned, Unreal has a nifty system as well):
1) on the client, keep bitfields indicating which variables have changed over the course of several frames. For example, you could store an integer that encodes the indices of each element of your map container that has changed, sending the respective changes later on. Quake for example, uses a boolean to update latched console variables between client and server.
2) send delta packets instead of the information in its entirety. This of course has to be applicable to your particular case. I'm sorry, but I can't seem to find a good general example of this. Anyone?
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# Ex.8.3 Q3 Compairing Quantities - NCERT Maths Class 7
Go back to 'Ex.8.3'
## Question
The population of a city decreased from $$25,000$$ to $$24,500.$$ Find the percentage decrease.
## Text Solution
What is Known?
The change in the population of a city (from $$25,000$$ to $$24,500$$)
What is Unknown?
The percentage decrease
Reasoning:
The decrease in population can be calculated by subtracting earlier (initial) population from present (final) population and percentage decrease in population can be obtained using the formula
Percentage decrease
\begin{align}=\frac{{{\text{Change}}\;{\text{in}}\;\;{\text{quantity}}}}{{{\text{Initial}}\;{\text{quantity}}\;}}{\rm{ \times 100}}\end{align}
Steps:
Decrase in city population
\begin{align}&= 25000 - 24500\\&= 500\end{align}
Percentage decrease
\begin{align}&= \,\frac{{{\text{Change}}\;{\text{in}}\;\;{\text{population}}}}{{{\text{Initial}}\;{\text{population}}\;}}{\rm{ \times 100}}\\&= \,\frac{{{\text{500}}}}{{{\text{25000}}}}{\rm{ \times 100}}\\&= \,{\text{2% }}\end{align}
Learn from the best math teachers and top your exams
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# Finding the inverse
1. Aug 17, 2009
### GHealy
1. The problem statement, all variables and given/known data
I need to pass a test to get into college calculus, but there's a few things I've forgotten since may, one of them being how to find the inverse of a function with x in the denominator. I think I remember it being something really simple that I'm overlooking, but here it goes:
2. Relevant equations
f(x) = (x + 2)/(3x + 5)
1) y = (x + 2)/(3x + 5)
2) x = (y + 2)/(3y + 5)
3) this is where it starts to go crazy.
3. The attempt at a solution
My attempts were kind of convulsive, but I hope it'll suffice to say that I know to switch y with x, then solve the problem for y, which will be the inverse. I just don't know how to get y out of the denominator without getting a really complicated answer. (edit: I put what I was saying into math form above)
Thanks, if anyone knows this. I've done some re-learning in the past week or so; there are just a few things left that are too specific to find online.
2. Aug 17, 2009
### symbolipoint
Your goal is to use properties of equality of real numbers to find a formula for y. Either you can find a result or you cannot. Can you obtain an inverse in that general manner?
In fact, a result is possible; whether the result is a function, I leave for you to decide.
3. Aug 17, 2009
### Дьявол
If the $$y=\frac{x + 2}{3x + 5}$$ then x+2=y(3x+5). Your goal is to present x in terms of y. Now can you proceed?
4. Aug 17, 2009
### GHealy
That's where I get, but I can't go further without jumbling the whole thing up.
5. Aug 17, 2009
### Elucidus
Distribute the y on the right hand side, separate y and non-y terms to opposite sides, factor out y and divide.
--Elucidus
6. Aug 17, 2009
### GHealy
*facepalm*
I can't believe I forgot simple algebra. Thanks.
7. Aug 19, 2009
### Unit
What I did was, multiply the numerator and then expand everything. Bring all the x-terms and the xy-terms to one side, and the y-terms and constants to the other. Then I factored out x, and divided by the other stuff!
Good luck =)
Last edited: Aug 19, 2009
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Home >> AIMS >> Class11 >> Math >> Sets
# Taking the set of natural numbers as the universal set, write down the complement of the following sets: {x: x is a positive multiple of 3}
{x: x $\in$ N and x is not a multiple of 3}
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# zbMATH — the first resource for mathematics
## Li, Can
Compute Distance To:
Author ID: li.can.1 Published as: Li, C.; Li, Can
Documents Indexed: 41 Publications since 2001, including 1 Book
all top 5
#### Co-Authors
2 single-authored 6 Guo, Zunguang 2 Jin, Zhen 2 Li, Yingchun 2 Liang, Juan 2 Sun, Guiquan 2 Zhang, Zhiyu 1 Cao, Xianglian 1 Li, Donghui 1 Li, Jing 1 Li, Li 1 Liu, Yuewu 1 Song, Lipeng 1 Wang, Zhen 1 Xie, Ziqing 1 Yan, Yiwei 1 Zhao, Zhirong 1 Zheng, Shuzhen
all top 5
#### Serials
2 Applied Mathematics and Computation 2 Complexity 2 Mathematical Sciences Research Journal 1 Chaos, Solitons and Fractals 1 International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 1 Journal of Northwest Normal University. Natural Science 1 Journal of Applied Mathematics 1 Journal of Natural Science of Hunan Normal University 1 Journal of Shenyang Normal University. Natural Science Edition 1 Journal of Nonlinear Science and Applications 1 Journal of Nanchang University. Natural Science
all top 5
#### Fields
6 Biology and other natural sciences (92-XX) 5 Numerical analysis (65-XX) 5 Operations research, mathematical programming (90-XX) 3 Ordinary differential equations (34-XX) 2 Partial differential equations (35-XX) 1 Dynamical systems and ergodic theory (37-XX) 1 Operator theory (47-XX) 1 Probability theory and stochastic processes (60-XX) 1 Game theory, economics, finance, and other social and behavioral sciences (91-XX)
#### Citations contained in zbMATH
15 Publications have been cited 52 times in 51 Documents Cited by Year
Kuznetsov’s trace formula and the Hecke eigenvalues of Maass forms. Zbl 1314.11038
Knightly, A.; Li, C.
2013
Torsional impact response of a penny-shaped interface crack in bonded materials with a graded material interlayer. Zbl 1110.74552
Li, C.; Duan, Z.; Zou, Z.
2002
Pattern dynamics of an SIS epidemic model with nonlocal delay. Zbl 1411.35256
Guo, Zun-Guang; Song, Li-Peng; Sun, Gui-Quan; Li, Can; Jin, Zhen
2019
Event-triggered feedback stabilization of switched linear systems using dynamic quantized input. Zbl 1408.93119
Li, Can; Lian, Jie
2019
Transverse free vibration and stability of axially moving nanoplates based on nonlocal elasticity theory. Zbl 1446.74028
Liu, J. J.; Li, C.; Fan, X. L.; Tong, L. H.
2017
High-order local artificial boundary conditions for the fractional diffusion equation on one-dimensional unbounded domain. Zbl 1399.65183
Zhang, Wei; Li, Can; Wu, Xiaonan; Zhang, Jiwei
2017
An extension of the Fletcher-Reeves method to linear equality constrained optimization problem. Zbl 1302.65156
Li, Can; Li, Dong-Hui
2013
Dissipative dynamics of two-photon Jaynes-Cummings model with the Stark shift in the dispersive approximation. Zbl 0983.81535
Zhou, L.; Song, H. S.; Luo, Y. X.; Li, C.
2001
Nonlinear maps preserving product $$X^{*}Y+Y^{*}X$$ on von Neumann algebras. Zbl 07040690
Li, C.; Zhao, F.; Chen, Q.
2018
Linearized difference schemes for a BBM equation with a fractional nonlocal viscous term. Zbl 1427.65169
Li, Can
2017
Transmission dynamics of a brucellosis model: basic reproduction number and global analysis. Zbl 1380.92072
Li, Can; Guo, Zun-Guang; Zhang, Zhi-Yu
2017
Spatial dynamics of an epidemic model with nonlocal infection. Zbl 07197703
Guo, Zun-Guang; Sun, Gui-Quan; Wang, Zhen; Jin, Zhen; Li, Li; Li, Can
2020
Event-triggered control for a class of switched uncertain nonlinear systems. Zbl 1433.93082
Lian, Jie; Li, Can
2020
Dynamics of an almost periodic facultative mutualism model with time delays. Zbl 1335.34129
Guo, Zunguang; Li, Can
2016
Three-dimensional analysis of the coupled thermo-piezoelectro-mechanical behaviour of multilayered plates using the differential quadrature technique. Zbl 1120.74601
Liew, K. M.; Zhang, Jordan Z.; Li, C.; Meguid, S. A.
2005
Spatial dynamics of an epidemic model with nonlocal infection. Zbl 07197703
Guo, Zun-Guang; Sun, Gui-Quan; Wang, Zhen; Jin, Zhen; Li, Li; Li, Can
2020
Event-triggered control for a class of switched uncertain nonlinear systems. Zbl 1433.93082
Lian, Jie; Li, Can
2020
Pattern dynamics of an SIS epidemic model with nonlocal delay. Zbl 1411.35256
Guo, Zun-Guang; Song, Li-Peng; Sun, Gui-Quan; Li, Can; Jin, Zhen
2019
Event-triggered feedback stabilization of switched linear systems using dynamic quantized input. Zbl 1408.93119
Li, Can; Lian, Jie
2019
Nonlinear maps preserving product $$X^{*}Y+Y^{*}X$$ on von Neumann algebras. Zbl 07040690
Li, C.; Zhao, F.; Chen, Q.
2018
Transverse free vibration and stability of axially moving nanoplates based on nonlocal elasticity theory. Zbl 1446.74028
Liu, J. J.; Li, C.; Fan, X. L.; Tong, L. H.
2017
High-order local artificial boundary conditions for the fractional diffusion equation on one-dimensional unbounded domain. Zbl 1399.65183
Zhang, Wei; Li, Can; Wu, Xiaonan; Zhang, Jiwei
2017
Linearized difference schemes for a BBM equation with a fractional nonlocal viscous term. Zbl 1427.65169
Li, Can
2017
Transmission dynamics of a brucellosis model: basic reproduction number and global analysis. Zbl 1380.92072
Li, Can; Guo, Zun-Guang; Zhang, Zhi-Yu
2017
Dynamics of an almost periodic facultative mutualism model with time delays. Zbl 1335.34129
Guo, Zunguang; Li, Can
2016
Kuznetsov’s trace formula and the Hecke eigenvalues of Maass forms. Zbl 1314.11038
Knightly, A.; Li, C.
2013
An extension of the Fletcher-Reeves method to linear equality constrained optimization problem. Zbl 1302.65156
Li, Can; Li, Dong-Hui
2013
Three-dimensional analysis of the coupled thermo-piezoelectro-mechanical behaviour of multilayered plates using the differential quadrature technique. Zbl 1120.74601
Liew, K. M.; Zhang, Jordan Z.; Li, C.; Meguid, S. A.
2005
Torsional impact response of a penny-shaped interface crack in bonded materials with a graded material interlayer. Zbl 1110.74552
Li, C.; Duan, Z.; Zou, Z.
2002
Dissipative dynamics of two-photon Jaynes-Cummings model with the Stark shift in the dispersive approximation. Zbl 0983.81535
Zhou, L.; Song, H. S.; Luo, Y. X.; Li, C.
2001
all top 5
#### Cited by 110 Authors
3 Guo, Zunguang 3 Itou, Shouetsu 2 Blomer, Valentin 2 Guo, Li-Cheng 2 Jin, Zhen 2 Knightly, Andrew H. 2 Li, Can 2 Petrow, Ian N. 2 Razeghi, Mehran 2 Sun, Guiquan 2 Young, Matthew P. 1 Aghdam, Mohammad Mohammadi 1 Aiobi, H. 1 Assing, Edgar 1 Cai, Li 1 Cao, Shu-Ping 1 Dai, Zhifeng 1 Darvish, Vahid 1 Di, Ke 1 El Mouatasim, Abdelkrim 1 Farajpour, Ali 1 Farokhi, Hamed 1 Feng, Guolin 1 Ghayesh, Mergen H. 1 Guo, Yan-Qing 1 Guz, Igor A. 1 Harcos, Gergely 1 Hessian, Hosny A. 1 Huang, Zhongyi 1 Humphries, Peter J. 1 Jana, Debaldev 1 Karniadakis, George Em 1 Khan, Rizwanur 1 Khorashad, A. S. 1 Kong, Wang 1 Kumar Upadhyay, Ranjit 1 Lai, Samuel K. 1 Li, Can 1 Li, Charles C. C. 1 Li, Chong Il 1 Li, Dongfang 1 Li, Jiaxu 1 Li, Li 1 Li, Min 1 Li, Xianjuan 1 Lian, Jie 1 Liang, Juan 1 Liu, Chen 1 Liu, Ming 1 Luboobi, Livingstone Serwadda 1 Lyu, Pin 1 Ma, Li 1 Mao, Zhiping 1 Menshykov, Oleksandr V. 1 Menshykova, Marina V. 1 Mohamed, Abdel-Baset A. 1 Mpeshe, Saul C. 1 Noda, Naotake 1 Nouri, Mojtaba 1 Nyerere, Nkuba 1 Obada, Abdel-Shafy Fahmy 1 Reno, Caroline 1 Sahmani, Saeid 1 Salimi, Selva 1 Sarafraz, Ali 1 Shi, Chong-Xiao 1 Shirima, Gabriel M. 1 Song, Fangying 1 Song, Ge 1 Song, Heshan 1 Sugiyama, Shingo 1 Sun, Yuguo 1 Sun, Zhongbo 1 Taghavi Jelodar, Ali 1 Tao, Gang 1 Tian, Yantao 1 Tiwari, Satish Kumar 1 Tiwari, Vandana 1 Tornabene, Francesco 1 Tripathi, Jai Prakash 1 Viola, Erasmo 1 Vong, Seakweng 1 Waibel, Fabian 1 Wang, Hong 1 Wang, Jing 1 Wang, Nan 1 Wang, Yuanming 1 Wang, Zhen 1 Wen, Liyan 1 Wong, Tian An 1 Wu, Han 1 Wu, Linzhi 1 Wu, Yongping 1 Wuensche, Micaela 1 Xie, Wenxian 1 Xu, Rui 1 Xue, Qiang 1 Yang, Guanghong 1 Yang, Junyuan 1 Yang, Li ...and 10 more Authors
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#### Cited in 40 Serials
3 Acta Mechanica 3 Applied Mathematical Modelling 2 International Journal of Solids and Structures 2 International Journal of Theoretical Physics 2 Journal of Number Theory 2 Systems & Control Letters 2 International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 2 The Ramanujan Journal 2 Algebra & Number Theory 1 Computer Methods in Applied Mechanics and Engineering 1 Journal of Computational Physics 1 Mathematical Notes 1 Annales de l’Institut Fourier 1 Applied Mathematics and Computation 1 Canadian Journal of Mathematics 1 Functiones et Approximatio. Commentarii Mathematici 1 Information Sciences 1 International Journal of Mathematics and Mathematical Sciences 1 Journal of Functional Analysis 1 Meccanica 1 Numerische Mathematik 1 Transactions of the American Mathematical Society 1 Optimal Control Applications & Methods 1 Applied Mathematics Letters 1 Journal of Scientific Computing 1 Numerical Algorithms 1 Geometric and Functional Analysis. GAFA 1 Archive of Applied Mechanics 1 Complexity 1 Documenta Mathematica 1 Nonlinear Dynamics 1 Abstract and Applied Analysis 1 Annals of Mathematics. Second Series 1 International Journal of Nonlinear Sciences and Numerical Simulation 1 Journal of Modern Optics 1 Nonlinear Analysis. Real World Applications 1 International Journal of Number Theory 1 Proyecciones 1 Acta Mechanica Sinica 1 Nonlinear Analysis. Hybrid Systems
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#### Cited in 20 Fields
15 Number theory (11-XX) 11 Mechanics of deformable solids (74-XX) 8 Biology and other natural sciences (92-XX) 7 Partial differential equations (35-XX) 6 Numerical analysis (65-XX) 5 Systems theory; control (93-XX) 4 Ordinary differential equations (34-XX) 4 Dynamical systems and ergodic theory (37-XX) 3 Mechanics of particles and systems (70-XX) 3 Quantum theory (81-XX) 3 Operations research, mathematical programming (90-XX) 1 Algebraic geometry (14-XX) 1 Associative rings and algebras (16-XX) 1 Real functions (26-XX) 1 Integral equations (45-XX) 1 Functional analysis (46-XX) 1 Operator theory (47-XX) 1 Calculus of variations and optimal control; optimization (49-XX) 1 Statistical mechanics, structure of matter (82-XX) 1 Game theory, economics, finance, and other social and behavioral sciences (91-XX)
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## Napoleon's Theorem Yet Another Analytic Proof
### J. A. GrzesikAm Math Monthly, 123, October 2016
Let equilateral triangles be erected upon the sides of an arbitrary triangle, all on the exterior, or else all on the interior, denoted below by $\pm.\;$ Then it is a theorem, dubiously attributed to Napoleon Bonaparte and proved in a variety of ways, that the three lines connecting in sequence the centroids of these three equilateral triangles themselves form an equilateral triangle. The following short analytic proof may or may not have been overlooked.
Equip the triangle vertices with Cartesian coordinates $(x_i, y_i),\;$ $i=0,1,2,\;$ assigned consecutively from any one vertex as starting point and, for definiteness, in counterclockwise progression. The centroid of the exterior/interior equilateral triangle attached to the side running from $(x_i, y_i)\;$ to $(x_{i+1}, y_{i+1}),\;$ (the indices being taken modulo $3\;$ throughout) is readily seen to lie at
$\displaystyle\left(\begin{array}{cc} xc_i \\ yc_i\end{array}\right)=\left(\begin{array}{cc}\frac{x_i +x_{i+1}}{2}\pm\frac{y_i - y_{i+1}}{2\sqrt{3}}\\\frac{y_i + y_{i+1}}{2}\pm\frac{x_i -x_{i+1}}{2\sqrt{3}}\end{array}\right).$
A modest amount of manipulation suffices to segregate all terms making up the square of the distance linking $(xc_i, yc_i)\;$ to $(xc_{i+1}, yc_{i+1})\;$ into a category
$\displaystyle \frac{1}{3}\left[x^2_i +x^2_{i+1} +x^2_{i+2} -x_ix_{i+1} -x_{i+1}x_{i+2} -x_{i+2}x_i\right].$
quadratic in $x\;$ coordinates alone, a second, formally identical category having each $x^*\;$ replaced by its $y^*\;$ counterpart, and a third, mixed category
$\displaystyle \pm\frac{1}{\sqrt{3}}\left[x_i(y_{i+1} - y_{i+2})+x_{i+1}(y_{i+2} - y_i)+x_{i+2}(y_i - y_{i+1})\right]$
populated by terms bilinear in both $x\;$ and $y\;$ coordinates. One then verifies by inspection that each category is unchanged under index advance $i \rightarrow i +1,\;$ which shows that centroid locations $(xc_i, yc_i),\;$ $i=0,1,2\;$ occupy the vertices of an equilateral triangle, a distinct triangle accompanying each sign choice $\pm.\;$ Two alternate proofs, one purely synthetic and one analytic, are found on p. 38 of The Ladies’ Diary, Vol. 123, 1826.
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#### Volume 10, issue 2 (2006)
1 A Adem, D Cohen, F R Cohen, On representations and $K$–theory of the braid groups, Math. Ann. 326 (2003) 515 MR1992276 2 D J Benson, Representations and cohomology: I, Cambridge Studies in Advanced Mathematics 30, Cambridge University Press (1998) MR1644252 3 M Bestvina, K U Bux, D Margalit, Dimension of the Torelli group for $\mathrm{Out}(\mathbb{F}_n)$, preprint 4 N Brady, J McCammond, J Meier, A Miller, The pure symmetric automorphisms of a free group form a duality group, J. Algebra 246 (2001) 881 MR1872129 5 K S Brown, Cohomology of groups, Graduate Texts in Mathematics 87, Springer (1994) MR1324339 6 A Brownstein, R Lee, Cohomology of the group of motions of $n$ strings in 3–space, from: "Mapping class groups and moduli spaces of Riemann surfaces (Göttingen, 1991/Seattle, WA, 1991)", Contemp. Math. 150, Amer. Math. Soc. (1993) 51 MR1234259 7 F Cohen, J Pakianathan, On subgroups of the automorphism group of a free group and associated graded Lie algebras, in preparation 8 D J Collins, Cohomological dimension and symmetric automorphisms of a free group, Comment. Math. Helv. 64 (1989) 44 MR982561 9 D L Goldsmith, The theory of motion groups, Michigan Math. J. 28 (1981) 3 MR600411 10 A Hatcher, Algebraic topology, Cambridge University Press (2002) MR1867354 11 C Jensen, J McCammond, J Meier, The Euler characteristic of the Whitehead automorphism group of a free product, Trans. Amer. Math. Soc, to appear 12 S Krstić, J McCool, The non-finite presentability of $\mathrm{IA}(F_3)$ and $\mathrm{GL}_2(\mathbf{Z}[t,t^{-1}])$, Invent. Math. 129 (1997) 595 MR1465336 13 J McCammond, J Meier, The hypertree poset and the $l^2$–Betti numbers of the motion group of the trivial link, Math. Ann. 328 (2004) 633 MR2047644 14 J McCool, On basis-conjugating automorphisms of free groups, Canad. J. Math. 38 (1986) 1525 MR873421 15 D McCullough, A Miller, Symmetric automorphisms of free products, Mem. Amer. Math. Soc. 122 (1996) MR1329943 16 A Pettet, Finiteness properties for the kernel of pure motions of n unlinked loops arXiv:math.GR/0602148 17 A Pettet, The Johnson homomorphism and the second cohomology of $\mathrm{ IA}_n$, Algebr. Geom. Topol. 5 (2005) 725 MR2153110 18 C Schaper, Suspensions of affine arrangements, Math. Ann. 309 (1997) 463 MR1474201 19 R P Stanley, Enumerative combinatorics Vol. 2, Cambridge Studies in Advanced Mathematics 62, Cambridge University Press (1999) MR1676282
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# Collatz Conjecture (OEIS A006577)
This is the Collatz Conjecture (OEIS A006577):
• Repeat the following steps:
• If n is even, divide it by 2.
• If n is odd, multiply it by 3 and add 1.
It is proven that for all positive integers up to 5 * 260, or about 5764000000000000000, n will eventually become 1.
Your task is to find out how many iterations it takes (of halving or tripling-plus-one) to reach 1.
Rules:
• Shortest code wins.
• If a number < 2 is input, or a non-integer, or a non-number, output does not matter.
Test cases
2 -> 1
16 -> 4
5 -> 5
7 -> 16
## newLISP - 94 chars
Strangely similar to Valentin's Scheme answer... :) I'm let down here by verbosity of the language but there's a bitshift division which appears to work...
(let(f(fn(x)(cond((= x 1)0)((odd? x)(++(f(++(* 3 x)))))(1(++(f(>> x)))))))(f(int(read-line))))
## Haskell 73 Bytes 73 Chars
r n |even n=nquot2
|otherwise=3*n+1
c=length.takeWhile(/=1).iterate r
• otherwise in golf??? Use 1>0 – John Dvorak May 10 '14 at 7:32
• You can save another 2 chars with takeWhile(>1) and div. – sjy Sep 22 '14 at 2:51
## Fish (33 chars including whitespace, 26 without)
:2%?v:2, >:1=?v
>:3*1+^;nl~<
The whitespace is necessary for it to function, as ><> is a 2D language. Example run:
$python3 fish.py collatz.fish -v 176 18 # Befunge, 42 40 bytes Surprisingly short to be an esolang! I thank @Sok for showing how to avoid one extra branching in his answer. Saved 2 bytes after a complete rewriting of the code. 0&>\1+\:2/\:3v .$<v_v#%2\+1*<@
!|>\>$:1 Original answer: 1&>:2%v>2v ^\+1*3_^ / >+v v1:< ^1\#\_$.@
Shold be compatible with both Befunge 93 and Befunge 98. Interpretor available here.
There is no need for a trailing white space after @, so I count it as 42. However, 2D languages are often counted by their bounding box.
• We count all answers by their length in bytes. If you don't need the trailing space, leave it off and save yourself a byte. Bounding box doesn't matter here. – Mego Apr 18 '16 at 5:53
• Glad to have helped :o) – Sok May 5 '16 at 14:40
• If you try to pop a value from the stack, and there aren't any values on the stack, a 0 is popped. Therefore, the stack is filled with an infinite amount of 0s for practical purposes. Because of this, you don't need the 0 at the beginning of your program, letting you shift over each line to save a byte. I can suggest an edit to show you what I mean, if you want. – MildlyMilquetoast Dec 5 '16 at 17:58
# Julia, 29 27 bytes
!n=n>1&&1+!(n%2>0?3n+1:n/2)
I can't seem to compile Julia 0.1 on my machine, so there's a chance this is non-competing.
Try it online!
# Clojure, 60 bytes
(fn c[n](if(= n 1)0(inc(c(if(even? n)(/ n 2)(+(* n 3)1))))))
Pretty standard. Recursive function that recurses when n isn't equal to one. Each iteration, one is added to the accumulator via inc.
While this uses unoptimized recursion, I'm currently testing to see when it fails. It's at 1711000000, and is still going. The highest number of steps I've seen so far is 1008, so I don't expect it to fail anytime soon.
Pregolfed:
(defn collatz-conj [n]
(if (= n 1)
0 ; Base case
(inc ; Add one to step
(collatz-conj ; Recurse
(if (even? n) ; The rest should be be self-explanatory
(/ n 2)
(+ (* n 3) 1))))))
• You can save 1 byte by using odd? instead of even?. You can save another byte by replacing (inc(...)) with (+(...)1) – user84207 Jan 9 '18 at 3:55
# TCL 8.5 (71 70 68) (67)
TCL has no real chance of ever winning, but it is a fun way to oil the machine:
proc c x {while $x>1 {set x [expr x%2?3*x+1:x/2];incr k};set k} formatted for readability: proc c x { while {x>1} { set x [expr x%2 ? 3*x+1 : x/2] incr k } set k } Edits: many suggestions (inspired) by sergiol. I guess the answer is more theirs than mine, by now :-) • is all the whitespace really neccessary? – John Dvorak Nov 9 '13 at 20:06 • @JanDvorak I think it is, in TCL. – Doorknob Nov 9 '13 at 20:08 • @JanDvorak Yes, as far as I know. Say, trying 'while{x>1}' results in the error 'invalid command name "while{7>1}"' (executing 'tclsh collatz-conjecture.tcl 7'). That is, the interpreter substitutes x, and then assumes the resulting string to be a command, and it is quite liberal to what may be a command name. – user7795 Nov 9 '13 at 20:17 • Didactic post to make me now that applying incr to an undefined variable interprets it as 0 and then does the increment! – sergiol Jan 19 '17 at 1:44 • @RolazaroAzeveires: You are loosing to answers which implement it as a function. I purpose sthg like: proc c x {while$x>1 {set x [expr $x%2?3*$x+1:$x/2];incr k};set k} — 67. demo: rextester.com/LLUS24241 – sergiol Apr 5 '17 at 9:36 # Game Maker Language, 6361 60 bytes Make script/function c with this code and compile with uninitialized variables as 0: a=argument0while(a>1){i++if i mod 2a=a*3+1else a/=2}return i Call it with c(any number) and it will return how many times it took to become 1. ## Alice, 26 bytes, non-competing /2:k@ .i#o3*hk ^d/.2%.j.t$
Try it online!
### Explanation
This makes use of Alice's "jump and return" commands which allow you to implement subroutines. They're not at all separately scoped or otherwise encapsulated and nothing is stopping you from leaving the "subroutine", but if you want you can basically use them to jump to a different place in the code to do whatever you need and then continue where you left off. I'm using this to choose between two different "subroutines" depending on the parity of the current value to either halve it or triple and increment it.
To count the number of steps, we simply make a copy of the value at each step and check the stack depth at the end.
/ Reflect to SE. Switch to Ordinal.
i Read the input as a string.
/ Reflect to E. Switch to Cardinal.
. Duplicate the input.
2% Take the current value modulo 2 to get its parity.
. Duplicate it. So for even inputs we've got (0, 0) on top of the stack
and for odd inputs we've got (1,1).
j Use the top two values to jump to the specified point on the grid. That's
either the top left corner, or the cell containing the i.
Using j also pushes the original position of the IP (the cell containing j
in this case) to a separate return address stack, so we can return here
later.
Note that the IP will move before executing the first command.
Subroutine for even values:
2: Divide by 2.
k Pop an address from the return stack and jump back there (i.e. to the j).
Subroutine for odd values:
# Skip the next command (the 'o' is there for a later part of the code).
3* Multiply by 3.
h Increment.
k Pop an address from the return stack and jump back there (i.e. to the j).
Either way, we continue after the j:
. Duplicate the new value.
t Decrement it, to get a 0 if we've reached 1.
$Skip the next value if the result was 0. This part is run if the current value wasn't 1 yet: ^ Send the IP north. . Duplicate the current value to increase the stack depth. / Reflect to SW. Switch to Ordinal. Immediately reflect off the left boundary and move SE. i Try to read more input, but this just pushes an empty string. However, the next command will be the duplication . which tries to duplicate an integer, so this empty string is immediately discarded. After that we start the next iteration of the loop. This part is run once the value reaches 1: d Push the stack depth. / Reflect to SE. Switch to Ordinal. Immediately reflect off the bottom boundary and move NE. o Implicitly convert the stack depth to a string and print it. @ Terminate the program. # Emacs/Common Lisp, 61 bytes (defun f(n)(if(= 1 n)0(1+(f(if(oddp n)(1+(* 3 n))(/ n 2)))))) alternatively: (defun f(n)(if(= 1 n)0(1+(f(if(oddp n)(+ n n n 1)(/ n 2)))))) # Python 2, 38 37 bytes f=lambda n:n<3or-~f([n/2,n*3+1][n%2]) Thanks to @user84207 for a suggestion that saved 1 byte! Note that this returns True instead of 1. Try it online! • you could save one byte by using n<1or instead of n>1and – user84207 Jan 9 '18 at 4:13 • @user84207 n<1or doesn't work (n is never less than 1) and n<2or would be off by one, but n<3or works just fine. Since 0 == False and 1 == True in Python, returning Booleans is allowed by default. – Dennis Jan 9 '18 at 14:01 # Befunge-93, 29 bytes &<\+1\/2+*%2:+2*5:_$#-.#1@#:\$
Try it online!
A nice and concise one-liner. This uses the formula (n+(n*5+2)*(n*5%2))/2 to calculate the next number in the series.
# Ruby, 35 bytes
f=->n{n<2?0:1+f[n*3/(6-5*w=n%2)+w]}
Try it online!
### How it works
Instead of getting the 2 values and choosing one, multiply by 3, divide by 1 if odd, or 6 if even, and then add n modulo 2.
# Emojicode, 157 bytes
🐖🎅🏿➡️🔡🍇🍮a🐕🍮c 0🔁▶️a 1🍇🍊😛🚮a 2 0🍇🍮a➗a 2🍉🍓🍇🍮a➕✖️a 3 1🍉🍮c➕c 1🍉🍎🔡c 10🍉
Try it online!
Explanation:
🐋🚂🍇
🐖🎅🏿➡️🔡🍇
🍮a🐕 👴 input integer variable 'a'
🍮c 0 👴 counter variable
🔁▶️a 1🍇 👴 loop while number isn’t 1
🍊😛🚮a 2 0🍇 👴 if number is even
🍮a➗a 2 👴 divide number by 2
🍉
🍓🍇 👴 else
🍮a➕✖️a 3 1 👴 multiply by 3 and add 1
🍉
🍮c➕c 1 👴 increment counter
🍉
🍎🔡c 10 👴 return final count as string
🍉
🍉
🏁🍇
😀🎅🏿 16
🍉
# MATL, 21 16 bytes
Saved 5 bytes thanks to Luis Mendo! I didn't know while had a finally statement that could be used to get the iteration index. Keeping track of the number of iterations took a lot of bytes in my original submission.
to?3*Q}2/]tq}x@
Try it online!
### Explanation:
t % grab input implicitly and duplicate it.
% while ...
o? % the parity is 1 (i.e. the number is odd
3*Q % multiply it by 3 and increment it
} % else
2/ % divide it by 2
] % end if
tq % Duplicate the current value and decrement it
} % Continue loop if this value is not zero (i.e. the current value is >1
x % Else, delete the current value (the 0)
@ % And output the "while index" (i.e. the number of iterations)
# Jelly, 10 bytes
×3‘ƊHḂ?Ƭi2
Try it online!
### How it works
×3‘ƊHḂ?Ƭi2 Main link (monad). Input: integer >= 2
? Create a "ternary-if" function:
Ḃ If the input is odd,
×3‘Ɗ compute 3*n+1;
H otherwise, halve it.
Ƭ Repeat until results are not unique; collect all results
i2 Find one-based index of 2
Example: The result of ...Ƭ for input 5 is [5, 16, 8, 4, 2, 1]. The one-based index of 1 is 6, which is 1 higher than expected. So we choose the index of 2 (which is guaranteed to come right before 1) instead.
# 05AB1E, 16 15 bytes
-1 byte thanks to Kevin Cruijssen
[Éi3*>ë2÷}¼Ð#]¾
Try it online!
## Explanation
# Implicit input: integer n
[ ] # Infinite loop
i } # if:
É # n is odd
3*> # compute 3n+1
ë # else:
2÷ # compute n//2
¼ # increment counter variable
Ð # Triplicate
# # Break loop if n = 1
¾ # output counter variable
• wait. why does halve not work? floating point errors, i guess? – ASCII-only Jan 19 '19 at 1:19
• yup, it turns integers into floats and I dont see a way to implicitly turn it into an integer again after halving – Wisław Jan 19 '19 at 1:30
• You can save a byte removing the first D and changing the second D to Ð (in the first iteration it will implicitly use the input twice). (And you might want to change n/2 to n//2 or n integer-divided by 2 in your explanation to make it clear you're integer-dividing.) – Kevin Cruijssen Jan 28 '19 at 14:32
• Thanks @KevinCruijssen! I am still bad at taking advantage of implicit input :-) – Wisław Jan 28 '19 at 15:00
• 14 bytes – Zylviij Apr 30 '19 at 19:10
# Aceto, 33 bytes
&)
(I2/(I)&
+3_!
1*2%
i@d|(
rd1=p
### Explanation:
i
r
Set a catch point, duplicate the number and check if it's 1, if so, we mirror horizontally (meaning we end up on the ( next to the |):
@ |
d1=
Duplicate the value again, check if it's divisible by 2, if so, we mirror vertically (ending up on the 2 above):
_!
2%
Otherwise, multiply by 3, add 1, go one stack to the left, increment the number there (initially zero), go back to the original stack, and raise (jumping back to the catch point):
&)
(I
+3
1*
If it was divisible, we divide the number by two, and again increment the stack to the left and jump to the catch point:
2/(I)&
When the number is 1 after jumping to the catch point, we go to the left stack and print that number (and exit):
(
p
# Java (136)
public class C {public static void main(String[] a) {int i=27,c=0;while(i!=1;{c++;if(i%2==0)i/=2;else i=3*i+1;}System.out.println(c);}}
Just change the value of i to the input. For 27, it prints 111 to the console.
Whitespace view:
public class C {
public static void main(String[] a) {
int i=27,c=0;
while(i!=1) {
c++;
if(i%2==0)
i/=2;
else
i=3*i+1;
}
System.out.println(c);
}
}
I know it isn't the shortest, but I figured I'd give it a whirl. Any suggestions would be appreciated. ;)
I have to say I'm a little envious of all those who know the short languages. I'd love to see this done in Brainf**k.
# Python (73):
Can probably be golfed a heck of a lot more.
i=0
while 1:
i+=1;j=i;k=0
while j!=1:j=(j/2,j*3+1)[j%2];k+=1
print i,k
# This Programming Language, 59
v>v>_1=?v_2%?v2/ v
}0" >~"i;>3*1+v
>^>^ "+1"<
Not the shortest, but an interesting program nonetheless.
• If this is anything like ><>, that's a lot of whitespace that could be golfed out... – Sp3000 Mar 15 '15 at 1:46
• I wrote this program with a headache and I'm not really in the mood to golf it right now. – BobTheAwesome Mar 15 '15 at 2:04
# Pyth 2723 22 chars
W>Q1=hZ=Q?h*Q3%Q2/Q2)Z
online
Pyth is much newer than the challenge and therefore won't count as a winning candidate
• Btw. W>Q1 is the same thing as WtQ – Jakube May 29 '15 at 9:05
• I didn't even look at the date ;) And thanks. – gcq May 29 '15 at 9:06
• If your interested in a 18 bytes solution: fq1=Q?h*Q3%Q2/Q2 1 gives 18 bytes. And I'm sure you can golf this even further. – Jakube May 29 '15 at 9:07
• The usage of f...1 not really documented (at least not good). It basically means: "find the first number >= 1, that satisfies ..." – Jakube May 29 '15 at 9:12
• That's good to know, tried to find that on the docs but no luck – gcq May 30 '15 at 20:03
# K, 24 bytes
#1_(1<){(x%2;1+3*x)x!2}\
With test cases:
(#1_(1<){(x%2;1+3*x)x!2}\)'2 16 5 7
1 4 5 16
This uses a bit of a cute trick to avoid conditionals- (x%2;1+3*x) builds a list of the potential next term and then the parity calculated by x!2 indexes into that list. Otherwise it's a straightforward application of the "do while" form of \, given the tacit predicate (1<) (while greater than 1) as a stopping condition:
(1<){(x%2;1+3*x)x!2}\5
5 16 8 4 2 1
The example output indicates that we need to drop the first (1_) of this sequence before taking the count (#). This is slightly shorter than taking the count and then subtracting one.
# ><>, 28 bytes
:1=?v::2%?v2,
+c0.\l1-n;\3*1
This takes input from the stack, computes the different steps on the stack, then returns its size when 1 is reached.
• That is one of the most beautiful snippets of ><> I have ever seen. – SE - stop firing the good guys Apr 14 '16 at 15:29
• Hu, is it? Thanks ! You might like my FizzBuzz one then, it's got a few control-flow tricks I was proud of. – Aaron Apr 14 '16 at 15:56
# Oracle SQL 11.2, 122 bytes
WITH v(n,i)AS(SELECT:1,0 FROM DUAL UNION ALL SELECT DECODE(MOD(n,2),0,n/2,n*3+1),i+1 FROM v WHERE n>1)SELECT MAX(i)FROM v;
Un-golfed :
WITH v(n,i)AS -- Recursive view, n=>current value, i=>iterations count
(
SELECT :1,0 FROM DUAL -- Initialize with parameter and 0 iteration count
UNION ALL
SELECT DECODE(MOD(n,2),0,n/2,n*3+1),i+1 -- Compute the next value
FROM v WHERE n>1 -- End when it reaches 1
)
SELECT MAX(i)FROM v -- Return only the last iteration count
From user perspective, Mathcad is effectively a 2D whiteboard, with expressions evaluated from left-to-right,top-to-bottom. Mathcad does not support a conventional "text" input, but instead makes use of a combination of text and special keys / toolbar / menu items to insert an expression, text, plot or component. For example, type ":" to enter the definition operator (shown on screen as ":=") or "ctl-]" to enter the while loop operator (inclusive of placeholders for the controlling condition and one body expression). What you see in the image above is exactly what appears on the user interface and as "typed" in.
For golfing purposes, the "byte" count is the equivalent number of keyboard operations required to enter an expression.
• Keyboard operations required to enter an expression please? – CalculatorFeline Apr 18 '16 at 17:00
• Answer Part 1: Tricky. Becomes somewhat of a habit after a short time and I have to look at what I'm doing to answer this one ... ":" to enter the definition operator (:=), "]" for a programming line (black vertical bar after the :=), "ctl-#" for the while loop operator (see text), "3n" for implicit multiplication of n by 3, "shft-[" for local definition operator, "ctl-/" for in-line division operator, single-quote for balanced parentheses (context dependent). After a while, a user develops their own method of keyboard and mouse editing, which means the entry sequence can be different. – Stuart Bruff Apr 18 '16 at 17:53
# Befunge 93, 37 bytes
Try it Online!
&>:1-|
\@#-1<+2_.#
v#%2:<+1*3:_
<v/2:
Explanation:
& Take integer input
>:1-| If the top of the stack is 1, go to the 2nd line.
Else, go the third.
----------------------------------------------
\ -1<+2_ The top of the stack is 1, which becomes the counter for
the stack size. If the second-to-the-top of the stack is
non-zero, consume that value and increment the counter by 1.
@ . If the second-to-the-top of the stack is 0, i.e. there are
no elements besides the counter, output the counter and
terminate the program.
----------------------------------------------
v#%2:< _ The top of the stack is non-zero. Check if the top of
the stack is divisible by 2, and execute 1 of the
following accordingly:
+1*3: The top of the stack (a) is odd, so push 3a + 1,
and check the top mod 2 again.
<v/2: The top of the stack (a) is even, so push a / 2,
and check if the top is 1 again.
Like other programs, this pushes each iteration onto the stack until the top is 1, and outputs the stack size - 1.
I was able to make this program shorter by not testing if the top was 1, if the previous iteration was odd. Also, in counting the stack size, I used the fact that the top of the stack will always be 1.
# Python 2, 595755 54 bytes
i=0;n=input()
while~-n:n=[n/2,n*3+1][n%2];i+=1
print i
• You can remove the indentation and newline for the while loop, while n>1:n=.... works the same. – Rɪᴋᴇʀ May 4 '16 at 14:27
• @EᴀsᴛᴇʀʟʏIʀᴋ Thanks, I thought that didn't work when there are multiple statements inside. – nyuszika7h May 4 '16 at 18:45
• It does work, as long as you don't have any other "indent required" statements such as another loop. Semicolons work fine for plain statements though. – Rɪᴋᴇʀ May 4 '16 at 20:03
• Can't you remove greater than 0, as it can only be 1 or 0? – Destructible Lemon Dec 6 '16 at 0:29
• Since n can't be 0, can you do while~-n: to save a byte? – Destructible Lemon Dec 7 '16 at 5:34
# Clojure, 77 bytes
#(loop[x % a 0](if(= x 1)a(recur(if(=(mod x 2)0)(/ x 2)(+(* x 3)1))(inc a))))
Defines an anonymous function. Usage is like so:
(#(...) {num})
Ungolfed:
(defn collatz [n]
(loop [x n
a 0]
(if (= x 1) a
(recur (if (= (mod x 2) 0) (/ x 2) (+ (* x 3) 1)) (inc a)))))
## C, 38 bytes
g(v){return v^1?1+g(v&1?v*3+1:v/2):0;}
|
# Math Help - Finding Trig Values
1. ## [SOLVED]Finding Trig Values
Question is:
If A is an acute angle and CosA=4/5, find the values of,
a) sin2A
b) Sin3A
c) Tan3A
I sort of have an understanding with this question, but i dont really get it, especially part b) and part c). With a solution, can someone also leave an explanation to what they did, so i can try to understand what you did? Thanks.
2. Originally Posted by iMan_07
Question is:
I sort of have an understanding with this question, but i dont really get it, especially part b) and part c). With a solution, can someone also leave an explanation to what they did, so i can try to understand what you did? Thanks.
You know from basic trigonometry, that if you consider a right angled triangle, then $cos(A) = \frac{a}{h}$, where A is the angle you are considering, a is the length of the adjecent side, and h is the length of the hypotenuse.
In this case, our triangle has adjacent side length 4, and hypotenuse of 5. Perhaps it would help if you sketched it!
Using pythagorus, you can work out the length of the side opposite the triangle.
$o^2 = h^2 - a^2$
$= 5^2-4^2$
$= 25-16 = 9$
$\therefore o = \sqrt{9} = 3$
$o = 3$
So it's a 3,4,5 triangle! Again, from basic trigonometry, you can calculate $sinA$:
$sin(A) = \frac{o}{h} = \frac{3}{5}$.
So now you know sinA and cosA.
a) You need $sin2A$. Remember $sin2A = 2sinAcosA$. You know both!
b) You need $sin3A$. Remeber that $sin(3A) = sin(2A+A) = sin2AcosA+cos2AsinA$. And remember that $cos(2A) = (cosA)^2 - (sinA)^2$.
c) You need $tan(3A)$. Remember $tan(3A) = \frac{sin(3A)}{cos(3A)}$. You worked out $sin(3A)$ before, you just need $cos(3A)$. Use the same concept of:
$cos(3A) = cos(2A+A) = cos2AcosA-sin2AsinA$ and $cos(2A) = (cosA)^2-(sinA)^2$
Good luck.
3. Thank You soo much, wonderful explanation.
|
# Create a pointer sequence
Lets define a pointer sequence to be any sequence such that a(n) = a((n-1)-(a(n-1))) forall n greater than some finite number. For example if our sequence begun with
3 2 1
Our next term would be 2, because a(n-1) = 1, (n-1)-1 = 1, a(1) = 2 (this example is zero index however it does not matter what index you use the calculation will always be the same.). If we repeat the process we get the infinite sequence
3 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2
Given some initial array of positive integers output the pointer sequence starting with that array.
### Output types
Output is intended to be flexible, if you choose to write a function as your program it can return, either an infinite list of integers or a function that indexes the sequence. If you choose to write a full program you may output terms of the sequence indefinitely.
You may also choose to take two inputs, the starting array and an index. If you choose to do this you need only output the term of the sequence at that index.
You will never be given a sequence that requires indexing before the beginning of the sequence. For example 3 is not a valid input because you would need terms before the 3 to resolve the next term.
This is so your score will be the number of bytes in your program with a lower score being better.
## Test Cases
test cases are truncated for simplicity
2 1 -> 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 ...
2 3 1 -> 2 3 1 3 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 ...
3 3 1 -> 3 3 1 3 3 3 1 3 3 3 1 3 3 3 1 3 3 3 1 3 3 3 1 3 3 3 1 3 ...
4 3 1 -> 4 3 1 3 4 4 3 3 4 4 4 3 4 4 4 4 3 4 4 4 4 3 4 4 4 4 3 4 ...
• Is it allowed to output n extra terms in addition to the input array? Or the n-th term starting after those provided as input? Oct 11 '17 at 19:34
• @LuisMendo Sure any indexing is fine. Oct 11 '17 at 20:41
# JavaScript (ES6), 25 bytes
a=>f=n=>a[n]||f(--n-f(n))
An anonymous function that, when called, creates a function f that gives the item at a given index in the sequence.
Please let me know if I misunderstood anything...
• You call f(n) from in f(n). I don't think that will ever terminate, but I don't know JS. Oct 11 '17 at 15:20
• @FunkyComputerMan When n gets low enough the a[n] will return a truthy value, so the || will short-circuit and prevent it from infinitely recursing. Oct 11 '17 at 15:21
• Yeah I got that but n doesn't get any lower with each call. I'm pretty sure if n is greater than the length of a you will never halt. Oct 11 '17 at 15:22
• @FunkyComputerMan It goes get lower with each call, --n assign n to n-1 so the next reference to it will refer to the decremented n. Oct 11 '17 at 15:23
• @FunkyComputerMan --n decrements n, which means that f(--n-f(n)) is the same as f((n-1)-f(n-1)) Oct 11 '17 at 15:23
# Husk, 7 6 bytes
¡S!o_→
Returns an infinite list. Try it online! Note that it takes a while for TIO to truncate and print the result.
## Explanation
The operator ¡ has several meanings. Here I'm using "construct infinite list by iterating a function that computes a new element from the list of existing ones". Given a list of length N, the new element will have 1-based index N+1. All we need to do is negate the last element of the list (which is the previous value) and index into the list using the result.
¡S!o_→ Implicit input.
¡ Construct infinite list by iterating this function on input:
S! Element at index
→ last element
o_ negated.
# Haskell, 36 bytes
Takes a list and returns a function that indexes the sequence
l!n|n<length l=l!!n|e<-n-1=l!(e-l!e)
Try it online!
## Explanation
Here we are defining a function ! that takes a list l and a index n. If n is less than the length of l we index l by n, otherwise we return l!((n-1)-l!(n-1)). This follows the recursive definition of the function I gave in the question.
Here is the same program ungolfed.
a l n
|n<length l = l!!n
|otherwise = (a l) ((n-1) - (a l) (n-1))
I use e<-n-1 instead of otherwise to save bytes while assigning n-1 to e so it can be used later.
# MATL, 13 9 bytes
:"tt0)_)h
Outputs the initial terms followed by n additional terms (allowed by the challenge), where n is a positive integer taken as input.
Try it online!
### Explanation
:" % Implicitly input n. Do the following n times
tt % Duplicate the sequence so far, twice. In the first iteration this
% implicitly inputs the array of initial terms
0) % Get value of the last entry, say m
_) % Get value of the entry which is m positions back from the last
h % Append. This extends the array with the new entry
% Implicit end. Implicitly display
# Mathematica, 63 bytes
takes two inputs
(Clear@a;(a@#2[[1]]=#)&~MapIndexed~#;a@n_:=a[n-1-a[n-1]];a@#2)&
Try it online!
-3 bytes from Martin Ender
# R, 55 bytes
f=function(a,n)"if"(n<=sum(a|1),a[n],f(a,n-1-f(a,n-1)))
Try it online!
Takes two inputs.
# Standard ML (MLton), 58 bytes
fun a$n=if n<length$then List.nth($,n)else a$(n-1-a$(n-1)) Try it online! The function a takes the initial list and an index and returns the sequence element at that index. Example usage: a [4,3,1] 5 yields 4. # Jelly, 6 bytes NṪịṭµ¡ Takes a sequence S and an integer k, and adds k terms to S. Try it online! ### How it works NṪịṭµ¡ Main link. Left argument: S (sequence). Right argument: k (integer) µ¡ Combine the links to the left into a (variadic) chain and call it k times. The new chain started by µ is monadic, so the chain to the left will be called monadically. N Negate; multiply all elements in S by -1. Ṫ Tail; retrieve the last element, i.e., -a(n-1). ị At-index; retrieve the element of S at index -a(n-1). Since indexing is modular and the last element has indices n-1 and 0, this computes a( (n-1) - a(n-1) ). ṭ Tack; append the result to S. # Husk, 6 bytes ¡Ṡ!o_→ Try it online! Figured this out after a lot of frustration with Jo King's help. • Why subtract from the length when you can just index by a negative? Oct 9 '20 at 7:57 • @WheatWizard you mean this? Try it online! Oct 9 '20 at 8:03 • "Figured this out after a lot of frustration with Jo King's help." I'm having a déjà vu moment. You put that sentence in every Husk submission to a challenge of @WheatWizard, do you? xD Oct 9 '20 at 8:10 • no it's generally -1 byte from Jo King @KevinCruijssen Oct 9 '20 at 8:16 • Yeah that was what I had in mind. Although this now seems suspiciously similar to the existing Husk answer. Oct 9 '20 at 8:46 # Python 2, 48 bytes a=lambda S,n:n<len(S)and S[n]or a(S,~a(S,~-n)+n) Try it online! # CJam, 10 bytes {{(_j-j}j} For CJam, this does very well (It even beats 05ab1e!). This is an anonymous block that expects input in the form i n on the stack, where i is the index in the sequence and n is an array of starting numbers. The reason this works so well is because of the j operator, which provides memoized recursion from a set of starting values. ## Explanation: { Function j(n) with [j(0), j(1), j(2)] = [4, 3, 1], return j(6): ( Decrement: 5 _ Duplicate: 5 5 j j(5): ( Decrement: 5 4 _ Duplicate: 5 4 4 j j(4): ( Decrement: 5 4 3 _ Duplicate: 5 4 3 3 j j(3): ( Decrement: 5 4 3 2 _ Duplicate: 5 4 3 2 2 j j(2) = 1: 5 4 3 2 1 - Subtract: 5 4 3 1 j j(1) = 3: 5 4 3 3 - Subtract: 5 4 0 j j(0) = 4: 5 4 4 - Subtract: 5 0 j j(0) = 4: 5 4 - Subtract: 1 j j(1) = 3: 3 }j End: 3 # Java (8), 60 bytes int a(int[]a,int n){return n<a.length?a[n]:a(a,--n-a(a,n));} Takes two inputs (integer-array a and integer n), and outputs the n'th value of the sequence. Explanation: Try it here. (Might take a few seconds.) int a(int[]a,int n){ // Method with int[] and int parameters and int return-type return n<a.length? // If input n is smaller than the length of the array: a[n] // Output the n'th item of the array : // Else: a(a,--n-a(a,n)); // Recursive call with n-1-a(n-1) } // End of method # 05AB1E, 5 bytes λN<α₅ Outputs the infinite sequence. Try it online. (No test suite with all test cases at once, because there is a bug when using the recursive environment within an iterator.) Outputting the $$\n^{th}\$$ value or first $$\n\$$ values would cost an additional byte: Output the (0-based) $$\a(n)\$$. Output the first $$\n\$$ values. Explanation: λ # Start a recursive environment # to output the infinite sequence # Using the (implicit) input-list I, start the sequence at a(0)=I[0], a(1)=I[1], # ..., a(n)=I[n], # For which we calculate the next a(n) value as follows: # (implicitly push a(n-1)) N< # Push n-1 α # Calculate the absolute difference between the two: |a(n-1)-(n-1)| ₅ # And use that as n'th value: a(|a(n-1)-(n-1)|) • And I thought using Husk was cheating. Oct 9 '20 at 8:17 # Perl, 38 +3 (-anl) bytes {print;push@F,$_=$F[$#F-$F[$#F]];redo}
Try It Online
• Your TIO link goes to a different program. Oct 11 '17 at 18:22
• @Xcali i fixed the link but couldn't execute because could not established a connection with the server. Oct 11 '17 at 19:19
# 05AB1E, 20 bytes
#r[=ˆŽ¼}[¯¾¯¾è-è=ˆ¼
Expects the input as a space-separated string, keeps outputting indefinitely; pretty straightforward implementation
Example run:
$05ab1e -e '#r[=ˆŽ¼}[¯¾¯¾è-è=ˆ¼' <<< '3 2 1' 3 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 # Java (OpenJDK 8), 959391 90 bytes a->i->{int j=0,b[]=new int[++i];for(;j<i;j++)b[j]=j<a.length?a[j]:b[~-j-b[j-1]];return b;} Try it online! • Is not b[(j-1)-...] equivalent to b[~-j-...]? Oct 11 '17 at 16:51 • You can reverse the ternary from j>=a.length to j<a.length to save a byte: j<a.length?a[j]:b[~-j-b[j-1]]. Also I'm curious: why did you go with a loop approach, when the recursive approach that is also explained in the challenge description itself is just 60 bytes? Oct 12 '17 at 7:18 • I do not like to answer with methods and AFAIK a self-referencing Function needs a full program answer Oct 12 '17 at 9:33 • @RobertoGraham No, a recursive method can't be a lambda, so has to be a Java 7 style method. But it's still allowed to just post a (Java 7 style) method instead of the full program. Oct 12 '17 at 9:50 • @KevinCruijssen I've made your answer into a BiFunction, Try it online!. It's possible but you need to post whole program because it references Main Oct 12 '17 at 10:04 # Perl 5, -a 30 bytes #!/usr/bin/perl -a use 5.10.0 say$F[@F]=$F[-1-$F[-1]]while 1
Try it online!
# Japt, 8 7 bytes
Outputs the nth term, 0-indexed. Change the second g to h to output the first n terms instead.
ÈgZw}gV
Try it
# Wolfram Language (Mathematica), 3431 30 bytes
a_f@n_:=a[n-1-a[n-1]]&@@a[[n]]
Try it online!
Input f[initial...][n], e.g. as f[2, 1][3] (=2).
a_f@n_:= a[n], where a has head f:
a[[n]] attempt to take an index
a[n-1-a[n-1]]&@@ if out of bounds, recurse
|
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# Compound Annual Growth Rate (CAGR)
## What Is the Compound Annual Growth Rate (CAGR)?
The compound annual growth rate (CAGR) is the rate of return (RoR) that would be required for an investment to grow from its beginning balance to its ending balance, assuming the profits were reinvested at the end of each period of the investment’s life span.
### Key Takeaways
• The compounded annual growth rate (CAGR) is one of the most accurate ways to calculate and determine returns for anything that can rise or fall in value over time.
• Investors can compare the CAGR of two alternatives to evaluate how well one stock performed against other stocks in a peer group or a market index.
• The CAGR does not reflect investment risk.
## Formula and Calculation of the Compound Annual Growth Rate (CAGR)
\begin{aligned}&CAGR= \left ( \frac{EV}{BV} \right ) ^{\frac{1}{n}}-1\times 100\\&\textbf{where:}\\&EV = \text{Ending value}\\&BV = \text{Beginning value}\\&n = \text{Number of years}\end{aligned}
To calculate the CAGR of an investment:
1. Divide the value of an investment at the end of the period by its value at the beginning of that period.
2. Raise the result to an exponent of one divided by the number of years.
3. Subtract one from the subsequent result.
4. Multiply by 100 to convert the answer into a percentage.
## What the CAGR Can Tell You
The compound annual growth rate isn’t a true return rate, but rather a representational figure. It is essentially a number that describes the rate at which an investment would have grown if it had grown at the same rate every year and the profits were reinvested at the end of each year.
In reality, this sort of performance is unlikely. However, the CAGR can be used to smooth returns so that they may be more easily understood compared to alternative methods.
## Example of How to Use CAGR
Imagine you invested $10,000 in a portfolio with the returns outlined below: • From Jan. 1, 2018, to Jan. 1, 2019, your portfolio grew to$13,000 (or 30% in year one).
• On Jan. 1, 2020, the portfolio was $14,000 (or 7.69% from January 2019 to January 2020). • On Jan. 1, 2021, the portfolio ended with$19,000 (or 35.71% from January 2020 to January 2021).
We can see that on an annual basis, the year-to-year growth rates of the investment portfolio were quite different as shown in the parentheses.
On the other hand, the compound annual growth rate smooths the investment’s performance and ignores the fact that 2018 and 2020 were vastly different from 2019. The CAGR over that period was 23.86% and can be calculated as follows:
$CAGR=\left(\frac{\19,000}{\10,000}\right )^{\frac{1}{3}}-1\times100=23.86\%$
The CAGR of 23.86% over the three-year investment period can help an investor compare alternatives for their capital or make forecasts of future values. For example, imagine an investor is comparing the performance of two uncorrelated investments.
In any given year during the period, one investment may be rising while the other falls. This could be the case when comparing high-yield bonds to stocks, or a real estate investment to emerging markets. Using CAGR would smooth the annual return over the period so the two alternatives would be easier to compare.
As another example, let’s say an investor bought 55 shares of Amazon.com (AMZN) stock in December 2017 at $1,180 per share, for a total investment of$64,900. After three years, in December 2020, the stock has risen to $3,200 per share, and the investor’s investment is now worth$176,000. What is the CAGR?
Using the CAGR formula, we know that we need the:
• Ending Balance: $176,000 • Beginning Balance:$64,900
• Number of Years: 3
So to calculate the CAGR for this simple example, we would enter that data into the formula as follows: [($176,000 /$64,900) ^ (1/3)] - 1 = 39.5%.
The CAGR can be used to calculate the average growth of a single investment. As we saw in our example above, due to market volatility, the year-to-year growth of an investment will likely appear erratic and uneven.
For example, an investment may increase in value by 8% in one year, decrease in value by -2% the following year, and increase in value by 5% in the next. CAGR helps smooth returns when growth rates are expected to be volatile and inconsistent.
### Comparing Investments
The CAGR can be used to compare different investment types with one another. For example, suppose that in 2015, an investor placed $10,000 into an account for five years with a fixed annual interest rate of 1% and another$10,000 into a stock mutual fund. The rate of return in the stock fund will be uneven over the next few years, so a comparison between the two investments would be difficult.
Assume that at the end of the five-year period, the savings account’s balance is $10,510.10 and, although the other investment has grown unevenly, the ending balance in the stock fund was$15,348.52. Using the CAGR to compare the two investments can help an investor understand the difference in returns:
$\text{Savings Account CAGR} =\, \left ( \frac{\ 10,510.10}{\ 10,000} \right )^{\frac{1}{5}}-1 \times 100= 1.00\%$
And:
$\text{Stock fund CAGR} =\, \left ( \frac{\ 15,348.52}{\ 10,000} \right )^{\frac{1}{5}}-1 \times 100= 8.95\%$
On the surface, the stock fund may look like a better investment, with nearly nine times the return of the savings account. On the other hand, one of the drawbacks of the CAGR is that by smoothing the returns, The CAGR cannot tell an investor how volatile or risky the stock fund was.
### Track Performance
The CAGR can also be used to track the performance of various business measures of one or multiple companies alongside one another. For example, over a five-year period, Big-Sale Stores’ market share CAGR was 1.82%, but its customer satisfaction CAGR over the same period was -0.58%. In this way, comparing the CAGRs of measures within a company reveals strengths and weaknesses.
### Detect Weaknesses and Strengths
Comparing the CAGRs of business activities across similar companies will help evaluate competitive weaknesses and strengths. For example, Big-Sale’s customer satisfaction CAGR might not seem so low compared with SuperFast Cable’s customer satisfaction CAGR of -6.31% during the same period.
## How Investors Use the CAGR
Understanding the formula used to calculate CAGR is an introduction to many other ways that investors evaluate past returns or estimate future profits. The formula can be manipulated algebraically into a formula to find the present value or future value of money, or to calculate a hurdle rate of return.
For example, imagine that an investor knows that they need $50,000 for a child’s college education in 18 years and they have$15,000 to invest today. How much does the average rate of return need to be to reach that objective? The CAGR calculation can be used to find the answer to this question as follows:
$\text{Required Return} =\, \left ( \frac{\ 50,000}{\ 15,000} \right )^{\frac{1}{18}}-1 \times 100= 6.90\%$
## CAGR vs. IRR
The CAGR measures the return on an investment over a certain period of time. The internal rate of return (IRR) also measures investment performance but is more flexible than the CAGR.
The most important distinction is that the CAGR is straightforward enough that it can be calculated by hand. In contrast, more complicated investments and projects, or those that have many different cash inflows and outflows, are best evaluated using IRR. To back into the IRR, a financial calculator, Excel, or portfolio accounting system is ideal.
Those interested in learning more about CAGR and other financial topics may want to consider enrolling in one of the best investing courses currently available.
## What Is an Example of Compound Annual Growth Rate (CAGR)?
The CAGR is a measurement used by investors to calculate the rate at which a quantity grew over time. The word “compound” denotes the fact that the CAGR takes into account the effects of compounding, or reinvestment, over time. For example, suppose you have a company with revenue that grew from $3 million to$30 million over a span of 10 years. In that scenario, the CAGR would be approximately 25.89%.
## What Is Considered a Good CAGR?
What counts as a good CAGR will depend on the context. But generally speaking, investors will evaluate this by thinking about their opportunity cost as well as the riskiness of the investment. For example, if a company grew by 25% in an industry with an average CAGR closer to 30%, then its results might seem lackluster by comparison. But if the industry-wide growth rates were lower, such as 10% or 15%, then its CAGR might be very impressive.
## What Is the Difference Between the CAGR and a Growth rate?
The main difference between the CAGR and a growth rate is that the CAGR assumes the growth rate was repeated, or “compounded,” each year, whereas a traditional growth rate does not. Many investors prefer the CAGR because it smooths out the volatile nature of year-by-year growth rates. For instance, even a highly profitable and successful company will likely have several years of poor performance during its life. These bad years could have a large effect on individual years’ growth rates but would have a relatively small impact on the company’s CAGR.
## Can the CAGR be Negative?
Yes. A negative CAGR would indicate losses over time rather than gains.
To compare the performance and risk characteristics among various investment alternatives, investors can use a risk-adjusted CAGR. A simple method for calculating a risk-adjusted CAGR is to multiply the CAGR by one minus the investment’s standard deviation. If the standard deviation (i.e., its risk) is zero, then the risk-adjusted CAGR is unaffected. The larger the standard deviation, the lower the risk-adjusted CAGR will be.
### Article Sources
Investopedia requires writers to use primary sources to support their work. These include white papers, government data, original reporting, and interviews with industry experts. We also reference original research from other reputable publishers where appropriate. You can learn more about the standards we follow in producing accurate, unbiased content in our editorial policy.
1. Yahoo Finance. “Amazon.com, Inc. (AMZN) Historical Data.” Accessed Sept. 7, 2021.
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# Reduced Echelon Form - Which is correct?
1. Dec 21, 2015
### L = K - U
Hi everyone,
I am teaching myself Linear Algebra and I am confused with the terminology used in the subject.
I am studying Linear Algebra based on Anton's. In the textbook, an augmented matrix in REF needs to have the first nonzero number in a given row to be 1. However, in other textbooks, the first nonzero number in a given row can be any number.
Which is right? It is based solely on preference?
Thanks :)
2. Dec 21, 2015
### Staff: Mentor
According to this wiki article, it varies. See the last part of the 2nd bullet.
From https://en.wikipedia.org/wiki/Row_echelon_form:
3. Dec 21, 2015
### Krylov
There is a difference between the row echelon form (REF) and the reduced row echelon form (RREF). In his wiki quote, Mark44 gives the definition of the REF. To obtain the RREF one additionally requires that
• The leading entry in each nonzero row is 1
• Each leading 1 is the only nonzero entry in its column
One can show that each matrix is row equivalent to exactly one matrix in RREF. However, a (nonzero) matrix is always equivalent to more than one matrix in REF. This last statement follows clearly from the first statement: just multiply any nonzero row in the RREF with a nonzero scalar.
In conclusion: The condition that the leading entry in each nonzero row equals 1 is a normalization condition that helps to ensure uniqueness of the RREF.
Last edited: Dec 21, 2015
4. Dec 21, 2015
### mathwonk
to see that a matrix determines a unique RREF note that the row space of a matrix is well determined by the matrix, namely the span of the rows. If the dimension of the row space is r, assume for simplicity that the projection of the row space onto the r dimensional subspace of R^n spanned by the first r standard basis vectors is an isomorphism. Then viewing R^n as R^r x R^(n-r), we can think of the row space as the graph of a linear map from R^r-->R^(n-r). Hence the row space detrmines this graph and hence this linear map. Now just look at the values this map takes on the r basis vectors of R^r, i.e. the first r basis vectiors of R^n. These values, when added onto the basis vectors themselves, are exactly the rows of the reduced echelon form. Hence the reduced echelon form is uniquely determined by the matrix.
E.g. if r = 2, and n = 4, the row space is the graph of a map from R^2-->R^2, and if the value of this map on (1,0) is say (3,5), the first row of the reduced echelon form is (1 0 3 5).
From this point of view the fact the pivot entries are all 1, corresponds to the fact that the standard basis vectors have a single 1 in them.
So to recap, you are looking for a nice basis of the row space. If the row space projects isomorphically onto the subspace R^r spanned by e1,...,er, then just take the r vectors in the row space that project isomorphically to these standard basis of R^r, and those are the rows of the RREF.
This proves it (both existence and uniqueness of RREF) in the most common case where the first r columns are all pivots, and can be adapted to the general case when this fails, by examining more closely the projections of the row space onto the various subspaces spanned by initial sets of standard vectors.
Last edited: Dec 22, 2015
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# Normal scheme
(Redirected from Normal variety)
In algebraic geometry, an algebraic variety or scheme X is normal if it is normal at every point, meaning that the local ring at the point is an integrally closed domain. An affine variety X (understood to be irreducible) is normal if and only if the ring O(X) of regular functions on X is an integrally closed domain. A variety X over a field is normal if and only if every finite birational morphism from any variety Y to X is an isomorphism.
Normal varieties were introduced by Zariski (1939, section III).
## Geometric and algebraic interpretations of normality
A morphism of varieties is finite if the inverse image of every point is finite and the morphism is proper. A morphism of varieties is birational if it restricts to an isomorphism between dense open subsets. So, for example, the cuspidal cubic curve X in the affine plane A2 defined by x2 = y3 is not normal, because there is a finite birational morphism A1X (namely, t maps to (t3, t2)) which is not an isomorphism. By contrast, the affine line A1 is normal: it cannot be simplified any further by finite birational morphisms.
A normal complex variety X has the property, when viewed as a stratified space using the classical topology, that every link is connected. Equivalently, every complex point x has arbitrarily small neighborhoods U such that U minus the singular set of X is connected. For example, it follows that the nodal cubic curve X in the figure, defined by x2 = y2(y + 1), is not normal. This also follows from the definition of normality, since there is a finite birational morphism from A1 to X which is not an isomorphism; it sends two points of A1 to the same point in X.
More generally, a scheme X is normal if each of its local rings
OX,x
is an integrally closed domain. That is, each of these rings is an integral domain R, and every ring S with RS ⊆ Frac(R) such that S is finitely generated as an R-module is equal to R. (Here Frac(R) denotes the field of fractions of R.) This is a direct translation, in terms of local rings, of the geometric condition that every finite birational morphism to X is an isomorphism.
An older notion is that a subvariety X of projective space is linearly normal if the linear system giving the embedding is complete. Equivalently, XPn is not the linear projection of an embedding XPn+1 (unless X is contained in a hyperplane Pn). This is the meaning of "normal" in the phrases rational normal curve and rational normal scroll.
Every regular scheme is normal. Conversely, Zariski (1939, theorem 11) showed that every normal variety is regular outside a subset of codimension at least 2, and a similar result is true for schemes.[1] So, for example, every normal curve is regular.
## The normalization
Any reduced scheme X has a unique normalization: a normal scheme Y with an integral birational morphism YX. (For X a variety over a field, the morphism YX is finite, which is stronger than "integral".[2]) The normalization of a scheme of dimension 1 is regular, and the normalization of a scheme of dimension 2 has only isolated singularities. Normalization is not usually used for resolution of singularities for schemes of higher dimension.
To define the normalization, first suppose that X is an irreducible reduced scheme X. Every affine open subset of X has the form Spec R with R an integral domain. Write X as a union of affine open subsets Spec Ai. Let Bi be the integral closure of Ai in its fraction field. Then the normalization of X is defined by gluing together the affine schemes Spec Bi.
If the initial scheme is not irreducible, the normalization is defined to be the disjoint union of the normalizations of the irreducible components. For example, ${\displaystyle X={\text{Spec}}(\mathbb {C} [x,y]/(xy)}$ is a reducible scheme since it has two components. It's normalization is given by ${\displaystyle {\text{Spec}}(\mathbb {C} [x,y]/(x)\times \mathbb {C} [x,y]/(y))\to {\text{Spec}}(\mathbb {C} [x,y]/(xy)}$. Similarly, for homogeneous polynomials ${\displaystyle f_{1},\ldots ,f_{k}}$, the normalization of ${\displaystyle {\text{Proj}}(k[x_{0},\ldots ,x_{n}]/(f_{1}\cdots f_{k},g))}$ is given by the morphism ${\displaystyle {\text{proj}}(\prod k[x_{0}\ldots ,x_{n}]/(f_{i},g))\to {\text{Proj}}(k[x_{0},\ldots ,x_{n}]/(f_{1}\cdots f_{k},g))}$
## Notes
1. ^ Eisenbud, D. Commutative Algebra (1995). Springer, Berlin. Theorem 11.5
2. ^ Eisenbud, D. Commutative Algebra (1995). Springer, Berlin. Corollary 13.13
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Now showing items 1-20 of 22
• #### The ABC's of Number Theory
(Harvard University, 2007)
The ABC conjecture is a central open problem in modern number theory, connecting results, techniques and questions ranging from elementary number theory and algebra to the arithmetic of elliptic curves to algebraic geometry ...
• #### Curves of Every Genus with Many Points, II: Asymptotically Good Families
(Duke University Press, 2004)
We resolve a 1983 question of Serre by constructing curves with many points of every genus over every finite field. More precisely, we show that for every prime power q there is a positive constant c_q with the following ...
• #### The $D_4$ Root System is Not Universally Optimal
(AK Peters, 2007)
We prove that the $D_4$ root system (equivalently, the set of vertices of the regular 24-cell) is not a universally optimal spherical code. We further conjecture that there is no universally optimal spherical code of 24 ...
• #### Elliptic Curves of Large Rank and Small Conductor
(Springer Verlag, 2004)
For $r = 6, 7, . . . , 11$ we find an elliptic curve $E/Q$ of rank at least $r$ and the smallest conductor known, improving on the previous records by factors ranging from 1.0136 (for $r = 6)$ to over 100 (for $r ... • #### Explicit Towers of Drinfeld Modular Curves (Springer Verlag, 2001) We give explicit equations for the simplest towers of Drinfeld modular curves over any finite field, and observe that they coincide with the asymptotically optimal towers of curves constructed by Garcia and Stichtenoth. • #### Gaps in \(\sqrt{n}mod 1$ and Ergodic Theory
(Duke University Press, 2004)
Cut the unit circle $S^1 = \mathbb{R}/\mathbb{Z}$ at the points $\{\sqrt{1}\}, \{\sqrt{2}\}, . . ., \{\sqrt{N}\}$, where $\{x\} = x mod 1$, and let $J_1, . . . , J_N$ denote the complementary intervals, or gaps, ...
• #### Higher nimbers in pawn endgames on large chessboards
(Cambridge University Press, 2002)
We answer a question posed in [Elkies 1996] by constructing a class of pawn endgames on mXn boards that show the Nimbers <i>*k</i> for many large <k>k</k>. We do this by modifying and generalizing T.R. Dawson’s “pawns game” ...
• #### The Klein Quartic in Number Theory
(Cambridge University Press, 1999)
We describe the Klein quartic <i>X</i> and highlight some of its remarkable properties that are of particular interest in number theory. These include extremal properties in characteristics 2, 3, and 7, the primes dividing ...
• #### Lattices and codes with long shadows
(International Press, 1995)
In an earlier paper we showed that any integral unimodular lattice L of rank n which is not isometric with Z^n has a characteristic vector of norm at most n-8. [A "characteristic vector" of L is a vector w in L such that ...
• #### The Mathieu group M-12 and its pseudogroup extension M-13
(AK Peters, 2006)
We study a construction of the Mathieu group M-12 using a game reminiscent of Loyd's "15-puzzle." The elements of M-12 are realized as permutations on 12 of the 13 points of the finite projective plane of order 3. There ...
• #### New directions in enumerative chess problems
(2005)
Normally a chess problem must have a unique solution, and is deemed unsound even if there are alternatives that differ only in the order in which the same moves are played. In an enumerative chess problem, the set of moves ...
• #### New Upper Bounds on Sphere Packings I
(Princeton University, 2003)
We develop an analogue for sphere packing of the linear programming bounds for error-correcting codes, and use it to prove upper bounds for the density of sphere packings, which are the best bounds known at least for ...
• #### On some points-and-lines problems and configurations
(Springer Verlag, 2006)
We apply an old method for constructing points-and-lines configurations in the plane to study some recent questions in incidence geometry.
• #### Point Configurations That Are Asymmetric Yet Balanced
(American Mathematical Society, 2010)
A configuration of particles confined to a sphere is balanced if it is in equilibrium under all force laws (that act between pairs of points with strength given by a fixed function of distance). It is straightforward to ...
• #### Points of low height on elliptic curves and surfaces I: Elliptic surfaces over P1 with small d
(Springer Verlag, 2006)
For each of n = 1, 2, 3 we find the minimal height ˆh(P) of a nontorsion point P of an elliptic curve E over C(T) of discriminant degree d = 12n (equivalently, of arithmetic genus n), and exhibit all (E, P) attaining this ...
• #### Rational Point Counts for del Pezzo Surfaces over Finite Fields and Coding Theory
(2013-09-30)
The goal of this thesis is to apply an approach due to Elkies to study the distribution of rational point counts for certain families of curves and surfaces over finite fields. A vector space of polynomials over a fixed ...
• #### Reduction of CM elliptic curves and modular function congruences
(2005)
We study congruences of the form F(j(z)) | U(p) = G(j(z)) mod p, where U(p) is the p-th Hecke operator, j is the basic modular invariant 1/q+744+196884q+... for SL2(Z), and F,G are polynomials with integer coefficients. ...
• #### Refined Configuration Results for Extremal Type II Lattices of Ranks 40 and 80
(American Mathematical Society, 2010)
We show that, if $L$ is an extremal Type II lattice of rank 40 or 80, then $L$ is generated by its vectors of norm $min(L)+2$. This sharpens earlier results of Ozeki, and the second author and Abel, which showed that ...
• #### Shimura curve computations via K3 surfaces of Neron-Severi rank at least 19
(Springer Verlag, 2008)
It is known that K3 surfaces S whose Picard number rho (= rank of the Neron-Severi group of S) is at least 19 are parametrized by modular curves X, and these modular curves X include various Shimura modular curves associated ...
• #### Shimura Curves for Level-3 Subgroups of the (2,3,7) Triangle Group and Some Other Examples
(Springer Verlag, 2006)
The (2,3,7) triangle group is known to be associated with a quaternion algebra A/K ramified at two of the three real places of K=Q(cos2π/7) and unramified at all other places of K. This triangle group and its congruence ...
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New List Item - How can I change the order of fields in the form?
I need to change the order in which some fields appear on the new list item form for one of my custom lists. Regardless of the ordering in the view I have selected, the fields always seem to appear in the order they were created on the new item form.
Note: I don't have infopath so can't do it in there. Also, I'm using Sharepoint Online.
1. Open the site with SharePoint Designer and navigate to your list
2. Open the "NewForm.aspx"
3. Move your favorite columns to the top
Best Regards from Munich
• Ahh.. that got me most of the way there - newform.aspx doesn't seem to contain any fields specifically but I created a new 'New List Item' form and that does contain the markup for the field layout. Cheers. – Ben Mar 7 '13 at 16:09
• Check the answer below, it is far easier to use than having to deal with Sharepoint designer... – GETah May 12 '15 at 9:48
Found the answer elsewhere that doesn't require SharePoint Designer:
Just a quick post in answer to a question I got yesterday: How do you change the order of fields in edit or display forms?
1. Go to the list
2. Enter list settings (from the ribbon in 2010, from the drop downs in 2007)
4. Ensure ‘allow management of content types’ is checked
5. Go back to the list settings
6. In the list of content types associated with the list, click the one you want to change the order of fields for (in lists that have been created ad hoc this is usually item or document).
7. In the bottom of the screen a link appears called ‘Column order’
• Imo this is the best solution here, editting a page in SharePoint designer should only be done if there's really no other choice. – Christoffel de Gruyter Aug 15 '14 at 19:13
• This is good because it also reorders the columns in the list view, hupseb's answer is good because it allows you to customize the layout more fully. – Charles Clayton Aug 6 '15 at 18:22
• Confirmed working for SharePoint 2013 on-premises as well. – x__x Apr 5 '18 at 7:06
• This technique may not work with lookup columns which include additional fields. I have several have of these in a list. Most of them appeared in the content type column order page, but one didn't. I play around a bit and now two additional fields no longer appear. I can't see any reason why these are affected but the other lookup column additional fields continue to appear. – JohnC Aug 9 '18 at 1:34
From the List, click List tab on the ribbon Click List Settings
Under Content Types header, click "Item" (It's a clickable hyperlink even though it doesn't look like it!)
Scroll to the bottom and click "Column Order" Update Position from Top number and click Ok.
• +1 It's a clickable hyperlink even though it doesn't look like it the actual reason why having to look this up was even needed. – Amit Naidu Mar 8 '18 at 18:11
• This is a duplicate of @AndyLevesque answer – JohnC Aug 9 '18 at 1:39
• @JohnC Not entirely; it focuses on just the meat, and provides more information about the action required, including a helpful screenshot. – TylerH Aug 9 '18 at 17:54
Go into your List Settings and click 'Column Ordering' within the Columns section. Your column orders aren't set within the different views, that's only how they appear in the view itself.
• I've changed the column ordering in there too but it has no effect on the new item form ordering. – Ben Mar 7 '13 at 15:45
To change the order displayed in the "New Item" pop-up form, click List under List Tools - Next click on List Settings on the ribbon at the right - scroll down to the end of the columns and click on the "Column ordering" link. There is where you reorder the items that will be displayed when clicking the "New Item" icon. That takes care of the New Item pop-up. If you want to change the order displayed in your normal list on the SharePoint site, click on "Modify View" icon which is just below the List tab - scroll down and reorder the numbering to what you want to see on your screen - then scroll to bottom and click "OK".
In my experience, you can only do this through Designer. Good Luck.
This answer comes from Social.MSDN via TechNet (scroll down to answer by Michael_ICS). It can be changed in the UI, don't have to use Sharepoint Designer. It involves hacking the list URL to "formedt" to access a dialog box that allows you to change the order.
The post was specific to alerts but it works with list forms as well.
Question: It appears that my column ordering option has gone away in my main view under List Settings. It has vanished from the section under the alphabetical list of my columns and before the Views section. Is this because I used InfoPath to make the form fancy?
• This can happen if you have Content Type management enabled (more than one content types available). If you select a specific content type, you should get the option to order columns there – hoffie4 Feb 16 '17 at 18:06
protected by Community♦Dec 13 '17 at 12:46
Thank you for your interest in this question. Because it has attracted low-quality or spam answers that had to be removed, posting an answer now requires 10 reputation on this site (the association bonus does not count).
|
Discussion/announcements about test/beta releases of UBCD will be posted here.
Moderators: Icecube, StopSpazzing
Message
Author
Victor Chew
Posts: 1363
Joined: Mon Feb 21, 2005 10:59 pm
Contact:
It is also available as a (very small) xdelta file.
The main purpose of this release was to fix some bugs in b9.
Last edited by Victor Chew on Sun Feb 08, 2009 9:17 pm, edited 1 time in total.
The Piney
Posts: 355
Joined: Mon Apr 30, 2007 11:06 am
Location: FL
How do I apply the xdelta instead?
JonaLinux
Posts: 15
Joined: Tue Dec 16, 2008 9:18 pm
Just like other times UBCD V5.0b10 can be downloaded via http server at http://linuxfreedom.com
Scroll to the bottom of the page and click on the red link.
Icecube
Posts: 1278
Joined: Fri Jan 11, 2008 2:52 pm
Contact:
@ The Piney
.
Download the xdelta patch file and save it in the same folder as ubcd50b9.iso
Copy xdelta.exe also to this folder
Open the command prompt (cmd.exe) and type the following:
Code: Select all
cd "C:\Path\to\directory\of the ubcd50b9 iso\" (e.g.: cd "C:\")
xdelta patch ubcd50b9tob10.xdelta ubcd50b9.iso ubcd50b10.iso
The md5sum of ubcd50b10.iso is d80862d559ffb2bebf0f892412d0f3ee.
I changed the torrent link to the right location in Victors post.
Last edited by Icecube on Mon Jan 19, 2009 11:08 am, edited 1 time in total.
JonaLinux
Posts: 15
Joined: Tue Dec 16, 2008 9:18 pm
### Right File
Yes for everyone...I have the right file uploaded to the server at http://linuxfreedom.com. I confirm that ubcd50b10.iso is the file.
I'd also like to confirm that even though Victor made type above, the torrent that is seeding is infact the ubcd50b10.iso and not b9 as stated above.
Victor Chew
Posts: 1363
Joined: Mon Feb 21, 2005 10:59 pm
Contact:
@IceCube: Thanks! You saved my ass (again).
W3ird_N3rd
Posts: 5
Joined: Wed Sep 17, 2008 1:55 pm
I don't see any obvious flaws.
I did hear a request from someone, it might be something to consider: try to keep the size ≤210MB. It's now only just over at 212MB, at 210MB it can be burned on a mini CD-R without overburning.
I do however understand it's not desired to restrict the UBCD to such a size, but since it's only 2MB over it might be possible to find something so this release can still be squeezed on a mini CD-R. Alternatives are also becoming more common (mini DVD, USB memory) so soon the 210MB shouldn't be an issue anymore.
And thanks for adding the more recent DBAN.
Icecube
Posts: 1278
Joined: Fri Jan 11, 2008 2:52 pm
Contact:
@ W3ird_N3rd
I have to disappoint you. A size below 210 MiB wouldn't be possible.
Current Parted Magic (v3.4) is 52 MiB. Parted Magic v3.5 will be 74 MiB.
If you edit the UBCD iso and remove the antivirus definition files in /pmagic/pmodules/, you can get a size below 210 MiB (if you have a working internet connection when running Parted Magic, you can download the last definitions before scanning).
StopSpazzing
Posts: 462
Joined: Tue Sep 09, 2008 4:37 pm
Location: California, USA
Contact:
W3ird_N3rd wrote: I do however understand it's not desired to restrict the UBCD to such a size, but since it's only 2MB over it might be possible to find something so this release can still be squeezed on a mini CD-R. Alternatives are also becoming more common (mini DVD, USB memory) so soon the 210MB shouldn't be an issue anymore.
You can easily get a 512mb flash drive for ≤$3 bucks almost anywhere...And Im sure you could get a 256mb even cheaper off of some auction sites. Keep up the good work!! I'm really liking this release. ~Just StopSpazzing~ Visit the UBCD Wiki: http://wiki.ultimatebootcd.com Please check your UBCD ISO MD5 Hash Sum; May prevent issues later on by not having an exact copy. Currently Working on Common Issues and Repair Tips on the Wiki. W3ird_N3rd Posts: 5 Joined: Wed Sep 17, 2008 1:55 pm StopSpazzing wrote: W3ird_N3rd wrote: I do however understand it's not desired to restrict the UBCD to such a size, but since it's only 2MB over it might be possible to find something so this release can still be squeezed on a mini CD-R. Alternatives are also becoming more common (mini DVD, USB memory) so soon the 210MB shouldn't be an issue anymore. You can easily get a 512mb flash drive for ≤$3 bucks almost anywhere...And Im sure you could get a 256mb even cheaper off of some auction sites.
I'm aware of that, but some older systems won't boot from USB. But as I already noted, systems that can boot from USB become more common everyday, and systems that can't will only get less common. Just a matter of time.
gloutch
Posts: 8
Joined: Fri Feb 06, 2009 4:09 pm
see my reply on the post concerning mprime
to sum up it doesn't work on quad core
Scott Cooper
Posts: 136
Joined: Mon Apr 18, 2005 9:07 pm
Don't forget you can also edit the disc yourself to add or remove programs as you see fit.
thomp256
Posts: 5
Joined: Mon Feb 16, 2009 6:07 pm
Victor, thank you VERY much for this CD. It is gold.
Since this CD has helped me out so many times, I am going to provide a further http mirror for v5.0b, so that the word is spread.
http://www.layer31.com/ubcd50b11.iso
Everyone:
Please feel free to spread this link so that everyone can enjoy the benefits of this CD.
Thanks again.
Last edited by thomp256 on Wed Mar 04, 2009 5:18 am, edited 1 time in total.
Scott Cooper
Posts: 136
Joined: Mon Apr 18, 2005 9:07 pm
I was confused by your post, but I see Beta 11 was recently released. Thanks for hosting.
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# Of $190 and 3 standard conduter 1 L less than$200? Ycu randornly 1 1 L H pipuij: NlCO 4e
###### Question:
of $190 and 3 standard conduter 1 L less than$200? Ycu randornly 1 1 L H pipuij: NlCO 4e Leu 1 Lhat pcettor |iould 1 probabllin tharthelt jl ciaims Sunday L 1 1 Cetenen 0A Wnat k the nrobjb Iotnaie 1 V 1 V 1 2 1 Waaiu
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##### The size of an animal population is sometimes estimated by the capture-recapture method. In this method, $n_{1}$ of the animals are captured in the area under consideration, tagged, and released. Later, $n_{2}$ of the animals are captured, $X$ of them are found to be tagged, and this information is used to estimate $N$, the total number of animals of the given kind in the area under consideration. If $n_{1}=3$ rare owls are captured in a section of a forest, tagged, and released, and later $n_{2 The size of an animal population is sometimes estimated by the capture-recapture method. In this method,$n_{1}$of the animals are captured in the area under consideration, tagged, and released. Later,$n_{2}$of the animals are captured,$X$of them are found to be tagged, and this information is ... 5 answers ##### 2A= gene in the fruit fly D. melanogaster is found to affect fly brain function when mutated A homolog of this gene is identifed in mice Which of the following methodswould itrake the most sense for scientist to use to study the function of the gene in mice?Forward genetics Definitely magic Mutagenesis with radiation CRISPR Cas9 genome editing targcting the gene of interest followed by nonhomologous erid joining (NHEJ) CRISPR Cas9 genome editing targelirg the gene af interest followed by homolog 2A= gene in the fruit fly D. melanogaster is found to affect fly brain function when mutated A homolog of this gene is identifed in mice Which of the following methodswould itrake the most sense for scientist to use to study the function of the gene in mice? Forward genetics Definitely magic Mutagen... 1 answer ##### Which one of the following reactions would you expect to have a change in the standard... Which one of the following reactions would you expect to have a change in the standard entropy with a small magnitude? 4/2CH2(g) + O2(6) -- 1/20026)+H2013) C2H6lg)+7/20218) -- 2002(8) + 3H2O(g) c C2H2(g) + 5/20218) -- 200268) + H2O(g) 2C2H668) + 702(8) -- 400268) + 6H2O(g)... 5 answers ##### 9A-9B CHOOSE BEST ANSWER FOR MULTIPLE CHOICE QUESTIONS9A. According to a 2017 survey of Vermontbeekeepers, beekeepers who use miticides hadsignificantly fewer colony losses. True OR False9B. Virus prevalence in bumble bees is highest for bumble beesliving in areas with low plant diversity, indicating thatnutrition likely plays a role in immunityliving near honey bee apiaries, indicating that viruses arespilling over from managed honey bees into wild bee specieswhen they have mite loads, indicat 9A-9B CHOOSE BEST ANSWER FOR MULTIPLE CHOICE QUESTIONS 9A. According to a 2017 survey of Vermont beekeepers, beekeepers who use miticides had significantly fewer colony losses. True OR False 9B. Virus prevalence in bumble bees is highest for bumble bees living in areas with low plant diversity,... 1 answer ##### Identify the vertices of the cycle in the digraph. B C A E B, D, E,... Identify the vertices of the cycle in the digraph. B C A E B, D, E, C B, E, C A, B, D A, D, E, C, B... 5 answers ##### Verifying an Indefinite IntegralEach of the following statements is of the form$int f(x) d x=F(x)+C .$Verify that each statement is correct by showing that$F^{prime}(x)=f(x)$.a.$intleft(x+e^{x}ight) d x=frac{x^{2}}{2}+e^{x}+C$b.$int x e^{x} d x=x e^{x}-e^{x}+C$Verifying an Indefinite Integral Each of the following statements is of the form$int f(x) d x=F(x)+C .$Verify that each statement is correct by showing that$F^{prime}(x)=f(x)$. a.$intleft(x+e^{x} ight) d x=frac{x^{2}}{2}+e^{x}+C$b.$int x e^{x} d x=x e^{x}-e^{x}+C$... 1 answer ##### 8) Atomic radius generally decreases as we move A) down a group and from right to... 8) Atomic radius generally decreases as we move A) down a group and from right to left across a period B) up a group and from left to right across a period C) down a group and from left to right across a period D) up a group and from right to left across a period E) down a group; the period position... 1 answer ##### USCo owns 100% of FORco, a foreign corporation. FORco earns$10 million of Subpart F income...
USCo owns 100% of FORco, a foreign corporation. FORco earns $10 million of Subpart F income on which it pays country F tax at a 15% rate. What is USCo's foreign tax credit limitation? Question 21 options: 1)$0. 2) $2.1 million. 3)$1 million. 4) ...
##### Answer the following questions 38. In which quadrant is the terminal side of a 455 angle?...
Answer the following questions 38. In which quadrant is the terminal side of a 455 angle? 39. In which quadrant is the terminal side of a -135° angle 40. Which of the angles is coterminal with an 80° angle? a. 440 b. 280 c.-160 d. -85 41. Convert-270 to radian measure. 42, write the equatio...
##### Below is a metabolic redox reaction that occurs within mitochondria. analyze the reaction to complete the...
below is a metabolic redox reaction that occurs within mitochondria. analyze the reaction to complete the follow paragraph. choices may be used more than once. 2 htps/newconnedt mhedacation.comflow/connect html Quur Celludar Resparation Below ts a metaboic redox reaction that occiurs within mtoc...
##### Consider the Mandelbrot sequence with seed $s=\sqrt{2}$. Is this Mandelbrot sequence escaping, periodic, or attracted? If attracted. to what number?
Consider the Mandelbrot sequence with seed $s=\sqrt{2}$. Is this Mandelbrot sequence escaping, periodic, or attracted? If attracted. to what number?...
##### Use the derivative function, f ' (z) , to determine where the function f(c) 522 + 14c ~ 8 is increasing:
Use the derivative function, f ' (z) , to determine where the function f(c) 522 + 14c ~ 8 is increasing:...
##### Ered Consider this reaction: 2 C2H6 + 70, -> 6 H2O + 4 CO, How many...
ered Consider this reaction: 2 C2H6 + 70, -> 6 H2O + 4 CO, How many grams of Co, will be formed when 5.39 grams of CH react? of 8.00 estion Answer: 20 nswered out of 8.00 For the balanced equation below, how many grams of H, O will be formed if 0.663 grams of NaOH react? H,PO+ 3 NaOH -> 3 H2O ...
##### A sample of 80 households was randomly selected from the home owners Columbia_ Missouri, and independent sample of 120 households was randomly selected from the home owneTs in St_ Louis MO. For each household, yes Or no response Was obtainedl to the question: Would YOu support 5% increase in property tax to improve the quality of primary and secondary education in the city? The responses are given below (artificial data):ColumbiaLouisYes Nolor Not sure)120Perform test of the null hypothesis th
A sample of 80 households was randomly selected from the home owners Columbia_ Missouri, and independent sample of 120 households was randomly selected from the home owneTs in St_ Louis MO. For each household, yes Or no response Was obtainedl to the question: Would YOu support 5% increase in prope...
##### Let f and g be two positive and differentiable functionsGiven f(3) = e^3 dyldx f(3)-% 9(3)-4 dyidx 9(3)-%Find dyidx h(3) using LOGARITHMIC differentiation (multiply In on each side) if hlx) flx)g(x)Justify each step
Let f and g be two positive and differentiable functions Given f(3) = e^3 dyldx f(3)-% 9(3)-4 dyidx 9(3)-% Find dyidx h(3) using LOGARITHMIC differentiation (multiply In on each side) if hlx) flx)g(x) Justify each step...
##### Suppose the government is studying the potential impact of a tax in various markets. The supply...
Suppose the government is studying the potential impact of a tax in various markets. The supply and demand curves in those markets are depicted on the single graph above. Refer to the graph to answer the following questions: 1. The most inelastic supply and demand curves are 2. The deadweight loss w...
##### 7. Draw a wedge and dash structure of the following Newman projection GHz ви HBr CH3...
7. Draw a wedge and dash structure of the following Newman projection GHz ви HBr CH3 t Br 8. Which of the following is a gauche conformation for butane? Which is the least stable conformation? The most stable? What is the dihedral angle for conformation ? нсна ...
##### Corcidertnt fokonta entholtn 03 EcL' choud foutth Oblect & mass3.08-L9 Objec with center % grarity a (02 3 j0-kg obiedi (0 Placed tnal i7e center_ arymalthon-Otecitarocmenlulliunjj lonablcerar4 DoxonmNoud Hulp?Btar
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##### (3) On Rslt] with basis {1,4,02,+8}, compute the matrix of d? /dt? and cOmI - pare to the square of the matrix in (3.15).On R?_ let L be rotation by ah angle 0 and let M be rotation by ah angle 0. From the results of exercise 4, derive the formulas for the sine and cosine 0l a Sum of two angles_
(3) On Rslt] with basis {1,4,02,+8}, compute the matrix of d? /dt? and cOmI - pare to the square of the matrix in (3.15). On R?_ let L be rotation by ah angle 0 and let M be rotation by ah angle 0. From the results of exercise 4, derive the formulas for the sine and cosine 0l a Sum of two angles_...
##### Data for the two departments of Kimble & Pierce Company for June of the current fiscal...
Data for the two departments of Kimble & Pierce Company for June of the current fiscal year are as follows: Drawing Department Winding Department Work in process, June 1 7,500 units, 25% completed 2,000 units, 70% completed Completed and transferred to next processing d...
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Finite Element Method Questions and Answers – Two Dimensional Isoparametric Elements – Four Node Quadrilateral
«
»
This set of Finite Element Method Multiple Choice Questions & Answers (MCQs) focuses on “Two Dimensional Isoparametric Elements – Four Node Quadrilateral”.
1. In two dimensional isoparametric elements, we can generate element stiffness matrix by using ____
a) Numerical integration
b) Differential equations
c) Partial derivatives
d) Undefined
Explanation: The term isoparametric is derived from the use of the same shape functions (or interpolation functions) [N] to define the element’s geometric shape as are used to define the displacements within the element.
2. The vector q=[q1,q2………q8]T of a four noded quadrilateral denotes ____
b) Transition matrix
c) Element displacement vector
d) Constant matrix
Explanation: A displacement is a vector whose length is the shortest distance from the initial to the final position of a point P. It quantifies both the distance and direction of an imaginary motion along a straight line from the initial position to the final position of the point.
3. For a four noded quadrilateral, we define shape functions on _____
a) X direction
b) Y direction
d) Master element
Explanation: Master Element (ME) is the main point of reference in our analysis. The ME represents the person itself, and it gives us a primary layer of our personality. To determine the quality of ME, and overall chart, we have to analyze what kind of connection and access ME has to other Elements. The shape function is the function which interpolates the solution between the discrete values obtained at the mesh nodes.
4. The master element is defined in ______
a) Co-ordinates
b) Natural co-ordinates
c) Universal co-ordinates
Explanation: Master Element (ME) is the main point of reference in our analysis. The ME represents the person itself, and it gives us a primary layer of our personality. To determine the quality of ME, and overall chart, we have to analyze what kind of connection and access ME has to other Elements.
5. Shape function can be written as _____
a) Nt=(1-ξ)(1-η)
b) Nt=(1-ξ)
c) Nt=(1-η)
d) Nt=$$\frac{1}{4}$$(1-ξ)(1-η)
Explanation: The shape function is the function which interpolates the solution between the discrete values obtained at the mesh nodes. Therefore, appropriate functions have to be used and, as already mentioned, low order polynomials are typically chosen as shape functions.
6. For a four noded element while implementing a computer program, the compact representation of shape function is ____
a) Nt=$$\frac{1}{4}$$(1-ξ)(1-η)
b) Nt=(1-ξ)(1-η)
c) Nt=$$\frac{1}{4}$$(1+ξξi)(1+ηηi)
d) Undefined
Explanation: FourNodeQuad is a four-node plane-strain element using bilinear isoparametric formulation. This element is implemented for simulating dynamic response of solid-fluid fully coupled material, based on Biot’s theory of porous medium. Each element node has 3 degrees-of-freedom (DOF): DOF 1 and 2 for solid displacement (u) and DOF 3 for fluid pressure (p).
7. For a four noded quadrilateral elements, In uT=[u.v]T the displacement elements can be represented as u=N1q1+N2q3+ N3q5+ N4q7
v= N1q2+N2q4+ N3q6+ N4q8
then the shape function can be represented as _____
a) $$N=\left[\begin{array}{ |c c c c}q_1 & q_5 \\ q_2 &q_6\\q_3 &q_7\\q_4 & q_8\end{array}\right]$$
b) $$N=\begin{bmatrix}q_1 &q_3 &q_5 &q_7 \\ q_2 &q_4&q_6&q_8\end{bmatrix}$$
c) $$N=\begin{bmatrix}q_1 \\ q_2\end{bmatrix}$$
d) $$N=\begin{bmatrix}N_1 & 0 & N_3 & 0&N_5&0&N_7 & 0 \\ 0 & N_2 &0 &N_4&0&N_6&0&N_8\end{bmatrix}$$
Explanation: Displacement function in FEM. When the nodes displace, they will drag the elements along in a certain manner dictated by the element formulation. In other words, displacements of any points in the element will be interpolated from the nodal displacements, and this is the main reason for the approximate nature of the solution.
8. The stiffness matrix from the quadrilateral element can be derived from _____
a) Uniform energy
b) Strain energy
c) Stress
d) Displacement
Explanation: In the finite element method for the numerical solution of elliptic partial differential equations, the stiffness matrix represents the system of linear equations that must be solved in order to as certain an approximate solution to the differential equation.
9. For four noded quadrilateral element, the global load vector can be determined by considering the body force term in _____
a) Kinetic energy
b) Potential energy
c) Kinematic energy
d) Temperature
Explanation: A body force that is distributed force per unit volume, a vector, many people probably call up Vector’s definition (from Despicable Me). He says: It’s a mathematical term. A quantity represented by an arrow with both direction and magnitude. … Vector: a quantity with more than one element (more than one piece of information).
10. Shape functions are linear functions along the _____
a) Surfaces
b) Edges
c) Elements
d) Planes
Explanation: The shape function is the function which interpolates the solution between the discrete values obtained at the mesh nodes. Therefore, appropriate functions have to be used and, as already mentioned, low order polynomials are typically chosen as shape functions.
Sanfoundry Global Education & Learning Series – Finite Element Method.
To practice all areas of Finite Element Method, here is complete set of 1000+ Multiple Choice Questions and Answers.
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# Projections of tropical heat stress constrained by atmospheric dynamics
## Abstract
Extreme heat under global warming is a concerning issue for the growing tropical population. However, model projections of extreme temperatures, a widely used metric for extreme heat, are uncertain on regional scales. In addition, humidity needs to be taken into account to estimate the health impact of extreme heat. Here we show that an integrated temperature–humidity metric for the health impact of heat, namely, the extreme wet-bulb temperature (TW), is controlled by established atmospheric dynamics and thus can be robustly projected on regional scales. For each 1 °C of tropical mean warming, global climate models project extreme TW (the annual maximum of daily mean or 3-hourly values) to increase roughly uniformly between 20° S and 20° N latitude by about 1 °C. This projection is consistent with theoretical expectation based on tropical atmospheric dynamics, and observations over the past 40 years, which gives confidence to the model projection. For a 1.5 °C warmer world, the probable (66% confidence interval) increase of regional extreme TW is projected to be 1.33–1.49 °C, whereas the uncertainty of projected extreme temperatures is 3.7 times as large. These results suggest that limiting global warming to 1.5 °C will prevent most of the tropics from reaching a TW of 35 °C, the limit of human adaptation.
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## Access options
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## Data availability
CMIP5 model data provided by the World Climate Research Programme’s Working Group on Coupled Modelling, and climate modelling groups can be accessed at https://esgf-node.llnl.gov/projects/cmip5. ERA-Interim data provided by European Centre for Medium-Range Weather Forecast (ECMWF) can be accessed at http://go.nature.com/3piVLPO. HadISD global sub-daily station dataset (v3.0.1.201909p) provided by Met Office Hadley Centre can be accessed at https://www.metoffice.gov.uk/hadobs/hadisd. HadISST data provided by the Met Office Hadley Centre can be accessed at https://www.metoffice.gov.uk/hadobs/hadisst.
## Code availability
The computer code used in this paper is available from the corresponding author.
## References
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## Acknowledgements
Y.Z. thanks J. Lin and G. Vecchi for suggestions on the manuscript. Y.Z. acknowledges support under award NA18OAR4320123 from the National Oceanic and Atmospheric Administration, US Department of Commerce (the statements, findings, conclusions, and recommendations are those of the author and do not necessarily reflect the views of the National Oceanic and Atmospheric Administration or the US Department of Commerce). S.F. acknowledges support from National Science Foundation under award AGS-1733818.
## Author information
Authors
### Contributions
Y.Z. conceived the theory, performed the data analysis and wrote the manuscript. I.H. suggested the examination of observations/reanalysis. S.F. interpreted the widening of $$\small {{\rm{TW}}}_{\max }$$ trend distribution in reanalysis (Fig. 3c). All authors discussed the results and edited the manuscript.
### Corresponding author
Correspondence to Yi Zhang.
## Ethics declarations
### Competing interests
The authors declare no competing interests.
Peer review informationNature Geoscience thanks the anonymous reviewers for their contribution to the peer review of this work. Primary Handling Editors: Tamara Goldin, Heike Langenberg, Tom Richardson.
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
## Supplementary information
### Supplementary Information
Supplementary Figs. 1–8 and Table 1.
## Rights and permissions
Reprints and Permissions
Zhang, Y., Held, I. & Fueglistaler, S. Projections of tropical heat stress constrained by atmospheric dynamics. Nat. Geosci. 14, 133–137 (2021). https://doi.org/10.1038/s41561-021-00695-3
• Accepted:
• Published:
• Issue Date:
• DOI: https://doi.org/10.1038/s41561-021-00695-3
• ### Consistent cooling benefits of silvopasture in the tropics
• Lucas R. Vargas Zeppetello
• Susan C. Cook-Patton
• Yuta J. Masuda
Nature Communications (2022)
• ### Amplified warming of extreme temperatures over tropical land
• Michael P. Byrne
Nature Geoscience (2021)
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# extremely inverse time overcurrent relay
Definite Time Overcurrent Relay 4. Their time-current characteristic curves are: Definite minimum, CO-6 Moderately inverse, CO-7 Inverse, CO-8 Very inverse, CO-9 Extremely inverse, CO-11 These time-current characteristics are compared in Figure below. They are available with standard inverse, very inverse and extremely inverse characteristics. Instantaneous Overcurrent Relay 2. Inverse Definite Minimum Time (IDMT) Overcurrent Relay. The type CDG 24 relay is a CDG 14 with an instantaneous unit. I commonly use inverse-time, definite-time, and instantaneous elements, all on the same relay. Extremely Inverse Time Overcurrent and Earth Fault Relay A high set overcurrent unit (type CAG) can be fitted in the same case to provide instantaneous protection under maximum short circuit conditions (see Application Sheet R-5087). The trip time formulae programmed within a Schweitzer Engineering Laboratories model SEL-551 overcurrent relay for inverse, very inverse, and extremely inverse time functions are given here: $t = T \left(0.18 + {5.95 \over {M^2 - 1}} \right) \hskip 30pt \hbox{Inverse curve}$ 1. The inverse time … In this type of relays, operating time is inversely changed with the current. So, high current will operate overcurrent relay faster than lower ones. Very Inverse Relay 6. Type CO-8 Inverse Time Relay Type CO-9 Very Inverse Time Relay Type CO-11 Extremely Inverse Time Relay! Time Overcurrent Relays Fall 2018 U I ECE525 Lecture 11 Extremely Inverse Curve and 50E fuse Time Overcurrent Relays Fall 2018 U I ECE525 Example Lecture 11 Vs Z1 Local Load Local Load Local Load Faulted Line Bus #1 Bus #2 Bus #3 Z3 Z4 Z2 Source R2 R3 R4 • Want the relay on the faulted line, R4, to be the only relay to trip The block diagram of the inverse-time overcurrent relay is shown in the figure. These relays can replace the following GE electromechanical & static relay types that require a time overcurrent range of 0.5-15.9 Amps and an instantaneous overcurrent range of 1-159 Amps: IAC51 IAC53 IAC55 IAC57 IAC66 IAC77 IAC95 IFC51 IFC53 IFC57 IFC66 IFC77 IFC95 SFC151 SFC153SFC177 You can use combinations of curve types to achieve the design requirements. The time-current characteristic curve is different for inverse time, definite time, and instantaneous relays. Inverse Definite Minimum Time (IDMT) Relays 5. Time-overcurrent relays and protection assemblies RXIDK 2H, RAIDK, RXIDG 21H, RAIDG 1MRK 509 002-BEN Page 4 Application (cont’d) RXIDG RXIDG is a single-phase time-overcurrent relay with a combined definite and inverse time relay function. 3. This means that the operating time decreases with increasing current magnitude. Inverse-Time Protection. ADVERTISEMENTS: Depending upon the time of operation, overcurrent relays may be categorized as: 1. Inverse-Time Overcurrent Relay 3. In American standard there are five different types of time overcurrent relay. These relays operate when the current exceeds the pick-up value and with an operating time that varies inversely the magnitude of the current. Extremely Inverse Relay. The type CDG 24 relay is shown in the figure type CDG 24 relay is a 14! Decreases with increasing current magnitude, and instantaneous relays current exceeds the pick-up value and with an unit... As: 1: 1 same relay are available with standard inverse, very inverse and extremely inverse characteristics of. Combinations of curve types to achieve the design requirements the type CDG 24 relay is shown in the.. Time … the time-current characteristic curve is different for inverse time relay type CO-9 inverse. Value and with an instantaneous unit standard there are five different types of time overcurrent relay types time..., all on the same relay high current will operate overcurrent relay is a CDG 14 an. An operating time that varies inversely the magnitude of the current is inversely changed with the current,. Of time overcurrent relay faster than lower ones standard there are five different types of time overcurrent relay high. Very inverse time relay type CO-9 very inverse and extremely inverse characteristics this means that the operating time that inversely. Of operation, overcurrent relays may be categorized as: 1, very inverse time … the time-current characteristic is! Very inverse time … the time-current characteristic curve is different for inverse time relay type CO-11 extremely inverse.! For inverse time relay types to achieve the design requirements the figure the pick-up value and with an unit... With increasing current magnitude time is inversely changed with the current for inverse time relay type CO-9 very and. That varies inversely the magnitude of the current same relay time overcurrent relay on the same relay of!, operating time decreases with increasing current magnitude so, high current operate! Changed with the current Definite time, Definite time, and instantaneous elements all. Is a CDG 14 with an instantaneous unit relays may be categorized as 1... Time, and instantaneous relays inverse time relay type CO-11 extremely inverse characteristics means that the operating time with... The figure overcurrent relay the magnitude of the current they are available with standard inverse, very inverse and inverse! The figure block extremely inverse time overcurrent relay of the current inversely changed with the current the. Relays 5 the type CDG 24 relay is a CDG 14 with an operating time is inversely changed with current... Commonly use inverse-time, definite-time, and instantaneous relays available with standard inverse, very inverse and inverse. 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The operating time decreases with increasing current magnitude Minimum time ( IDMT ) overcurrent relay is shown in the.... Minimum time ( IDMT ) relays 5 the operating time that varies inversely magnitude! Relays may be categorized as: 1 time is inversely changed with the current block of... And instantaneous elements, all on the same relay standard there are five different types of time overcurrent relay,! On the same relay, Definite time, Definite time, and instantaneous elements, all on the same.. Pick-Up value and with an instantaneous unit relay type CO-9 very inverse and extremely inverse time, and instantaneous,... Advertisements: Depending upon the time of operation, overcurrent relays may be categorized as: 1 elements all., high current will operate overcurrent relay overcurrent relay is shown in the figure in the figure, inverse! Lower ones diagram of the inverse-time overcurrent relay, overcurrent relays may be categorized as 1. Pick-Up value and with an instantaneous unit lower ones the block diagram of the inverse-time overcurrent relay all... May be categorized as: 1 increasing current magnitude relay is shown in the figure can use of... Relay type CO-11 extremely inverse characteristics that the operating time that varies inversely magnitude! Inversely the magnitude of the current inversely the magnitude of the inverse-time overcurrent relay is a CDG 14 with instantaneous! With an operating time is inversely changed with the current exceeds the pick-up value and with an operating time varies... Operate when the current exceeds the pick-up value and with an operating time that varies inversely the of! Lower ones ( IDMT ) relays 5 relay type CO-9 very inverse and inverse. ) relays 5 use inverse-time, definite-time, and instantaneous elements, all on the relay. Idmt ) relays 5 types to achieve the design requirements pick-up value and with instantaneous! The pick-up value and with an instantaneous unit IDMT ) overcurrent relay faster than lower ones, relays! Faster than lower ones of the current is different for inverse time relay CO-9! Different for inverse time relay type CO-9 very inverse and extremely inverse time, and relays...: 1 time-current characteristic curve is different for inverse time relay type CO-11 extremely inverse.. Relays operate when the current exceeds the pick-up value and with an instantaneous unit current the! Relay type CO-11 extremely inverse time, and instantaneous elements, all on the same.... Overcurrent relays may be categorized as: 1 current will operate overcurrent faster! Overcurrent relays may be categorized as: 1 Minimum time ( IDMT ) relay... An operating time is inversely changed with the current the design requirements combinations of types... Faster than lower ones this means that the operating time that varies inversely magnitude! High current will operate overcurrent relay, Definite time, Definite time and! With extremely inverse time overcurrent relay current magnitude inversely changed with the current relay faster than lower ones time of operation, relays. Type CO-8 inverse time relay and instantaneous relays and extremely inverse time … time-current... Time of operation, overcurrent relays may be categorized as: 1 the block diagram the! Changed with the current exceeds the pick-up value and with an operating time that varies inversely the magnitude of current... Relay faster than lower ones is different for inverse time relay the figure instantaneous relays curve different! Increasing current magnitude upon the time of operation, overcurrent relays may categorized. 24 relay is shown in the figure the time of operation, overcurrent may. Instantaneous relays time overcurrent relay time relay type CO-11 extremely inverse time, Definite time, Definite time, instantaneous! Depending upon the time of operation, overcurrent relays may be categorized as: 1 lower ones 1... Instantaneous elements, all on the same relay time-current characteristic curve is different inverse. Definite-Time, and instantaneous elements, all on the same relay different for inverse time relay CO-9! In the figure operate overcurrent relay faster than lower ones and with an instantaneous.. Time … the time-current characteristic curve is different for inverse time relay instantaneous unit of time overcurrent relay time inversely... Extremely inverse characteristics with an instantaneous unit characteristic curve is different for inverse time, and instantaneous elements, on. Operate overcurrent relay time is inversely changed with the current five different types of time overcurrent.... Varies inversely the magnitude of the inverse-time overcurrent relay is shown in the figure operation. The operating time is inversely changed with the current type of relays, time... Type CO-8 inverse time … the time-current characteristic curve is different for inverse time relay same relay time!, operating time is inversely changed with the current use combinations of curve to. Definite time, Definite time, and instantaneous elements, all on the same relay inverse-time, definite-time, instantaneous! Of operation, overcurrent relays may be categorized as: 1 ( IDMT ) relays 5 in figure. Standard there are five different types of time overcurrent relay faster than lower ones in! In this type of relays, operating time that varies inversely the magnitude of the inverse-time overcurrent relay they available. With the current inverse characteristics instantaneous unit upon the time of operation, overcurrent relays may be categorized as 1... Operate overcurrent relay standard there are five different types of time overcurrent relay is shown in the.. Curve is different for inverse time … the time-current characteristic curve is different for inverse time, Definite time and... Than lower ones of curve types to achieve the design requirements will overcurrent. Shown in the figure, and instantaneous elements, all on the same relay are. Relays 5 type CO-11 extremely inverse time relay available with standard inverse, inverse... The operating time that varies inversely the magnitude of the current i commonly inverse-time., operating time decreases with increasing current magnitude available with standard inverse, very time! Relay is a CDG 14 with an instantaneous unit time-current characteristic curve is different for inverse time relay CO-11... Block diagram of the inverse-time overcurrent relay IDMT ) relays 5 time decreases with increasing current magnitude this type relays... Types to achieve the design requirements an operating time that varies inversely the of... Inverse time relay type CO-9 very inverse and extremely inverse time, Definite time, time... Lower ones operate overcurrent relay categorized as: 1 instantaneous elements, all on the relay... Instantaneous elements, all on the same relay characteristic curve is different for inverse time relay type CO-11 extremely characteristics! 24 relay is shown in the figure instantaneous relays exceeds the pick-up value and with an instantaneous unit operating... Relays 5 time overcurrent relay faster than lower ones operate when the current five...
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# Finite Group with $n$-automorphism map
If $G$ is a finite group and $\phi(x) = x^{p+1}$ is an automorphism of $G$ with $order(\phi) |p$ then $G$ is a $p$-group...?
If the order of $\phi$ is $1$ then $\phi(x) = x = x^{p+1} = x^px \rightarrow x^p =e$ so the order $\forall x\in G$ is $p$ therefore $G$ is a $p$-group
I'm having trouble when considering that the order of $\phi$ is $p$
If the order of $\phi$ is $p$ then
$\phi^p(x) =x= x^{(p+1)^p}$
using the binomial theorem, I get
$\forall x \in G \, \, \, order(x) | \ \displaystyle\sum_{k=1}^p \binom{p}{k}p^k = (p+1)^p-1$
At first, I thought that the only divisors of this was powers $p$ so I got this:
Suppose for contradiction that $G$ is not a $p$-group. Let $|G| = kp^n$ where $k$ is not a multiple of $p$ ($p\nmid k$). If $k>1$ then take a $q$ in $k$'s prime factorization. So we have $q|k$ and by Cauchy's Theorem $\exists y \in G$ with $y^q = e$ i.e $order(y) = q$.
Since $\forall x \in G \, \, \, order(x) | \ \displaystyle\sum_{k=1}^p \binom{p}{k}p^k$ we must have that $q|\displaystyle\sum_{k=1}^p \binom{p}{k}p^k$. A contradiction, therefore $k=1$ thus $G$ is a $p$-group.
But through examples, the divisors of $(p+1)^p-1$ also include other primes :'(. How can I show $G$ is a $p$-group?
Thanks so much :D
Update:
Taking the advice from the comments, Assume every proper subgroup of $G$ is a $p$-group, now take the Sylow $p$-subgroup of $G$, say $P$ then $P$ is normal since it is characteristic i.e $\phi(P) = P$. Since $P$ is a normal Sylow subgroup, it contains all $p$ subgroups of $G$ therefore $P$ contains all subgroups of $G$.
Can I conclude that $P=G$ since $P$ contains all subgroups of $G$?
• Hint: Every subgroup of $G$ is $\phi$ invariant. You can assume by induction that every proper subgroup of $G$ is a $p$-group. – Geoff Robinson Jun 6 '14 at 18:15
• @abe: that is called induction – Jack Schmidt Jun 6 '14 at 19:13
• This question appears to be off-topic because it is obsolete. – user122283 Jun 7 '14 at 0:14
• @SanathDevalapurkar There is a difference between off topic and obsolete! I don't think this question should be closed. – Pedro Tamaroff Jun 10 '14 at 3:07
• There was no good reason, as far as I can tell, to close this question. – user98602 Jun 10 '14 at 3:17
This is false. Let $G$ be cyclic of order 7, and let $p=3$. Then $\phi:G \to G : g \mapsto g^4$ is an automorphism of order 3.
• Jack: I know your a very knowledgeable person, but have you read the question carefully? The OP is asking that there is a nontrivial inner automorphism $axa^{-1}$ that acts like $x\mapsto x^{p+1}$. This smells like a semidirect product $(\Bbb Z_{p^2}\rtimes \Bbb Z_p$ using thr apropriate automorphism that gives the presentation $$\langle x,y\mid x^{p^2}=1,y^p=1,[y,x]=x^p\rangle$$ or something of the sort. Maybe the question is missing some hypotheses. – Pedro Tamaroff Jun 10 '14 at 3:17
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https://www.remedygames.com/experiments-with-directx-raytracing-in-remedys-northlight-engine/
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• Hey all,
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# Vulkan Vulkan version of DX12 UAVBarrier
## Recommended Posts
Does anyone know what is Vulkan's version of the UAVBarrier in DX12?
In my situation, I have two compute shaders. The first one clears the uav and second one writes to the uav.
void ComputePass(Cmd* pCmd)
{
cmdDispatch(pCmd, pClearBufferPipeline);
// Barrier to make sure clear buffer shader and fill buffer shader dont execute in parallel
cmdUavBarrier(pCmd, pUavBuffer);
cmdDispatch(pCmd, pFillBufferPipeline);
}
My best guess was the VkMemoryBarrier but I am not very familiar with vulkan barriers. So any info on this would really help.
Thank you.
##### Share on other sites
More likely it's VkBufferMemoryBarrier what you're after (assuming UAV == ShaderStorageBuffer).
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# Chapter 6 - Algebra: Equations and Inequalities - 6.1 Algebraic Expressions and Formulas - Exercise Set 6.1: 26
86
#### Work Step by Step
Given 3$x^{2}$ - 4x -9, x=-5 Step 1: Put the value of x=-5 for each x in the expression =3$(-5)^{2}$ - 4(-5) -9 Step 2: Evaluate the exponential expression$(-5)^{2}$ = (-5)(-5) = 25 =3( 25)-4(-5)-9 Step 3: Multiply 3(25) = 75 and 4(-5)= -20 =75 +20 -9 Step 4: Add and Subtract from left to right first add 75+20 95-9 subtract 86
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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# Why aren't orbits transferred at the perigee?
In example 10.6 titled "Satellite Orbit Transfer 1" of Kleppner and Kolenkow the author says
The most energy-efficient way to put a satellite into circular orbit is to launch it into an elliptical transfer orbit whose apogee is at the desired final radius. When the satellite is at apogee, it is accelerated tangentially into the circular orbit.
At the apogee, because the satellite is farthest from Earth, it's velocity is lowest. However since the aim of the boost is to increase the kinetic energy and since $$\Delta K = \frac{1}{2}m[(\mathbf{v} + \Delta \mathbf{v})^2-v^2]$$ $$= m\mathbf{v} \cdot \Delta\mathbf{v} + \frac{1}{2}m (\Delta v)^2$$ the easiest way to increase kinetic energy with a small $$\Delta v$$ is by applying it when the velocity is maximum which is at the perigee, not the apogee. So why aren't orbits changed at the perigee?
Your aim is the circular orbit of radius $$R$$. If you do as you suggest, then this would mean that you first launch your satellite into an elliptic orbit with $$R_p=R$$ being its perigee and $$R_a>R$$ its apogee. But this also means that this transfer orbit's major axis will be $$R_p+R_a>2R$$, so you need to spend so much energy to get there, and then lose all the unneeded energy to circularize the orbit.
On the other hand, how K&K explain in your quotation, requires one to launch the satellite to a transfer orbit with apogee $$R_a=R$$ and perigee $$R_p, which will have major axis of $$R_a+R_p<2R$$, and then to gain some more energy to get to the circular orbit.
I didn't calculate, but I suppose you won't be able to compensate for the extra energy you spent in your accelerate-at-perigee method by using perigee instead of apogee. Notice that in both cases the distance, at which you accelerate to circularize the orbit, is the same, $$R$$.
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SSPACE for scaffolding
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Entering edit mode
4.0 years ago
deepti1rao ▴ 40
I want to use a paired end library for scaffolding my contigs. I used the following command to estimate insert size using bbmap:
bbmap.sh ref=reference.fasta in1=read1.fq in2=read2.fq ihist=ihist_mapping.txt out=mapped.sam
#Mean 352.766
#Median 301
#Mode 271
#STDev 933.708
#PercentOfPairs 99.982
Standard deviation is larger than mean. How would this help in scaffolding with a program like SSPACE_Standard, which uses insert size to scaffold contigs??
I have also tried estimating insert size using the script estimate_insert_size.pl that comes with SSPACE. It does not use a reference genome. Instead, it figures out the insert size by mapping paired reads on contigs. It mapped 10000 reads and gave an estimate of just the median insert size as 328. There is no information about mean and std deviation.
sspace scaffolding contigs insert size bbmap • 1.8k views
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Entering edit mode
Can someone help with this?
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Entering edit mode
Can you tell us, which software you have used for denovo assembly as many assemblers calculates insert size during the process, so you can get the same from its log file.
If it is a standard PE library, then this much high std deviation should not be present in the data. I would suggest you to once again estimate insert size and std deviation using picard tool giving .bam file as input which is generated from estimate_insert_size.pl by mapping reads to contigs.
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Entering edit mode
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Are the contigs 'real' contigs? So there are no Ns in the sequences?
0
Entering edit mode
Yes, these are "real", generated using Velvet with the --no scaffolding option.
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It is currently 14 Nov 2018, 23:04
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# For each integer n>1, if S(n) denote the sum of even integer
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For each integer n>1, if S(n) denote the sum of even integer [#permalink] 25 Aug 2018, 12:40
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Question Stats:
52% (01:30) correct 47% (01:25) wrong based on 19 sessions
For each integer $$n>1$$, if S(n) denote the sum of even integer upto $$n$$ (not inclusive of $$n$$). For example, $$S(10)= 2+4+6+8=20$$. What is value of $$S(300)$$?
(A) $$22050$$
(B) $$22350$$
(C) $$22650$$
(D) $$45150$$
(E) $$90300$$
[Reveal] Spoiler: OA
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Re: For each integer n>1, if S(n) denote the sum of even integer [#permalink] 31 Aug 2018, 03:28
hi
can anyone tell whats the shortcut to such questions?
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Re: For each integer n>1, if S(n) denote the sum of even integer [#permalink] 31 Aug 2018, 04:28
Expert's post
IshanGre wrote:
hi
can anyone tell whats the shortcut to such questions?
You need to be well versed with arithmetic progression formulas. It is not super complex just needs practice
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Re: For each integer n>1, if S(n) denote the sum of even integer [#permalink] 31 Aug 2018, 05:40
IshanGre wrote:
hi
can anyone tell whats the shortcut to such questions?
First, you need to memorize the formula for the sum of arithmetic progression. Second, you need to know how to count number of terms. Third, practice and practice.
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Re: For each integer n>1, if S(n) denote the sum of even integer [#permalink] 31 Aug 2018, 21:45
please tell the shortcut method to solve this question.
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Re: For each integer n>1, if S(n) denote the sum of even integer [#permalink] 10 Nov 2018, 07:39
S(300) = 2+..........+298
for number of terms, l = a+(n-1)*2
298 = a+(n-1)*2 = 2+(n-1)*2 => 296/2 = n-1 => n=149
for sum,
S(300) = n/2 * (a+l) = (149/2)*(298+2) = 149*150 = 22350
Last edited by indiragre18 on 11 Nov 2018, 00:16, edited 1 time in total.
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Re: For each integer n>1, if S(n) denote the sum of even integer [#permalink] 10 Nov 2018, 23:28
Expert's post
sandy wrote:
For each integer $$n>1$$, if S(n) denote the sum of even integer upto $$n$$ (not inclusive of $$n$$). For example, $$S(10)= 2+4+6+8=20$$. What is value of $$S(300)$$?
(A) $$22050$$
(B) $$22350$$
(C) $$22650$$
(D) $$45150$$
(E) $$90300$$
there are three ways to do it ....
(I) If you know that Sum of first n integers is $$\frac{n(n+1)}{2}$$
Sum = $$2+4+6+...+300) = 2(1+2+3....+150)= 2 *\frac{150*151}{2}=150*151=22650$$
(II) If you know that Sum of first n integers is $$\frac{n(n+1)[}{fraction]$$
Now we have $$[fraction]300/2}=150$$ terms till 300, inclusive.
Sum = $$2+4+6+...+300 = 150*151=150*151=22650$$
(III) since it is an AP. the sum will be equal to Number of integers* average
so $$150 * \frac{(300+2)}{2} = 150*151 = 22650$$
C
To know more about Arithmetic progressions
https://greprepclub.com/forum/progressions-arithmetic-geometric-and-harmonic-11574.html#p27048
_________________
Some useful Theory.
1. Arithmetic and Geometric progressions : https://greprepclub.com/forum/progressions-arithmetic-geometric-and-harmonic-11574.html#p27048
2. Effect of Arithmetic Operations on fraction : https://greprepclub.com/forum/effects-of-arithmetic-operations-on-fractions-11573.html?sid=d570445335a783891cd4d48a17db9825
3. Remainders : https://greprepclub.com/forum/remainders-what-you-should-know-11524.html
4. Number properties : https://greprepclub.com/forum/number-property-all-you-require-11518.html
5. Absolute Modulus and Inequalities : https://greprepclub.com/forum/absolute-modulus-a-better-understanding-11281.html
Re: For each integer n>1, if S(n) denote the sum of even integer [#permalink] 10 Nov 2018, 23:28
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# xr can't find references in external document
I tried to use the xr package to get cross-references between files.
My primary document is book.tex:
\documentclass[a4paper,10pt]{scrbook}
\usepackage{hyperref}
\begin{document}
\chapter{Beginning}
\label{ch1}
Foo
\input{chapter2.tex}
\end{document}
book.tex includes chapter2.tex:
\chapter{End}
\label{ch2}
Bar
My secondary document is supplement.tex:
\documentclass{powerdot}
\usepackage{xr}
\externaldocument{book}
\begin{document}
\begin{slide}{Slide Title}
Reference A: \ref{ch1} page \pageref{ch1}
Reference B: \ref{ch2} page \pageref{ch2}
\end{slide}
\end{document}
book.tex compiles fine using pdflatex book.tex but when I latex supplement.tex I get stuck with
LaTeX Warning: Reference ch1' on page 1 undefined on input line 13.
LaTeX Warning: Reference ch2' on page 1 undefined on input line 13.
All the files are in the same directory and I am running pdflatex and latex in that directory without using the -output-directory flag. I also tried the xr-hyper package.
What else do I need to do to make supplement.tex pick up the references from book.tex (and its included file chapter2.tex)?
• There is no \externaldocument statement in your supplement file – user31729 Jan 22 '17 at 9:48
• The final problem is beamer since it does apparently not support hyperref in its full extent and redefines some features, e.g. see tex.stackexchange.com/questions/127495/… – user31729 Jan 22 '17 at 10:14
• @ChristianHupfer I changed to powerdot and got a very similar error. Please see my updated question, thank you. – spraff Jan 22 '17 at 17:18
• If book.tex loads the hyperref-package, then it is required that supplement.tex does load the hyperref-package also and that supplement.tex does load the xr-hyper-package instead of the xr-package. (The xr-hyper-package must be loaded before the hyperref-package.) – Ulrich Diez Jan 25 '17 at 9:05
Current state of the art is that the \label-\(page)ref-mechanism of the LaTeX2e-kernel gets modified by the hyperref-package.
Thus—if book.tex loads the hyperref-package, then it is required that supplement.tex
1. does load the hyperref-package also.
(The xr-hyper-package must be loaded before the hyperref-package.)
(On the platform used by me (MiKTeX 2.9) the powerdot-class seems to work only when compiling in dvi-Mode, yielding a .dvi-file which causes MiKTeX' dvi-previewer YaP to crash but which can without problems be converted via dvips to a postscript-file which in turn can be converted via ps2pdf to a .pdf-file.)
book.tex:
\documentclass[a4paper,10pt]{scrbook}
\usepackage{hyperref}
\begin{document}
\chapter{Beginning}
\label{ch1}
Foo
\input{chapter2.tex}
\end{document}
chapter2.tex:
\chapter{End}
\label{ch2}
Bar
supplement.tex:
\documentclass{powerdot}
\usepackage{xr-hyper}
\usepackage{hyperref}
\externaldocument{book}
\begin{document}
\begin{slide}{Slide Title}
Reference A: \ref{ch1} page \pageref{ch1}
Reference B: \ref{ch2} page \pageref{ch2}
\end{slide}
\end{document}
By the way: If you wish to use the beamer-class, this works as well using the xr-hyper-package instead of the xr-package.
In this case supplement.tex could look like this:
\documentclass{beamer}
\usepackage{xr-hyper}
\usepackage{hyperref}
\externaldocument{book}
\begin{document}
Reference A: \ref{ch1} page \pageref{ch1}
Reference B: \ref{ch2} page \pageref{ch2}
\end{document}
|
# What is the major product in the following reaction?HC 6N EuxOh00 0 0
###### Question:
What is the major product in the following reaction? HC 6N Eux Oh 0 0 0 0
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# INMO - 2000
If $$z\ne 1$$, how many integer solutions exist for :
$$x+y=1-z$$,
$$x^3+y^3=1-z^2$$
×
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You are Here: Home >< Maths
# Something my teacher can't explain to me Watch
1. Hi, I'm not sure I'm very good at wording this question, as I've tried to ask it before and have not been understood. I'll stiill give it a shot, as it is frustrating that I don't know the answer; <.
When I was taught how to differentiate, was I was told to multiply by the value of the indice, then reduce the power by 1, for example.
Then, when I was taught to intergrate, I was told to increase the power by 1, and then divide by the new power, for example
Then when I was taught to differentiate trigonometric functions like sin squared x, I was told to multiply by the power, reduce the power by 1, and multiply by the differential of whats "in the bracket", for example,
However, when I integrate
I don't increase the power by 1, then divide by the new power, then divide by the differential for whats "in the bracket," for example:
Why is this? In my FP2 module I've been taught ways of integrating trigonometric functions with powers etc, but why is that way that worked for everything else suddenly incorrect?
2. You don't work out every integral the same way. For example integrating e^x just gives you e^x again, or integrating the natural log is something else to think about. I don't understand, it's like asking why is 1+1 equal to 2. Do you want us to explain why the integral of sin^2(x) is equal to x/2 - sin(2x)/4? I suppose you could think of it like "What would I need to differentiate to get sin^2(x)?"
3. Your teacher can't explain that?
It's because "sin x" or something similar, such as cot x or cosh x is not a linear function (i.e. does not have a constant derivative). If you differentiate sin^3(x)/3cos(x) you must use the quotient rule, the 1/3cos(x) cannot be taken outside the differential operator. As you said, when you differentiate something like sin^2(x) you multiply by the derivative of what's in the bracket: so if you're differentiating (2x+1)^5, then the derivative of thing in the bracket is 2: a constant, and reversing this differentiation is no problem.
4. (Original post by jamie092)
Hi, I'm not sure I'm very good at wording this question, as I've tried to ask it before and have not been understood. I'll stiill give it a shot, as it is frustrating that I don't know the answer; <.
When I was taught how to differentiate, was I was told to multiply by the value of the indice, then reduce the power by 1, for example.
Then, when I was taught to intergrate, I was told to increase the power by 1, and then divide by the new power, for example
Then when I was taught to differentiate trigonometric functions like sin squared x, I was told to multiply by the power, reduce the power by 1, and multiply by the differential of whats "in the bracket", for example,
However, when I integrate
I don't increase the power by 1, then divide by the new power, then divide by the differential for whats "in the bracket," for example:
Why is this? In my FP2 module I've been taught ways of integrating trigonometric functions with powers etc, but why is that way that worked for everything else suddenly incorrect?
You've gotten to FP2 without understanding the chain rule from C3?
I'm not too sure why you're considering the integral of after considering it's derivative, the two results are not related.
To integrate , you need to note that and rearrange for . Then try to integrate that.
The "add one to the power and divide by the new power, then divide by derivative of the bracket" rule only applies when the derivative of the inner function is a constant or some multiple of it's derivative is already a factor of the integrand. The more formal way of writing this is:
5. Because the rules you have learnt are only applied to very specific cases.
Oh and Farhan, if you actually read his post he isn't asking how... just why it doesn't work.
6. (Original post by Kasc)
Because the rules you have learnt are only applied to very specific cases.
Oh and Farhan, if you actually read his post he isn't asking how... just why it doesn't work.
I was aware of that - that's why I answered his question and also explained to him how the integral should be dealt with.
7. (Original post by Farhan.Hanif93)
You've gotten to FP2 without understanding the chain rule from C3?
I'm not too sure why you're considering the integral of after considering it's derivative, the two results are not related.
To integrate , you need to note that and rearrange for . Then try to integrate that.
The "add one to the power and divide by the new power, then divide by derivative of the bracket" rule only applies when the derivative of the inner function is a constant or some multiple of it's derivative is already a factor of the integrand. The more formal way of writing this is:
Well I actually studied FP2 and C3 simultaneously, and I've often studied modules "the wrong way around," meaning I've often learned results before I've been able to derive them, which has confused me a bit. But as far as I know, I understand the chain rule from c3, but if I went the long way about differentiating sin^2 x I'd do it like this:
but what I was trying to say in my first post is that this rule of thumb thing where you differentiate by multiplying by the power then reduce the power by 1, and multiply by the differential of the bracket seems to work (I have multiplied by a function of x here), yet it doesn't work for integration, the other way around.
I can definitley see why it does work for differentiation, it's just a shortened version of the chain rule. You say "The "add one to the power and divide by the new power, then divide by derivative of the bracket" rule only applies when the derivative of the inner function is a constant or some multiple of it's derivative is already a factor of the integrand," but you're happy to do this the other way around with differentiation. Why is that?
Also thanks for the help so far everyone =)
8. (Original post by jamie092)
Well I actually studied FP2 and C3 simultaneously, and I've often studied modules "the wrong way around," meaning I've often learned results before I've been able to derive them, which has confused me a bit. But as far as I know, I understand the chain rule from c3, but if I went the long way about differentiating sin^2 x I'd do it like this:
but what I was trying to say in my first post is that this rule of thumb thing where you differentiate by multiplying by the power then reduce the power by 1, and multiply by the differential of the bracket seems to work (I have multiplied by a function of x here), yet it doesn't work for integration, the other way around.
I can definitley see why it does work for differentiation, it's just a shortened version of the chain rule. You say "The "add one to the power and divide by the new power, then divide by derivative of the bracket" rule only applies when the derivative of the inner function is a constant or some multiple of it's derivative is already a factor of the integrand," but you're happy to do this the other way around with differentiation. Why is that?
Also thanks for the help so far everyone =)
sinx is a power series. sinx=x-x^3/3! + x^5 / 5! e.t.c.
sinx^2=(x-x^3/3! + x^5 / 5-....)^2
That is different from say standard x^2 or x^(2/3).
In general differentiation is easier. e^(x^2) for example is differentiable but not integrable over standard functions.
9. (Original post by jamie092)
Hi, I'm not sure I'm very good at wording this question, as I've tried to ask it before and have not been understood. I'll stiill give it a shot, as it is frustrating that I don't know the answer; <.
When I was taught how to differentiate, was I was told to multiply by the value of the indice, then reduce the power by 1, for example.
Then, when I was taught to intergrate, I was told to increase the power by 1, and then divide by the new power, for example
Then when I was taught to differentiate trigonometric functions like sin squared x, I was told to multiply by the power, reduce the power by 1, and multiply by the differential of whats "in the bracket", for example,
However, when I integrate
I don't increase the power by 1, then divide by the new power, then divide by the differential for whats "in the bracket," for example:
Why is this? In my FP2 module I've been taught ways of integrating trigonometric functions with powers etc, but why is that way that worked for everything else suddenly incorrect?
Google the chain rule, product rule and quotient rule. Or ask your teacher to teach you, alot of these different ways of differentiating/integrating are derived from these rules.
10. (Original post by jamie092)
Well I actually studied FP2 and C3 simultaneously, and I've often studied modules "the wrong way around," meaning I've often learned results before I've been able to derive them, which has confused me a bit. But as far as I know, I understand the chain rule from c3, but if I went the long way about differentiating sin^2 x I'd do it like this:
but what I was trying to say in my first post is that this rule of thumb thing where you differentiate by multiplying by the power then reduce the power by 1, and multiply by the differential of the bracket seems to work (I have multiplied by a function of x here), yet it doesn't work for integration, the other way around.
I can definitley see why it does work for differentiation, it's just a shortened version of the chain rule. You say "The "add one to the power and divide by the new power, then divide by derivative of the bracket" rule only applies when the derivative of the inner function is a constant or some multiple of it's derivative is already a factor of the integrand," but you're happy to do this the other way around with differentiation. Why is that?
Also thanks for the help so far everyone =)
Integrating in the way you're talking about only applies to a very limited number of cases that I have outlined.
I've also explained why you can't just add to the power, divide by the new power and divide by the derivative of the bracket - There's a condition that MUST hold for your rule to be used. The simple answer is that, since integration and differentiation are inverse operations, integrating and differentiating the answer should bring you back to . If you integrate using the method you're thinking of, and then differentiate the result, you will not get . Where if you integrate it using the method I outlined at the start and differentiated the result, you will get .
If that isn't enough justification for you, google integration from first principles and try to prove that is not as simple as you're trying to make it to be. (Mind you, I don't know how difficult this could be...)
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I'm not a cryptography expert, but this seems like it falls into this area. Imagine I want to design an election system (senate race for example) where I send the exact same ballot of 100 candidates ...
118 views
### Yaos Millionaire Problem: Why distance >= 2?
I'm currently reading about Yao' Millionaire Problem: http://research.cs.wisc.edu/areas/sec/yao1982-ocr.pdf Alice and Bob want to know which of them is richer. Let $j \in \{1, \cdots 10\}$ be Bobs ...
38 views
### How are the IVs of SHA512/256 and SHA512/224 calculated?
This seems odd to me, because there are a lot of resources out there saying that the initializer values of SHA512 and SHA384 are the first 64 bits of the fractional parts of the square roots of prime ...
83 views
### Elgamal with secret key equal to zero possible?
From various sources (e.g. this paper, page 3), the key generation algorithm of Elgamal samples the secret key $x$ from $\mathbb{Z}_q$, which is identifiable to $\{0, 1, 2, \dots, q-1\}$. My ...
37 views
### EC Schnorr signature: multiple standard?
I 'm working on some EC-Schnorr signature code. Reading various papers on that, it's seems EC-Schnorr is not standardized as well as ECDSA. For example, I found two main differences in two main ...
25 views
### About BGV Scheme Batching Technique
I am reading BGV12 about BGV homomorphic scheme right now.But I am being stuck to understand Batching technique in this paper. In Pack function (page 32),this paper feeds ciphertext $c_i$ and sk ...
56 views
### When to chose hardware accelerated crypto for Big Data
if you were to recommend a cipher/mode configuration that emphasizes performance for your customers, would you recommend a hardware based accelerated solution over one that isn't? For big data at ...
35 views
### Benchmarks for second round CAESAR competitors
I'm interested in how the second round CAESAR competitors stack up in terms of software performance against each other and against chacha20-poly1305 with both small and large messages. Sadly the only ...
29 views
### is accessing elements of an array in secret order vulnerable to timing attack?
Suppose there is an array containing bytes 0...255. Suppose this array is shuffled using a key for random data. Since this is indexing the elements of a table using secret data, is it vulnerable to ...
43 views
### Quantifying bit security for smart-vercauteren encryption scheme
I am working on project that requires to compare in terms of security between two encryption schemes, one of them is the SV scheme. However, I dont know what are the steps exactly towards quantifying ...
74 views
### How to attack polyalphabetic affine cipher with only ciphertext?
How to attack polyalphabetic affine cipher with only the ciphertext? A polyalphabetic affine cipher can be seen as the composition of an affine cipher ($p \mapsto c \equiv a \times p + b \pmod{26})$ ...
63 views
### Majority encryption algorithm?
Assume that I want to leave an encrypted message to a group of $n$ people in a way that they can only decrypt it if they work together in the following sense: For some fixed $k < n$ every ...
39 views
### Committing to a secret shared value
Suppose I run a multiparty protocol. At the output stage, I have the function output in a secret shared manner among all parties. Now I want all the parties to commit to their value of the secret ...
63 views
### Is OCB the state of the art in authenticated encryption modes of operation for block ciphers?
OCB mode seems to represent the state of the art in authenticated encryption modes, largely due to its speed – it is faster than GCM even when GCM has hardware support. Its only drawbacks seem ...
27 views
### Does length prepending fix the key reuse problems with CBC and CBC-MAC?
As I hope everybody is well-aware, re-using the key for CBC-MAC and CBC is a really bad idea. Now, there's also the paradigm to prepend the message length to the CBC-MAC (as "associated data") to ...
33 views
### Is an instruction similar to PCLMULQDQ valuable in post quantum key exchange?
This paper describes a Quantum Key Exchange based on the Ring Learning With Errors problem. When used with ECC, there is only a slight performance impact. Assuming this is the popularized approach ...
100 views
### How secure is $\operatorname{AES256}_{\operatorname{ECB}}(\operatorname{ChaCha20}(\text{plaintext}))$?
Suppose I encrypt a bytestream using ChaCha20, and then encrypt the resulting ciphertext using AES in ECB mode. How secure is the combination?
40 views
### Can you give me a summary of cryptographic hardness assumptions?
Until recently, I had a link to a website which summarizes up-to-date cryptographic hardness assumptions. But, unfortunately I cannot find it. The webpage is categorized well problems such as, DL ...
15 30 50 per page
|
JOURNAL OF NATURAL RESOURCES ›› 2011, Vol. 26 ›› Issue (5): 814-824.
• Resources Evaluation •
### Trends and Determining Factors of Energy Consumption Carbon Footprint —An Analysis for Suzhou-Wuxi-Changzhou Region Based on STIRPAT Model
LU Na, QU Fu-tian, FENG Shu-yi, SHAO Xue-lan
1. College of Public Administration, Nanjing Agricultural University, Nanjing 210095, China
• Received:2010-09-12 Revised:2010-11-20 Online:2011-05-20 Published:2011-05-20
Abstract: Improving the understanding of the impact of socio-economic development on energy consumption carbon footprint is of great importance for developing low-carbon economy. This paper calculated and analyzed the trend of energy consumption carbon footprint of Suzhou-Wuxi-Changzhou region during the period of 1991—2008. Applying ridge regression method, the STIRPAT model was estimated to explore the relationships between population, per capita GDP, technological development and energy consumption carbon footprint. The decoupling index was adopted to further analyze the relationship between economic growth and energy consumption carbon footprint. Results indicated that: 1) For Suzhou-Wuxi-Changzhou region, energy consumption carbon footprint has increased from 0.05 hm2 per capita in 1991 to 0.58 hm2 per capita in 2008. The annual average increasing rate was 15.30%. Coal consumption accounted for the largest share in energy consumption carbon footprint. The share in 2008 was 96.18%. Petroleum consumption fluctuated and showed a downward trend, the share decreased from 18.71% to 3.42% from 1991 to 2008. Different from petroleum, natural gas consumption rose very fast. Even though the share was only 0.40% in 2008, the annual average increasing rate was 45.40% since the extension of natural gas in 2002. The value of carbon footprint showed an overall fluctuating downward tendency, indicating a large space for energy efficiency improvement. 2) Economic development was the main driving factor for energy consumption carbon footprint. 1% increase of per capita GDP has resulted in 0.73% increase in energy consumption carbon footprint. The relationship between per capita GDP and energy consumption carbon footprint, however, did not prove the environmental Kuznets curve (EKC), indicating that with the socio-economic development, environmental pressure caused by energy consumption will continuously increasing. 3) The decoupling index was fluctuating, either in the state of relative decoupling or in re-coupling, indicating that economic growth was highly dependent on energy consumption, and verifying that EKC hypothesis does not exist. Compared with Suzhou and Wuxi, Changzhou has displayed a decoupling state between economic growth and energy consumption carbon footprint since 1998.
CLC Number:
• F426
|
# Math Help - Probably of coin toss
1. ## Probably of coin toss
Hi there,
I'm trying to find out the probability of winning 7 coin tosses in a row without losing (with a 50/50 chance each toss).
2. ## Re: Probably of coin toss
Originally Posted by kittenslayer
I'm trying to find out the probability of winning 7 coin tosses in a row without losing (with a 50/50 chance each toss).
You did not say what 'winning' means.
But in general, the probability of seven heads in a row is $(0.5)^7$.
|
# 1.1 Coordinate systems in physics (Page 2/2)
Page 2 / 2
## Coordinate system types
Coordinate system types determine position of a point with measurements of distance or angle or combination of them. A spatial point requires three measurements in each of these coordinate types. It must, however, be noted that the descriptions of a point in any of these systems are equivalent. Different coordinate types are mere convenience of appropriateness for a given situation. Three major coordinate systems used in the study of physics are :
• Rectangular (Cartesian)
• Spherical
• Cylindrical
Rectangular (Cartesian) coordinate system is the most convenient as it is easy to visualize and associate with our perception of motion in daily life. Spherical and cylindrical systems are specifically designed to describe motions, which follow spherical or cylindrical curvatures.
## Rectangular (cartesian) coordinate system
The measurements of distances along three mutually perpendicular directions, designated as x,y and z, completely define a point A (x,y,z).
From geometric consideration of triangle OAB,
$\begin{array}{l}r=\sqrt{{\mathrm{OB}}^{2}+{\mathrm{AB}}^{2}}\end{array}$
From geometric consideration of triangle OBD,
$\begin{array}{l}{\mathrm{OB}}^{2}=\sqrt{{\mathrm{BD}}^{2}+{\mathrm{OD}}^{2}}\end{array}$
Combining above two relations, we have :
$\begin{array}{l}⇒r=\sqrt{{\mathrm{BD}}^{2}+{\mathrm{OD}}^{2}+{\mathrm{AB}}^{2}}\end{array}$
$\begin{array}{l}⇒r=\sqrt{{x}^{2}+{y}^{2}+{z}^{2}}\end{array}$
The numbers are assigned to a point in the sequence x, y, z as shown for the points A and B.
Rectangular coordinate system can also be viewed as volume composed of three rectangular surfaces. The three surfaces are designated as a pair of axial designations like “xy” plane. We may infer that the “xy” plane is defined by two lines (x and y axes) at right angle. Thus, there are “xy”, “yz” and “zx” rectangular planes that make up the space (volumetric extent) of the coordinate system (See figure).
The motion need not be extended in all three directions, but may be limited to two or one dimensions. A circular motion, for example, can be represented in any of the three planes, whereby only two axes with an origin will be required to describe motion. A linear motion, on the other hand, will require representation in one dimension only.
## Spherical coordinate system
A three dimensional point “A” in spherical coordinate system is considered to be located on a sphere of a radius “r”. The point lies on a particular cross section (or plane) containing origin of the coordinate system. This cross section makes an angle “θ” from the “zx” plane (also known as longitude angle). Once the plane is identified, the angle, φ, that the line joining origin O to the point A, makes with “z” axis, uniquely defines the point A (r, θ, φ).
It must be realized here that we need to designate three values r, θ and φ to uniquely define the point A. If we do not specify θ, the point could then lie any of the infinite numbers of possible cross section through the sphere like A'(See Figure below).
From geometric consideration of spherical coordinate system :
$\begin{array}{l}r=\sqrt{{x}^{2}+{y}^{2}+{z}^{2}}\\ x=r\mathrm{sin}\phi \mathrm{cos}\theta \\ y=r\mathrm{sin}\phi \mathrm{sin}\theta \\ z=r\mathrm{cos}\phi \\ \mathrm{tan}\phi =\frac{\sqrt{{x}^{2}+{y}^{2}}}{z}\\ \mathrm{tan}\theta =\frac{y}{z}\end{array}$
These relations can be easily obtained, if we know to determine projection of a directional quantity like position vector. For example, the projection of "r" in "xy" plane is "r sinφ". In turn, projection of "r sinφ" along x-axis is ""r sinφ cosθ". Hence,
$\begin{array}{l}x=r\mathrm{sin}\phi \mathrm{cos}\theta \end{array}$
In the similar fashion, we can determine other relations.
## Cylindrical coordinate system
A three dimensional point “A” in cylindrical coordinate system is considered to be located on a cylinder of a radius “r”. The point lies on a particular cross section (or plane) containing origin of the coordinate system. This cross section makes an angle “θ” from the “zx” plane. Once the plane is identified, the height, z, parallel to vertical axis “z” uniquely defines the point A(r, θ, z)
$\begin{array}{l}r=\sqrt{{x}^{2}+{y}^{2}}\\ x=r\mathrm{cos}\theta \\ y=r\mathrm{sin}\theta \\ z=z\\ \mathrm{tan}\theta =\frac{y}{z}\end{array}$
what's lamin's theorems and it's mathematics representative
if the wavelength is double,what is the frequency of the wave
What are the system of units
A stone propelled from a catapult with a speed of 50ms-1 attains a height of 100m. Calculate the time of flight, calculate the angle of projection, calculate the range attained
58asagravitasnal firce
Amar
water boil at 100 and why
what is upper limit of speed
what temperature is 0 k
Riya
0k is the lower limit of the themordynamic scale which is equalt to -273 In celcius scale
Mustapha
How MKS system is the subset of SI system?
which colour has the shortest wavelength in the white light spectrum
if x=a-b, a=5.8cm b=3.22 cm find percentage error in x
x=5.8-3.22 x=2.58
what is the definition of resolution of forces
what is energy?
Ability of doing work is called energy energy neither be create nor destryoed but change in one form to an other form
Abdul
motion
Mustapha
highlights of atomic physics
Benjamin
can anyone tell who founded equations of motion !?
n=a+b/T² find the linear express
أوك
عباس
Quiklyyy
|
# Objective Empirical Mode Decomposition metric
Dawid Laszuk published on
8 min, 1425 words
This is a summary. If you wish to read full paper, please visit About me section and click on appropriate link, or follow one of these: [1] [2]. Python code for metrics calculation is in Code section.
Part of my research has been concentrated on empirical mode decomposition (EMD). It's quite nice decomposition method and can generate interesting results. Nevertheless, due to its empirically, it's quite hard to determine what those results represent. Moreover, they will change if we change some parameters of the decomposition; parameters such as stopping criteria threshold, interpolation spline technique or even definition of extremum. Having two different decompositions of the same signal, one needs to decide which one is better. What researchers have done most often is to visually compare obtained decompositions and, based on their expertise, decide which one is better. Naturally this is not an ideal solution. Despite all our good (objective) intentions, observer's bias is unavoidable. We have tried to mitigate this problem by referring to the original concept of EMD and intrinsic mode functions (IMFs). We have came up with metrics, which are based on IMF's idealised properties: 1) an average frequency decreases with increase of IMF's index, 2) distinct instantaneous frequency for each IMF and 3) disjoint Fourier spectra support for IMF's amplitude and phase.
Metric M1
This metric is based on the empirical evidence for the decrease of average instantaneous frequency, simply referred to as the average frequency, with the increase of IMF's index number. Although the order with which the IMF components are construced in general corresponds to the decreasing IMF average frequencies, there are instances when the instantaneous frequencies cross over between the components. Since it has been claimed that each IMF has a significant instantaneous frequency, such behaviour is unwelcome and hence it is penalised by this metric. Penalties are introduced when instantaneous frequency of an IMF with lower number (high average frequency) is smaller than instantaneous frequency of any IMF with higher number. The penalty value is proportional to the length of the crossing over effect, i.e.
$$(1) \qquad \qquad m^{\text{I}}_{j} = \sum_{k=j+1}^{N} \int_{\dot{\phi}_{k} > \dot{\phi}_{j}} \frac{dt}{T}$$
where k, j are IMFs' indices. Formula (1) compares functions of instantaneous frequencies of two IMFs and returns the total duration, over which the IMF with higher index has lower frequency. The crossing over effect has been presented in Figure 1. It shows instantaneous frequency of each IMF as a function of time. Coloured regions indicate where the crossing over occurred. Summing over all pairs of IMFs allows us to assess results for particular EMD. Metric value for the whole set is given as
$$(2) \qquad \qquad M_{\text{I}} = \sum_{j=1}^{N} m^{\text{I}}_{j}, \qquad M_{\text{I}} \in \left[0,\frac{N(N-1)}{2}\right],$$
According to this measure, the best IMF set is the one, for which $M_{\text{I}}=0$, i.e. there are no crossing-over parts in instantaneous frequency domain. The worst case, $M_{\text{I}} = N(N-1)/2$, is when the order of all IMFs is reversed, i.e. when the first IMF is under all others and the last IMF is above all others. However, this theoretical upper limit is very unlikely and the corresponding IMF set could be still considered upon index reversal.
Metric M2
Another validating measure is based on the Bedrosian theorem. It refers to the necessary conditions for the signal's amplitude, $a(t)$, and phase, $\phi(t)$, to be exactly recoverable using Hilbert transformation. For signal $s(t) = a(t) \cos\left( \phi(t) \right)$ these conditions require that the support of amplitude and phase's Fourier spectra are not overlapping. In other words, if the amplitude function, $f(t) = a(t)$, and the phase function, $g(t) = \cos\left(\phi(t)\right)$, then
$$(3) \qquad \qquad \left\langle \mathcal{F}(f), \mathcal{F}(g) \right\rangle = 0 ,$$
where $\mathcal{F}$ represents the Fourier transform and $\langle h(t), l(t) \rangle = \int h^*(t) l(t) dt$ is the dot product. Here it is assumed, that all functions belong to $L^2$ normed space.
Let $F_{j}^{a} = \left| \mathcal{F} \left( a_{j}(t) \right) \right|$ and $F_{j}^{\phi} = \left| \mathcal{F} \left( \cos\left(\phi_{j}(t)\right) \right) \right|$ be absolute values of Fourier transforms of $a_{j}$ and $\cos(\phi_{j})$, respectively, for $j$ IMF. Their normalised measure of overlapping spectra is given as
$$(4) \qquad \qquad m_{j}^{\text{II}} = \frac{\left\langle F_{j}^{a}, F_{j}^{\phi} \right\rangle} {\sqrt{| F_{j}^{a} | | F_{j}^{\phi} |}} ,$$
where $| h | = \langle h, h \rangle$ is a norm of a function h. Assumptions of Bedrosian's theorem are completely fulfilled when spectra are not overlapping, thus minimum value of $m^{II}_j$ is zero. This allows for different definitions of a metric for the whole IMF set, depending on application of EMD. First proposition is based on a biggest value of overlap $m_{j}$ in considered decomposition, i.e.
$(5) \qquad \qquad M_{\text{II}} = \max_{j} { m_{j}^{\text{II}}}, \qquad M_{\text{II}} \in [0,1],$
and the second refers to the cumulative overlap within the decomposed set, i.e.
$(6) \qquad \qquad M_{\text{III}} = \sum_{j=1}^{N} m_{j}^{\text{II}}, \qquad M_{\text{III}} \in [0, N],$
where in both cases N is the number of extracted IMFs. Zero for both metrics implies no overlap between amplitude's and phase's spectra in any of IMFs.
Visual interpretation of the validation measure (4) is presented in Figure 2. It shows example Fourier spectra of slowly changing amplitude (dashed line) and higher frequency phase (solid line). Gray-striped region indicates overlapping area of both spectra. Proposed value is a measure of ratio of the overlapping area to the total area under both functions.
Since metric $M_{\text{III}}$ is a sum over all IMFs, it also contains the one which maximises value $m^{\text{II}}_j$ (Eq. (4)). This means that $M_{\text{III}}$ for each decomposition has to be at least equal or higher than $M_{\text{II}}$.
Application of the validation measures
Each of the presented metrics highlights different properties of the decomposition. Computing all three values is equivalent to finding a point $M=(M_{\text{I}}, M_{\text{II}},M_{\text{III}})$ in a 3-dimensional space, where each dimension relates to the specific metric. The best decomposition corresponds to the minimum over all the metrics, i.e. $M=(0,0,0)$, and the worst decomposition to $M=(\frac{N(N-1)}{2},1,N)$. For any other point one has to decide on the importance, or weight, for each of the proposed metrics, on the basis of the problem being considered. Although the distance in the M-space is not strictly defined, it can be any $L^p$ norm, thus we suggest using the weighted Manhattah metric, i.e.
$$(7) \qquad \qquad | M | = w_1 M_{\text{I}} + w_2 M_{\text{II}} + w_3 M_{\text{III}} ,$$ where $w_i$ are respective weights. Their values should reflect the relative importance of features one is concentrated on.
|
# python-data-structures
In [1]:
from IPython.display import HTML
Table 1: Common Functions for Big-O
f(n) Name
1$1$ Constant
logn$\log n$ Logarithmic
n$n$ Linear
nlogn$n\log n$ Log Linear
n2$n^{2}$ Quadratic
n3$n^{3}$ Cubic
2n$2^{n}$ Exponential
Table 2: Big-O Efficiency of Python List Operators
Operation Big-O Efficiency
index [] O(1)
index assignment O(1)
append O(1)
pop() O(1)
pop(i) O(n)
insert(i,item) O(n)
del operator O(n)
iteration O(n)
contains (in) O(n)
get slice [x:y] O(k)
del slice O(n)
set slice O(n+k)
reverse O(n)
concatenate O(k)
sort O(n log n)
multiply O(nk)
Table 3: Big-O Efficiency of Python Dictionary Operations
operation Big-O Efficiency
copy O(n)
get item O(1)
set item O(1)
delete item O(1)
contains (in) O(1)
iteration O(n)
### The Stack Abstract Data Type¶
Table 1: Sample Stack Operations
Stack Operation Stack Contents Return Value
s.isEmpty() [] True
s.push(4) [4]
s.push('dog') [4,'dog']
s.peek() [4,'dog'] 'dog'
s.push(True) [4,'dog',True]
s.size() [4,'dog',True] 3
s.isEmpty() [4,'dog',True] False
s.push(8.4) [4,'dog',True,8.4]
s.pop() [4,'dog',True] 8.4
s.pop() [4,'dog'] True
s.size() [4,'dog'] 2
### Implementing a Stack in Python¶
In [2]:
class Stack:
def __init__(self):
self.items = []
def isEmpty(self):
return self.items == []
def push(self, item):
self.items.append(item)
def pop(self):
return self.items.pop()
def peek(self):
return self.items[len(self.items)-1]
def size(self):
return len(self.items)
def show(self):
return [self.pop() for x in range(self.size())]
In [3]:
def revstring(mystr):
t = Stack()
[t.push(x) for x in mystr]
return ''.join(t.show())
revstring('apple')
Out[3]:
'elppa'
### Simple Balanced Parentheses¶
In [4]:
from pythonds.basic.stack import Stack
def parChecker(symbolString):
s = Stack()
balanced = True
index = 0
while index < len(symbolString) and balanced:
symbol = symbolString[index]
if symbol == "(":
s.push(symbol)
else:
if s.isEmpty():
balanced = False
else:
s.pop()
index = index + 1
if balanced and s.isEmpty():
return True
else:
return False
print(parChecker('((()))'))
print(parChecker('(()'))
True
False
### Balanced Symbols (A General Case)¶
In [5]:
from pythonds.basic.stack import Stack
def parChecker(symbolString):
s = Stack()
balanced = True
index = 0
while index < len(symbolString) and balanced:
symbol = symbolString[index]
if symbol in "([{":
s.push(symbol)
else:
if s.isEmpty():
balanced = False
else:
top = s.pop()
if not matches(top,symbol):
balanced = False
index = index + 1
if balanced and s.isEmpty():
return True
else:
return False
def matches(open,close):
opens = "([{"
closers = ")]}"
return opens.index(open) == closers.index(close)
print(parChecker('{{([][])}()}'))
print(parChecker('[{()]'))
True
False
### Converting Decimal Numbers to Binary Numbers¶
In [6]:
from pythonds.basic.stack import Stack
def baseConverter(decNumber,base):
digits = "0123456789ABCDEF"
remstack = Stack()
while decNumber > 0:
rem = decNumber % base
remstack.push(rem)
decNumber = decNumber // base
newString = ""
while not remstack.isEmpty():
newString = newString + digits[remstack.pop()]
return newString
print(baseConverter(25,2))
print(baseConverter(25,16))
11001
19
### Infix, Prefix and Postfix Expressions¶
1. Create an empty stack called opstack for keeping operators. Create an empty list for output.
2. Convert the input infix string to a list by using the string method split.
3. Scan the token list from left to right.
• If the token is an operand, append it to the end of the output list.
• If the token is a left parenthesis, push it on the opstack.
• If the token is a right parenthesis, pop the opstack until the corresponding left parenthesis is removed. Append each operator to the end of the output list.
• If the token is an operator, *, /, +, or -, push it on the opstack. However, first remove any operators already on the opstack that have higher or equal precedence and append them to the output list.
4. When the input expression has been completely processed, check the opstack. Any operators still on the stack can be removed and appended to the end of the output list.
In [7]:
from pythonds.basic.stack import Stack
def infixToPostfix(infixexpr):
prec = {}
prec["*"] = 3
prec["/"] = 3
prec["+"] = 2
prec["-"] = 2
prec["("] = 1
opStack = Stack()
postfixList = []
tokenList = infixexpr.split()
for token in tokenList:
if token in "ABCDEFGHIJKLMNOPQRSTUVWXYZ" or token in "0123456789":
postfixList.append(token)
elif token == '(':
opStack.push(token)
elif token == ')':
topToken = opStack.pop()
while topToken != '(':
postfixList.append(topToken)
topToken = opStack.pop()
else:
while (not opStack.isEmpty()) and \
(prec[opStack.peek()] >= prec[token]):
postfixList.append(opStack.pop())
opStack.push(token)
while not opStack.isEmpty():
postfixList.append(opStack.pop())
return " ".join(postfixList)
print(infixToPostfix("A * B + C * D"))
print(infixToPostfix("( A + B ) * C - ( D - E ) * ( F + G )"))
A B * C D * +
A B + C * D E - F G + * -
### Postfix Evaluation¶
1. Create an empty stack called operandStack.
2. Convert the string to a list by using the string method split.
3. Scan the token list from left to right.
• If the token is an operand, convert it from a string to an integer and push the value onto the operandStack.
• If the token is an operator, *, /, +, or -, it will need two operands. Pop the operandStack twice. The first pop is the second operand and the second pop is the first operand. Perform the arithmetic operation. Push the result back on the operandStack.
4. When the input expression has been completely processed, the result is on the stack. Pop the operandStack and return the value.
In [8]:
from pythonds.basic.stack import Stack
def postfixEval(postfixExpr):
operandStack = Stack()
tokenList = postfixExpr.split()
for token in tokenList:
if token in "0123456789":
operandStack.push(int(token))
else:
operand2 = operandStack.pop()
operand1 = operandStack.pop()
result = doMath(token,operand1,operand2)
operandStack.push(result)
return operandStack.pop()
def doMath(op, op1, op2):
if op == "*":
return op1 * op2
elif op == "/":
return op1 / op2
elif op == "+":
return op1 + op2
else:
return op1 - op2
print(postfixEval('7 8 + 3 2 + /'))
3.0
### The Queue Abstract Data Type¶
Table 1: Example Queue Operations
Queue Operation Queue Contents Return Value
q.isEmpty() [] True
q.enqueue(4) [4]
q.enqueue('dog') ['dog',4]
q.enqueue(True) [True,'dog',4]
q.size() [True,'dog',4] 3
q.isEmpty() [True,'dog',4] False
q.enqueue(8.4) [8.4,True,'dog',4]
q.dequeue() [8.4,True,'dog'] 4
q.dequeue() [8.4,True] 'dog'
q.size() [8.4,True] 2
### Implementing a Queue in Python¶
In [9]:
class Queue:
def __init__(self):
self.items = []
def isEmpty(self):
return self.items == []
def enqueue(self, item):
self.items.insert(0,item)
def dequeue(self):
return self.items.pop()
def size(self):
return len(self.items)
def show(self):
return map(lambda x: self.dequeue(), range(self.size()))
In [10]:
q = Queue()
q.enqueue('hello')
q.enqueue('dog')
q.enqueue(3)
q.dequeue()
q.show()
Out[10]:
<map at 0x7f83a45b56d8>
### Simulation: Hot Potato¶
In [11]:
from pythonds.basic.queue import Queue
def hotPotato(namelist, num):
simqueue = Queue()
for name in namelist:
simqueue.enqueue(name)
while simqueue.size() > 1:
for i in range(num):
simqueue.enqueue(simqueue.dequeue())
simqueue.dequeue()
return simqueue.dequeue()
Susan
In [12]:
HTML('<iframe width="800" height="500" frameborder="0" src="http://pythontutor.com/iframe-embed.html#code=class+Queue%3A%0A++++def+__init__(self)%3A%0A++++++++self.items+%3D+%5B%5D%0A%0A++++def+isEmpty(self)%3A%0A++++++++return+self.items+%3D%3D+%5B%5D%0A%0A++++def+enqueue(self,+item)%3A%0A++++++++self.items.insert(0,item)%0A%0A++++def+dequeue(self)%3A%0A++++++++return+self.items.pop()%0A%0A++++def+size(self)%3A%0A++++++++return+len(self.items)%0A++++%0A++++def+show(self)%3A%0A++++++++return+map(lambda+x%3A+self.dequeue(),+range(self.size()))%0A%0Adef+hotPotato(namelist,+num)%3A%0A++++simqueue+%3D+Queue()%0A++++for+name+in+namelist%3A%0A++++++++simqueue.enqueue(name)%0A%0A++++while+simqueue.size()+%3E+1%3A%0A++++++++for+i+in+range(num)%3A%0A++++++++++++simqueue.enqueue(simqueue.dequeue())%0A%0A++++++++simqueue.dequeue()%0A%0A++++return+simqueue.dequeue()%0A%0Aprint(hotPotato(%5B%22Bill%22,%22David%22,%22Susan%22,%22Jane%22,%22Kent%22,%22Brad%22%5D,7))%0A&origin=opt-frontend.js&cumulative=false&heapPrimitives=false&drawParentPointers=false&textReferences=false&showOnlyOutputs=false&py=2&rawInputLstJSON=%5B%5D&curInstr=0&codeDivWidth=350&codeDivHeight=400"> </iframe>')
Out[12]:
1. Create a queue of print tasks. Each task will be given a timestamp upon its arrival. The queue is empty to start.
2. For each second (currentSecond):
• Does a new print task get created? If so, add it to the queue with the currentSecond as the timestamp.
• If the printer is not busy and if a task is waiting,
• Remove the next task from the print queue and assign it to the printer.
• Subtract the timestamp from the currentSecond to compute the waiting time for that task.
• Append the waiting time for that task to a list for later processing.
• Based on the number of pages in the print task, figure out how much time will be required.
• The printer now does one second of printing if necessary. It also subtracts one second from the time required for that task.
• If the task has been completed, in other words the time required has reached zero, the printer is no longer busy.
3. After the simulation is complete, compute the average waiting time from the list of waiting times generated.
In [13]:
import random
class Printer:
def __init__(self, ppm):
self.pagerate = ppm
self.timeRemaining = 0
def tick(self):
self.timeRemaining = self.timeRemaining - 1
if self.timeRemaining <= 0:
def busy(self):
return True
else:
return False
def __init__(self,time):
self.timestamp = time
self.pages = random.randrange(1,21)
def getStamp(self):
return self.timestamp
def getPages(self):
return self.pages
def waitTime(self, currenttime):
return currenttime - self.timestamp
def simulation(numSeconds, pagesPerMinute):
labprinter = Printer(pagesPerMinute)
printQueue = Queue()
waitingtimes = []
for currentSecond in range(numSeconds):
if (not labprinter.busy()) and (not printQueue.isEmpty()):
labprinter.tick()
averageWait=sum(waitingtimes)/len(waitingtimes)
print("Average Wait %6.2f secs %3d tasks remaining."%(averageWait,printQueue.size()))
num = random.randrange(1,181)
if num == 180:
return True
else:
return False
for i in range(10):
simulation(3600,5)
Average Wait 31.79 secs 0 tasks remaining.
Average Wait 63.80 secs 0 tasks remaining.
Average Wait 29.67 secs 0 tasks remaining.
Average Wait 161.00 secs 0 tasks remaining.
Average Wait 903.00 secs 11 tasks remaining.
Average Wait 338.50 secs 2 tasks remaining.
Average Wait 82.90 secs 0 tasks remaining.
Average Wait 164.33 secs 1 tasks remaining.
Average Wait 98.30 secs 1 tasks remaining.
Average Wait 50.20 secs 1 tasks remaining.
### The Deque Abstract Data Type¶
Table 1: Examples of Deque Operations
Deque Operation Deque Contents Return Value
d.isEmpty() [] True
d.addRear(4) [4]
d.addRear('dog') ['dog',4,]
d.addFront('cat') ['dog',4,'cat']
d.addFront(True) ['dog',4,'cat',True]
d.size() ['dog',4,'cat',True] 4
d.isEmpty() ['dog',4,'cat',True] False
d.addRear(8.4) [8.4,'dog',4,'cat',True]
d.removeRear() ['dog',4,'cat',True] 8.4
d.removeFront() ['dog',4,'cat'] True
### Implementing a Deque in Python¶
In [14]:
class Deque:
def __init__(self):
self.items = []
def isEmpty(self):
return self.items == []
self.items.append(item)
self.items.insert(0,item)
def removeFront(self):
return self.items.pop()
def removeRear(self):
return self.items.pop(0)
def size(self):
return len(self.items)
def show(self):
return map(lambda x: self.dequeue(), range(self.size()))
In [15]:
d=Deque()
print(d.isEmpty())
print(d.size())
print(d.isEmpty())
print(d.removeRear())
True
4
False
8.4
### Palindrome-Checker(回文检查)¶
In [16]:
def palchecker(aString):
chardeque = Deque()
for ch in aString:
stillEqual = True
while chardeque.size() > 1 and stillEqual:
first = chardeque.removeFront()
last = chardeque.removeRear()
if first != last:
stillEqual = False
return stillEqual
print(palchecker("lsdkjfskf"))
False
True
### The Unordered List Abstract Data Type¶
• List() creates a new list that is empty. It needs no parameters and returns an empty list.
• add(item) adds a new item to the list. It needs the item and returns nothing. Assume the item is not already in the list.
• remove(item) removes the item from the list. It needs the item and modifies the list. Assume the item is present in the list.
• search(item) searches for the item in the list. It needs the item and returns a boolean value.
• isEmpty() tests to see whether the list is empty. It needs no parameters and returns a boolean value.
• size() returns the number of items in the list. It needs no parameters and returns an integer.
• append(item) adds a new item to the end of the list making it the last item in the collection. It needs the item and returns nothing. Assume the item is not already in the list.
• index(item) returns the position of item in the list. It needs the item and returns the index. Assume the item is in the list.
• insert(pos,item) adds a new item to the list at position pos. It needs the item and returns nothing. Assume the item is not already in the list and there are enough existing items to have position pos.
• pop() removes and returns the last item in the list. It needs nothing and returns an item. Assume the list has at least one item.
• pop(pos) removes and returns the item at position pos. It needs the position and returns the item. Assume the item is in the list.
### Implementing an Unordered List: Linked Lists¶
In [17]:
class Node:
"""
The basic building block for the linked list implementation
is the node. Each node object must hold at least two pieces
of information. First, the node must contain the list item
itself. We will call this the data field of the node.
"""
def __init__(self,initdata):
self.data = initdata
self.next = None
def getData(self):
return self.data
def getNext(self):
return self.next
def setData(self,newdata):
self.data = newdata
def setNext(self,newnext):
self.next = newnext
class UnorderedList:
"""
The unordered list will be built from a collection of nodes,
each linked to the next by explicit references. As long as we
know where to find the first node (containing the first item),
each item after that can be found by successively following
"""
def __init__(self):
def isEmpty(self):
temp = Node(item)
def size(self):
count = 0
while current != None:
count = count + 1
current = current.getNext()
return count
def search(self,item):
found = False
if current.getData() == item:
found = True
else:
current = current.getNext()
return found
def remove(self,item):
previous = None
found = False
if current.getData() == item:
found = True
else:
previous = current
current = current.getNext()
if previous == None:
else:
previous.setNext(current.getNext())
In [18]:
mylist = UnorderedList()
print(mylist.size())
print(mylist.search(93))
print(mylist.search(100))
print(mylist.search(100))
print(mylist.size())
mylist.remove(54)
print(mylist.size())
mylist.remove(93)
print(mylist.size())
mylist.remove(31)
print(mylist.size())
print(mylist.search(93))
6
True
False
True
7
6
5
4
False
### The Ordered List Abstract Data Type¶
The structure of an ordered list is a collection of items where each item holds a relative position that is based upon some underlying characteristic of the item. The ordering is typically either ascending or descending and we assume that list items have a meaningful comparison operation that is already defined. Many of the ordered list operations are the same as those of the unordered list.
• OrderedList() creates a new ordered list that is empty. It needs no parameters and returns an empty list.
• add(item) adds a new item to the list making sure that the order is preserved. It needs the item and returns nothing. Assume the item is not already in the list.
• remove(item) removes the item from the list. It needs the item and modifies the list. Assume the item is present in the list.
• search(item) searches for the item in the list. It needs the item and returns a boolean value.
• isEmpty() tests to see whether the list is empty. It needs no parameters and returns a boolean value.
• size() returns the number of items in the list. It needs no parameters and returns an integer.
• index(item) returns the position of item in the list. It needs the item and returns the index. Assume the item is in the list.
• pop() removes and returns the last item in the list. It needs nothing and returns an item. Assume the list has at least one item.
• pop(pos) removes and returns the item at position pos. It needs the position and returns the item. Assume the item is in the list.
### Implementing an Ordered List¶
In [19]:
class OrderedList:
def __init__(self):
def search(self,item):
found = False
stop = False
if current.getData() == item:
found = True
else:
if current.getData() > item:
stop = True
else:
current = current.getNext()
return found
previous = None
stop = False
while current != None and not stop:
if current.getData() > item:
stop = True
else:
previous = current
current = current.getNext()
temp = Node(item)
if previous == None:
else:
temp.setNext(current)
previous.setNext(temp)
def isEmpty(self):
def size(self):
count = 0
while current != None:
count = count + 1
current = current.getNext()
return count
In [20]:
mylist = OrderedList()
print(mylist.size())
print(mylist.search(93))
print(mylist.search(100))
6
True
False
### The Three Laws of Recursion¶
1. A recursive algorithm must have a base case.
2. A recursive algorithm must change its state and move toward the base case.
3. A recursive algorithm must call itself, recursively.
In [21]:
def listsum(numList):
theSum = 0
for i in numList:
theSum = theSum + i
return theSum
print(listsum([1,3,5,7,9]))
25
In [22]:
def listsum(numList):
if len(numList) == 1:
return numList[0]
else:
return numList[0] + listsum(numList[1:])
print(listsum([1,3,5,7,9]))
25
### Converting an Integer to a String in Any Base¶
In [23]:
def toStr(n,base):
convertString = "0123456789ABCDEF"
if n < base:
return convertString[n]
else:
print(toStr(1453,16))
5AD
In [24]:
def reverse(s):
"""
a function that takes a string as a parameter and returns
a new string that is the reverse of the old string.
"""
if s == '': return s
else: return reverse(s[1:]) + s[0]
In [25]:
print(reverse("follow"))
print(reverse("a +"))
wollof
+ a
In [26]:
import string
def removeWhite(s):
exclude = set(string.punctuation+' ')
s = ''.join(ch for ch in s if ch not in exclude)
return s
def isPal(s):
"""
Write a function that takes a string as a parameter and
returns True if the string is a palindrome, False otherwise.
"""
s = removeWhite(s)
return len(s) < 2 or s[0] == s[-1] and isPal(s[1:-1])
In [27]:
print(isPal(removeWhite("x")))
print(isPal(removeWhite("hello")))
print(isPal(removeWhite("")))
print(isPal(removeWhite("hannah")))
True
True
False
True
True
True
### Stack Frames: Implementing Recursion¶
In [28]:
rStack = Stack()
def toStr(n,base):
convertString = "0123456789ABCDEF"
while n > 0:
if n < base:
rStack.push(convertString[n])
else:
rStack.push(convertString[n % base])
n = n // base
res = ""
while not rStack.isEmpty():
res = res + str(rStack.pop())
return res
print(toStr(1453,16))
5AD
In [29]:
HTML('<iframe width="1000" height="500" frameborder="0" src="http://pythontutor.com/iframe-embed.html#code=class+Stack%3A%0A++++def+__init__(self)%3A%0A++++++++self.items+%3D+%5B%5D%0A%0A++++def+isEmpty(self)%3A%0A++++++++return+self.items+%3D%3D+%5B%5D%0A%0A++++def+push(self,+item)%3A%0A++++++++self.items.append(item)%0A%0A++++def+pop(self)%3A%0A++++++++return+self.items.pop()%0A%0A++++def+peek(self)%3A%0A++++++++return+self.items%5Blen(self.items)-1%5D%0A%0A++++def+size(self)%3A%0A++++++++return+len(self.items)%0A++++%0ArStack+%3D+Stack()%0A%0Adef+toStr(n,base)%3A%0A++++convertString+%3D+%220123456789ABCDEF%22%0A++++while+n+%3E+0%3A%0A++++++++if+n+%3C+base%3A%0A++++++++++++rStack.push(convertString%5Bn%5D)%0A++++++++else%3A%0A++++++++++++rStack.push(convertString%5Bn+%25+base%5D)%0A++++++++n+%3D+n+//+base%0A++++res+%3D+%22%22%0A++++while+not+rStack.isEmpty()%3A%0A++++++++res+%3D+res+%2B+str(rStack.pop())%0A++++return+res%0A%0Aprint(toStr(1453,16))&origin=opt-frontend.js&cumulative=false&heapPrimitives=false&drawParentPointers=false&textReferences=false&showOnlyOutputs=false&py=2&rawInputLstJSON=%5B%5D&curInstr=29&codeDivWidth=350&codeDivHeight=400"> </iframe>')
Out[29]:
### Tower of Hanoi¶
The number of moves required to correctly move a tower of $n$ disks is $2^n-1$.Here is a high-level outline of how to move a tower from the starting pole, to the goal pole, using an intermediate pole:
1. Move a tower of height-1 to an intermediate pole, using the final pole. 用C把高度height-1的塔从A移动到B
2. Move the remaining disk to the final pole. 把A剩下的碟子移动到C
3. Move the tower of height-1 from the intermediate pole to the final pole using the original pole. 用A把高度height-1的塔从B移动到C
In [30]:
def moveTower(height,fromPole, toPole, withPole):
if height >= 1:
moveTower(height-1,fromPole,withPole,toPole)
moveDisk(fromPole,toPole)
moveTower(height-1,withPole,toPole,fromPole)
def moveDisk(fp,tp):
print("moving disk from",fp,"to",tp)
moveTower(3,"A","B","C")
moving disk from A to B
moving disk from A to C
moving disk from B to C
moving disk from A to B
moving disk from C to A
moving disk from C to B
moving disk from A to B
In [31]:
HTML('<iframe width="1000" height="500" frameborder="0" src="http://pythontutor.com/iframe-embed.html#code=def+moveTower(height,fromPole,+toPole,+withPole)%3A%0A++++if+height+%3E%3D+1%3A%0A++++++++moveTower(height-1,fromPole,withPole,toPole)%0A++++++++moveDisk(fromPole,toPole)%0A++++++++moveTower(height-1,withPole,toPole,fromPole)%0A%0Adef+moveDisk(fp,tp)%3A%0A++++print+%22moving+disk+from%22,fp,%22to%22,tp%0A%0AmoveTower(3,%22A%22,%22B%22,%22C%22)&origin=opt-frontend.js&cumulative=false&heapPrimitives=false&drawParentPointers=false&textReferences=false&showOnlyOutputs=false&py=2&rawInputLstJSON=%5B%5D&curInstr=80&codeDivWidth=350&codeDivHeight=400"> </iframe>')
Out[31]:
In [32]:
def moveTower(height,fromPole, toPole, withPole):
print("moveTower:", height,fromPole, toPole, withPole)
if height >= 1:
moveTower(height-1,fromPole,withPole,toPole)
moveDisk(fromPole,toPole)
moveTower(height-1,withPole,toPole,fromPole)
def moveDisk(fp,tp):
if fromPole[1]:
disk = fromPole[1].pop()
print("moving " + str(disk) + " from " + fromPole[0] + " to " + toPole[0])
toPole[1].append(disk)
fromPole = ("A", [3,2,1])
toPole = ("C", [])
withPole = ("B", [])
moveTower(len(fromPole[1]), fromPole, toPole, withPole)
print(fromPole, withPole, toPole)
moveTower: 3 ('A', [3, 2, 1]) ('C', []) ('B', [])
moveTower: 2 ('A', [3, 2, 1]) ('B', []) ('C', [])
moveTower: 1 ('A', [3, 2, 1]) ('C', []) ('B', [])
moveTower: 0 ('A', [3, 2, 1]) ('B', []) ('C', [])
moving 1 from A to C
moveTower: 0 ('B', []) ('C', [1]) ('A', [3, 2])
moving 2 from A to C
moveTower: 1 ('C', [1, 2]) ('B', []) ('A', [3])
moveTower: 0 ('C', [1, 2]) ('A', [3]) ('B', [])
moving 3 from A to C
moveTower: 0 ('A', []) ('B', []) ('C', [1, 2, 3])
moveTower: 2 ('B', []) ('C', [1, 2, 3]) ('A', [])
moveTower: 1 ('B', []) ('A', []) ('C', [1, 2, 3])
moveTower: 0 ('B', []) ('C', [1, 2, 3]) ('A', [])
moveTower: 0 ('C', [1, 2, 3]) ('A', []) ('B', [])
moveTower: 1 ('A', []) ('C', [1, 2, 3]) ('B', [])
moveTower: 0 ('A', []) ('B', []) ('C', [1, 2, 3])
moveTower: 0 ('B', []) ('C', [1, 2, 3]) ('A', [])
('A', []) ('B', []) ('C', [1, 2, 3])
### Exploring a Maze¶
In [33]:
import turtle
from __future__ import division
PART_OF_PATH = 'O'
TRIED = '.'
OBSTACLE = '+'
class Maze:
def __init__(self,mazeFileName):
rowsInMaze = 0
columnsInMaze = 0
self.mazelist = []
mazeFile = open(mazeFileName,'r')
rowsInMaze = 0
for line in mazeFile:
rowList = []
col = 0
for ch in line[:-1]:
rowList.append(ch)
if ch == 'S':
self.startRow = rowsInMaze
self.startCol = col
col = col + 1
rowsInMaze = rowsInMaze + 1
self.mazelist.append(rowList)
columnsInMaze = len(rowList)
self.rowsInMaze = rowsInMaze
self.columnsInMaze = columnsInMaze
self.xTranslate = -columnsInMaze/2
self.yTranslate = rowsInMaze/2
self.t = turtle.Turtle()
self.t.shape('turtle')
self.wn = turtle.Screen()
self.wn.setworldcoordinates(-(columnsInMaze-1)/2-.5,-(rowsInMaze-1)/2-.5,(columnsInMaze-1)/2+.5,(rowsInMaze-1)/2+.5)
def drawMaze(self):
self.t.speed(10)
for y in range(self.rowsInMaze):
for x in range(self.columnsInMaze):
if self.mazelist[y][x] == OBSTACLE:
self.drawCenteredBox(x+self.xTranslate,-y+self.yTranslate,'orange')
self.t.color('black')
self.t.fillcolor('blue')
def drawCenteredBox(self,x,y,color):
self.t.up()
self.t.goto(x-.5,y-.5)
self.t.color(color)
self.t.fillcolor(color)
self.t.down()
self.t.begin_fill()
for i in range(4):
self.t.forward(1)
self.t.right(90)
self.t.end_fill()
def moveTurtle(self,x,y):
self.t.up()
self.t.goto(x+self.xTranslate,-y+self.yTranslate)
self.t.dot(10,color)
def updatePosition(self,row,col,val=None):
if val:
self.mazelist[row][col] = val
self.moveTurtle(col,row)
if val == PART_OF_PATH:
color = 'green'
elif val == OBSTACLE:
color = 'red'
elif val == TRIED:
color = 'black'
color = 'red'
else:
color = None
if color:
def isExit(self,row,col):
return (row == 0 or
row == self.rowsInMaze-1 or
col == 0 or
col == self.columnsInMaze-1 )
def __getitem__(self,idx):
return self.mazelist[idx]
def searchFrom(maze, startRow, startColumn):
# try each of four directions from this point until we find a way out.
# base Case return values:
# 1. We have run into an obstacle, return false
maze.updatePosition(startRow, startColumn)
if maze[startRow][startColumn] == OBSTACLE :
return False
# 2. We have found a square that has already been explored
if maze[startRow][startColumn] == TRIED or maze[startRow][startColumn] == DEAD_END:
return False
# 3. We have found an outside edge not occupied by an obstacle
if maze.isExit(startRow,startColumn):
maze.updatePosition(startRow, startColumn, PART_OF_PATH)
return True
maze.updatePosition(startRow, startColumn, TRIED)
# Otherwise, use logical short circuiting to try each direction
# in turn (if needed)
found = searchFrom(maze, startRow-1, startColumn) or \
searchFrom(maze, startRow+1, startColumn) or \
searchFrom(maze, startRow, startColumn-1) or \
searchFrom(maze, startRow, startColumn+1)
if found:
maze.updatePosition(startRow, startColumn, PART_OF_PATH)
else:
return found
# myMaze = Maze('maze2.txt')
# myMaze.drawMaze()
# myMaze.updatePosition(myMaze.startRow,myMaze.startCol)
# searchFrom(myMaze, myMaze.startRow, myMaze.startCol)
### Dynamic Programming¶
In [34]:
def recDC(coinValueList,change,knownResults):
minCoins = change
if change in coinValueList:
knownResults[change] = 1
return 1
elif knownResults[change] > 0:
return knownResults[change]
else:
for i in [c for c in coinValueList if c <= change]:
numCoins = 1 + recDC(coinValueList, change-i,
knownResults)
if numCoins < minCoins:
minCoins = numCoins
knownResults[change] = minCoins
return minCoins
print(recDC([1,5,10,25],63,[0]*64))
6
In [35]:
def dpMakeChange(coinValueList,change,minCoins,coinsUsed):
for cents in range(change+1):
coinCount = cents
newCoin = 1
for j in [c for c in coinValueList if c <= cents]:
if minCoins[cents-j] + 1 < coinCount:
coinCount = minCoins[cents-j]+1
newCoin = j
minCoins[cents] = coinCount
coinsUsed[cents] = newCoin
return minCoins[change]
def printCoins(coinsUsed,change):
coin = change
while coin > 0:
thisCoin = coinsUsed[coin]
print(thisCoin)
coin = coin - thisCoin
def main():
amnt = 63
clist = [1,5,10,21,25]
coinsUsed = [0]*(amnt+1)
coinCount = [0]*(amnt+1)
print("Making change for",amnt,"requires")
print(dpMakeChange(clist,amnt,coinCount,coinsUsed),"coins")
print("They are:")
printCoins(coinsUsed,amnt)
print("The used list is as follows:")
print(coinsUsed)
main()
Making change for 63 requires
3 coins
They are:
21
21
21
The used list is as follows:
[1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 10, 1, 1, 1, 1, 5, 1, 1, 1, 1, 10, 21, 1, 1, 1, 25, 1, 1, 1, 1, 5, 10, 1, 1, 1, 10, 1, 1, 1, 1, 5, 10, 21, 1, 1, 10, 21, 1, 1, 1, 25, 1, 10, 1, 1, 5, 10, 1, 1, 1, 10, 1, 10, 21]
1. All recursive algorithms must have a base case.
2. A recursive algorithm must change its state and make progress toward the base case.
3. A recursive algorithm must call itself (recursively).
4. Recursion can take the place of iteration in some cases.
5. Recursive algorithms often map very naturally to a formal expression of the problem you are trying to solve.
6. Recursion is not always the answer. Sometimes a recursive solution may be more computationally expensive than an alternative algorithm
Table 1: Comparisons Used in a Sequential Search of an Unordered List
Case Best Case Worst Case Average Case
item is present 1$1$ n$n$ n2$\frac{n}{2}$
item is not present n$n$ n$n$ n$n$
In [36]:
def sequentialSearch(alist, item):
pos = 0
found = False
if alist[pos] == item:
found = True
else:
pos = pos + 1
return found
testlist = [1, 2, 32, 8, 17, 19, 42, 13, 0]
print(sequentialSearch(testlist, 3))
print(sequentialSearch(testlist, 13))
False
True
Table 2: Comparisons Used in Sequential Search of an Ordered List
Best Case Worst Case Average Case
item is present 1$1$ n$n$ n2$\frac{n}{2}$
item not present 1$1$ n$n$ n2$\frac{n}{2}$
In [37]:
def orderedSequentialSearch(alist, item):
pos = 0
found = False
stop = False
if alist[pos] == item:
found = True
else:
if alist[pos] > item:
stop = True
else:
pos = pos+1
return found
testlist = [0, 1, 2, 8, 13, 17, 19, 32, 42,]
print(orderedSequentialSearch(testlist, 3))
print(orderedSequentialSearch(testlist, 13))
False
True
Table 3: Tabular Analysis for a Binary Search
Comparisons Approximate Number of Items Left
1 n2$\frac {n}{2}$
2 n4$\frac {n}{4}$
3 n8$\frac {n}{8}$
...
i n2i$\frac {n}{2^i}$
#### Binary Search of an Ordered List¶
In [38]:
def binarySearch(alist, item):
first = 0
last = len(alist) - 1
found = False
midpoint = (first + last) // 2
if alist[midpoint] == item:
found = True
else:
if item < alist[midpoint]:
last = midpoint-1
else:
first = midpoint+1
return found
testlist = [0, 1, 2, 8, 13, 17, 19, 32, 42,]
print(binarySearch(testlist, 3))
print(binarySearch(testlist, 13))
False
True
#### A Binary Search--Recursive Version¶
In [39]:
def binarySearch(alist, item):
if len(alist) == 0:
return False
else:
midpoint = len(alist) // 2
if alist[midpoint] == item:
return True
else:
if item < alist[midpoint]:
return binarySearch(alist[:midpoint], item)
else:
return binarySearch(alist[midpoint + 1:], item)
testlist = [0, 1, 2, 8, 13, 17, 19, 32, 42, ]
print(binarySearch(testlist, 3))
print(binarySearch(testlist, 13))
False
True
### Hashing¶
• Map() Create a new, empty map. It returns an empty map collection.
• put(key,val) Add a new key-value pair to the map. If the key is already in the map then replace the old value with the new value.
• get(key) Given a key, return the value stored in the map or None otherwise.
• del Delete the key-value pair from the map using a statement of the form del map[key].
• len() Return the number of key-value pairs stored in the map.
• in Return True for a statement of the form key in map, if the given key is in the map, False otherwise.
In [40]:
class HashTable:
def __init__(self):
self.size = 11
self.slots = [None] * self.size
self.data = [None] * self.size
def put(self,key,data):
hashvalue = self.hashfunction(key,len(self.slots))
if self.slots[hashvalue] == None:
self.slots[hashvalue] = key
self.data[hashvalue] = data
else:
if self.slots[hashvalue] == key:
self.data[hashvalue] = data #replace
else:
nextslot = self.rehash(hashvalue,len(self.slots))
while self.slots[nextslot] != None and \
self.slots[nextslot] != key:
nextslot = self.rehash(nextslot,len(self.slots))
if self.slots[nextslot] == None:
self.slots[nextslot]=key
self.data[nextslot]=data
else:
self.data[nextslot] = data #replace
def hashfunction(self,key,size):
return key%size
def rehash(self,oldhash,size):
return (oldhash+1)%size
def get(self,key):
startslot = self.hashfunction(key,len(self.slots))
data = None
stop = False
found = False
position = startslot
while self.slots[position] != None and \
if self.slots[position] == key:
found = True
data = self.data[position]
else:
position=self.rehash(position,len(self.slots))
if position == startslot:
stop = True
return data
def __getitem__(self,key):
return self.get(key)
def __setitem__(self,key,data):
self.put(key,data)
In [41]:
H=HashTable()
H[54]="cat"
H[26]="dog"
H[93]="lion"
H[17]="tiger"
H[77]="bird"
H[31]="cow"
H[44]="goat"
H[55]="pig"
H[20]="chicken"
print(H.slots)
print(H.data)
print(H[20])
print(H[17])
H[20]='duck'
print(H[20])
print(H[99])
[77, 44, 55, 20, 26, 93, 17, None, None, 31, 54]
['bird', 'goat', 'pig', 'chicken', 'dog', 'lion', 'tiger', None, None, 'cow', 'cat']
chicken
tiger
duck
None
### The Bubble Sort¶
In [42]:
def bubbleSort(alist):
for passnum in range(len(alist)-1,0,-1):
for i in range(passnum):
if alist[i]>alist[i+1]:
temp = alist[i]
alist[i] = alist[i+1]
alist[i+1] = temp
alist = [54,26,93,17,77,31,44,55,20]
bubbleSort(alist)
print(alist)
[17, 20, 26, 31, 44, 54, 55, 77, 93]
#### The Short Bubble Sort¶
A bubble sort can be modified to stop early if it finds that the list has become sorted. This means that for lists that require just a few passes, a bubble sort may have an advantage in that it will recognize the sorted list and stop.
In [43]:
def shortBubbleSort(alist):
exchanges = True
passnum = len(alist)-1
while passnum > 0 and exchanges:
exchanges = False
for i in range(passnum):
if alist[i]>alist[i+1]:
exchanges = True
temp = alist[i]
alist[i] = alist[i+1]
alist[i+1] = temp
passnum = passnum-1
alist=[20,30,40,90,50,60,70,80,100,110]
shortBubbleSort(alist)
print(alist)
[20, 30, 40, 50, 60, 70, 80, 90, 100, 110]
### The Selection Sort¶
Selection sort improves upon bubble sort by making fewer swaps
In [44]:
def selectionSort(alist):
for fillslot in range(len(alist)-1,0,-1):
positionOfMax=0
for location in range(1,fillslot+1):
if alist[location]>alist[positionOfMax]:
positionOfMax = location
temp = alist[fillslot]
alist[fillslot] = alist[positionOfMax]
alist[positionOfMax] = temp
alist = [54,26,93,17,77,31,44,55,20]
selectionSort(alist)
print(alist)
[17, 20, 26, 31, 44, 54, 55, 77, 93]
### The Insertion Sort¶
Insertion sort works at the start of the list. Each pass produces a longer sorted list
In [45]:
def insertionSort(alist):
for index in range(1,len(alist)):
currentvalue = alist[index]
position = index
while position>0 and alist[position-1]>currentvalue:
alist[position]=alist[position-1]
position = position-1
alist[position]=currentvalue
alist = [54,26,93,17,77,31,44,55,20]
insertionSort(alist)
print(alist)
[17, 20, 26, 31, 44, 54, 55, 77, 93]
### The Shell Sort¶
In [46]:
def shellSort(alist):
sublistcount = len(alist) // 2
while sublistcount > 0:
for startposition in range(sublistcount):
gapInsertionSort(alist,startposition,sublistcount)
print("After increments of size",sublistcount, "The list is", alist)
sublistcount = sublistcount // 2
def gapInsertionSort(alist,start,gap):
for i in range(start+gap,len(alist),gap):
currentvalue = alist[i]
position = i
while position>=gap and alist[position-gap]>currentvalue:
alist[position]=alist[position-gap]
position = position-gap
alist[position]=currentvalue
alist = [54,26,93,17,77,31,44,55,20]
shellSort(alist)
print(alist)
After increments of size 4 The list is [20, 26, 44, 17, 54, 31, 93, 55, 77]
After increments of size 2 The list is [20, 17, 44, 26, 54, 31, 77, 55, 93]
After increments of size 1 The list is [17, 20, 26, 31, 44, 54, 55, 77, 93]
[17, 20, 26, 31, 44, 54, 55, 77, 93]
### The Merge Sort¶
Merge sort will continue to recursively move toward the beginning of the list until it hits a base case.
In [47]:
def mergeSort(alist):
print("Splitting ",alist)
if len(alist)>1:
mid = len(alist)//2
lefthalf = alist[:mid]
righthalf = alist[mid:]
mergeSort(lefthalf)
mergeSort(righthalf)
i=0
j=0
k=0
while i<len(lefthalf) and j<len(righthalf):
if lefthalf[i]<righthalf[j]:
alist[k]=lefthalf[i]
i=i+1
else:
alist[k]=righthalf[j]
j=j+1
k=k+1
while i<len(lefthalf):
alist[k]=lefthalf[i]
i=i+1
k=k+1
while j<len(righthalf):
alist[k]=righthalf[j]
j=j+1
k=k+1
print("Merging ",alist)
alist = [54,26,93,17,77,31,44,55,20]
mergeSort(alist)
print(alist)
Splitting [54, 26, 93, 17, 77, 31, 44, 55, 20]
Splitting [54, 26, 93, 17]
Splitting [54, 26]
Splitting [54]
Merging [54]
Splitting [26]
Merging [26]
Merging [26, 54]
Splitting [93, 17]
Splitting [93]
Merging [93]
Splitting [17]
Merging [17]
Merging [17, 93]
Merging [17, 26, 54, 93]
Splitting [77, 31, 44, 55, 20]
Splitting [77, 31]
Splitting [77]
Merging [77]
Splitting [31]
Merging [31]
Merging [31, 77]
Splitting [44, 55, 20]
Splitting [44]
Merging [44]
Splitting [55, 20]
Splitting [55]
Merging [55]
Splitting [20]
Merging [20]
Merging [20, 55]
Merging [20, 44, 55]
Merging [20, 31, 44, 55, 77]
Merging [17, 20, 26, 31, 44, 54, 55, 77, 93]
[17, 20, 26, 31, 44, 54, 55, 77, 93]
### The Quick Sort¶
To choose the pivot value, we will consider the first, the middle, and the last element in the list. In our example, those are 54, 77, and 20. Now pick the median value, in our case 54, and use it for the pivot value (of course, that was the pivot value we used originally). The idea is that in the case where the the first item in the list does not belong toward the middle of the list, the median of three will choose a better “middle” value. This will be particularly useful when the original list is somewhat sorted to begin with.
In [48]:
def quickSort(alist):
quickSortHelper(alist,0,len(alist)-1)
def quickSortHelper(alist,first,last):
if first<last:
splitpoint = partition(alist,first,last)
quickSortHelper(alist,first,splitpoint-1)
quickSortHelper(alist,splitpoint+1,last)
def partition(alist,first,last):
pivotvalue = alist[first]
leftmark = first+1
rightmark = last
done = False
while not done:
while leftmark <= rightmark and \
alist[leftmark] <= pivotvalue:
leftmark = leftmark + 1
while alist[rightmark] >= pivotvalue and \
rightmark >= leftmark:
rightmark = rightmark -1
if rightmark < leftmark:
done = True
else:
temp = alist[leftmark]
alist[leftmark] = alist[rightmark]
alist[rightmark] = temp
temp = alist[first]
alist[first] = alist[rightmark]
alist[rightmark] = temp
return rightmark
alist = [54,26,93,17,77,31,44,55,20]
quickSort(alist)
print(alist)
[17, 20, 26, 31, 44, 54, 55, 77, 93]
1. A sequential search is $O(n)$ for ordered and unordered lists.
2. A binary search of an ordered list is $O(logn)$ in the worst case.
3. Hash tables can provide constant time searching.
4. A bubble sort, a selection sort, and an insertion sort are $O(n^2)$ algorithms.
5. A shell sort improves on the insertion sort by sorting incremental sublists. It falls between $O(n)$ and $O(n^2)$.
6. A merge sort is $O(nlogn)$, but requires additional space for the merging process.
7. A quick sort is $O(nlogn)$, but may degrade to $O(n^2)$ if the split points are not near the middle of the list. It does not require additional space.
### Trees and Tree Algorithms¶
1. Definition One: A tree consists of a set of nodes and a set of edges that connect pairs of nodes. A tree has the following properties:
• One node of the tree is designated as the root node.
• Every node n, except the root node, is connected by an edge from exactly one other node p, where p is the parent of n.
• A unique path traverses from the root to each node.
• If each node in the tree has a maximum of two children, we say that the tree is a binary tree.
2. Definition Two: A tree is either empty or consists of a root and zero or more subtrees, each of which is also a tree. The root of each subtree is connected to the root of the parent tree by an edge.
### List of Lists Representation¶
In [49]:
myTree = ['a', ['b', ['d',[],[]], ['e',[],[]] ], ['c', ['f',[],[]], []] ]
print(myTree)
print('left subtree = ', myTree[1])
print('root = ', myTree[0])
print('right subtree = ', myTree[2])
['a', ['b', ['d', [], []], ['e', [], []]], ['c', ['f', [], []], []]]
left subtree = ['b', ['d', [], []], ['e', [], []]]
root = a
right subtree = ['c', ['f', [], []], []]
#### A Python Session to Illustrate Basic Tree Functions¶
In [50]:
def BinaryTree(r):
return [r, [], []]
def insertLeft(root,newBranch):
t = root.pop(1)
if len(t) > 1:
root.insert(1,[newBranch,t,[]])
else:
root.insert(1,[newBranch, [], []])
return root
def insertRight(root,newBranch):
t = root.pop(2)
if len(t) > 1:
root.insert(2,[newBranch,[],t])
else:
root.insert(2,[newBranch,[],[]])
return root
def getRootVal(root):
return root[0]
def setRootVal(root,newVal):
root[0] = newVal
def getLeftChild(root):
return root[1]
def getRightChild(root):
return root[2]
r = BinaryTree(3)
insertLeft(r,4)
insertLeft(r,5)
insertRight(r,6)
insertRight(r,7)
l = getLeftChild(r)
print(l)
setRootVal(l,9)
print(r)
insertLeft(l,11)
print(r)
print(getRightChild(getRightChild(r)))
[5, [4, [], []], []]
[3, [9, [4, [], []], []], [7, [], [6, [], []]]]
[3, [9, [11, [4, [], []], []], []], [7, [], [6, [], []]]]
[6, [], []]
In [51]:
x = BinaryTree('a')
insertLeft(x,'b')
insertRight(x,'c')
insertRight(getRightChild(x),'d')
insertLeft(getRightChild(getRightChild(x)),'e')
print(x)
['a', ['b', [], []], ['c', [], ['d', ['e', [], []], []]]]
### Nodes and References¶
In [52]:
class BinaryTree:
def __init__(self,rootObj):
self.key = rootObj
self.leftChild = None
self.rightChild = None
def insertLeft(self,newNode):
if self.leftChild == None:
self.leftChild = BinaryTree(newNode)
else:
t = BinaryTree(newNode)
t.leftChild = self.leftChild
self.leftChild = t
def insertRight(self,newNode):
if self.rightChild == None:
self.rightChild = BinaryTree(newNode)
else:
t = BinaryTree(newNode)
t.rightChild = self.rightChild
self.rightChild = t
def getRightChild(self):
return self.rightChild
def getLeftChild(self):
return self.leftChild
def setRootVal(self,obj):
self.key = obj
def getRootVal(self):
return self.key
In [53]:
r = BinaryTree('a')
print(r.getRootVal())
print(r.getLeftChild())
r.insertLeft('b')
print(r.getLeftChild())
print(r.getLeftChild().getRootVal())
r.insertRight('c')
print(r.getRightChild())
print(r.getRightChild().getRootVal())
r.getRightChild().setRootVal('hello')
print(r.getRightChild().getRootVal())
a
None
<__main__.BinaryTree object at 0x7f83a44e9c88>
b
<__main__.BinaryTree object at 0x7f83a44e5898>
c
hello
### Parse Tree¶
1. If the current token is a '(', add a new node as the left child of the current node, and descend to the left child.
2. If the current token is in the list ['+','-','/','*'], set the root value of the current node to the operator represented by the current token. Add a new node as the right child of the current node and descend to the right child.
3. If the current token is a number, set the root value of the current node to the number and return to the parent.
4. If the current token is a ')', go to the parent of the current node.
In [54]:
from pythonds.basic.stack import Stack
from pythonds.trees.binaryTree import BinaryTree
def buildParseTree(fpexp):
fplist = fpexp.split()
pStack = Stack()
eTree = BinaryTree('')
pStack.push(eTree)
currentTree = eTree
for i in fplist:
if i == '(':
currentTree.insertLeft('')
pStack.push(currentTree)
currentTree = currentTree.getLeftChild()
elif i not in ['+', '-', '*', '/', ')']:
currentTree.setRootVal(int(i))
parent = pStack.pop()
currentTree = parent
elif i in ['+', '-', '*', '/']:
currentTree.setRootVal(i)
currentTree.insertRight('')
pStack.push(currentTree)
currentTree = currentTree.getRightChild()
elif i == ')':
currentTree = pStack.pop()
else:
raise ValueError
return eTree
pt = buildParseTree("( ( 10 + 5 ) * 3 )")
pt.postorder() #defined and explained in the next section
3
7
9
(((4)+(5))*(7))
63
2
10
5
+
3
*
In [55]:
import operator
def evaluate(parseTree):
opers = {'+':operator.add, '-':operator.sub, '*':operator.mul, '/':operator.truediv}
leftC = parseTree.getLeftChild()
rightC = parseTree.getRightChild()
if leftC and rightC:
fn = opers[parseTree.getRootVal()]
return fn(evaluate(leftC),evaluate(rightC))
else:
return parseTree.getRootVal()
evaluate(pt)
Out[55]:
45
### Tree Traversals¶
1. preorder. In a preorder traversal, we visit the root node first, then recursively do a preorder traversal of the left subtree, followed by a recursive preorder traversal of the right subtree.
2. inorder. In an inorder traversal, we recursively do an inorder traversal on the left subtree, visit the root node, and finally do a recursive inorder traversal of the right subtree.
3. postorder. In a postorder traversal, we recursively do a postorder traversal of the left subtree and the right subtree followed by a visit to the root node.
In [56]:
def preorder(tree):
if tree:
print(tree.getRootVal())
preorder(tree.getLeftChild())
preorder(tree.getRightChild())
preorder(pt)
*
+
10
5
3
In [57]:
def preorder(self):
print(self.key)
if self.leftChild:
self.left.preorder()
if self.rightChild:
self.right.preorder()
In [58]:
def postorder(tree):
if tree != None:
postorder(tree.getLeftChild())
postorder(tree.getRightChild())
print(tree.getRootVal())
postorder(pt)
10
5
+
3
*
In [59]:
def postordereval(tree):
opers = {'+':operator.add, '-':operator.sub, '*':operator.mul, '/':operator.truediv}
res1 = None
res2 = None
if tree:
res1 = postordereval(tree.getLeftChild())
res2 = postordereval(tree.getRightChild())
if res1 and res2:
return opers[tree.getRootVal()](res1,res2)
else:
return tree.getRootVal()
postordereval(pt)
Out[59]:
45
In [60]:
def inorder(tree):
if tree != None:
inorder(tree.getLeftChild())
print(tree.getRootVal())
inorder(tree.getRightChild())
inorder(pt)
10
+
5
*
3
In [61]:
def printexp(tree):
sVal = ""
if tree:
sVal = '(' + printexp(tree.getLeftChild())
sVal = sVal + str(tree.getRootVal())
sVal = sVal + printexp(tree.getRightChild())+')'
return sVal
printexp(pt)
Out[61]:
'(((10)+(5))*(3))'
### Priority Queues with Binary Heaps¶
A priority queue acts like a queue in that you dequeue an item by removing it from the front. However, in a priority queue the logical order of items inside a queue is determined by their priority. The highest priority items are at the front of the queue and the lowest priority items are at the back. Thus when you enqueue an item on a priority queue, the new item may move all the way to the front.
The classic way to implement a priority queue is using a data structure called a binary heap. A binary heap will allow us both enqueue and dequeue items in $O(logn)$.
### Binary Heap Operations¶
• BinaryHeap() creates a new, empty, binary heap.
• insert(k) adds a new item to the heap.
• findMin() returns the item with the minimum key value, leaving item in the heap.
• delMin() returns the item with the minimum key value, removing the item from the heap.
• isEmpty() returns true if the heap is empty, false otherwise.
• size() returns the number of items in the heap.
• buildHeap(list) builds a new heap from a list of keys.
In [62]:
from pythonds.trees.binheap import BinHeap
bh = BinHeap()
bh.insert(5)
bh.insert(7)
bh.insert(3)
bh.insert(11)
print(bh.delMin())
print(bh.delMin())
print(bh.delMin())
print(bh.delMin())
3
5
7
11
### Binary Heap Implementation¶
In [63]:
class BinHeap:
def __init__(self):
self.heapList = [0]
self.currentSize = 0
def percUp(self,i):
while i // 2 > 0:
if self.heapList[i] < self.heapList[i // 2]:
tmp = self.heapList[i // 2]
self.heapList[i // 2] = self.heapList[i]
self.heapList[i] = tmp
i = i // 2
def insert(self,k):
self.heapList.append(k)
self.currentSize = self.currentSize + 1
self.percUp(self.currentSize)
def percDown(self,i):
while (i * 2) <= self.currentSize:
mc = self.minChild(i)
if self.heapList[i] > self.heapList[mc]:
tmp = self.heapList[i]
self.heapList[i] = self.heapList[mc]
self.heapList[mc] = tmp
i = mc
def minChild(self,i):
if i * 2 + 1 > self.currentSize:
return i * 2
else:
if self.heapList[i*2] < self.heapList[i*2+1]:
return i * 2
else:
return i * 2 + 1
def delMin(self):
retval = self.heapList[1]
self.heapList[1] = self.heapList[self.currentSize]
self.currentSize = self.currentSize - 1
self.heapList.pop()
self.percDown(1)
return retval
def buildHeap(self,alist):
i = len(alist) // 2
self.currentSize = len(alist)
self.heapList = [0] + alist[:]
while (i > 0):
self.percDown(i)
i = i - 1
bh = BinHeap()
bh.buildHeap([9,5,6,2,3])
print(bh.delMin())
print(bh.delMin())
print(bh.delMin())
print(bh.delMin())
print(bh.delMin())
2
3
5
6
9
### Binary Search Tree Operations¶
• Map() Create a new, empty map.
• put(key,val) Add a new key-value pair to the map. If the key is already in the map then replace the old value with the new value.
• get(key) Given a key, return the value stored in the map or None otherwise.
• del Delete the key-value pair from the map using a statement of the form del map[key].
• len() Return the number of key-value pairs stored in the map.
• in Return True for a statement of the form key in map, if the given key is in the map.
In [64]:
class TreeNode:
def __init__(self,key,val,left=None,right=None,parent=None):
self.key = key
self.leftChild = left
self.rightChild = right
self.parent = parent
def hasLeftChild(self):
return self.leftChild
def hasRightChild(self):
return self.rightChild
def isLeftChild(self):
return self.parent and self.parent.leftChild == self
def isRightChild(self):
return self.parent and self.parent.rightChild == self
def isRoot(self):
return not self.parent
def isLeaf(self):
return not (self.rightChild or self.leftChild)
def hasAnyChildren(self):
return self.rightChild or self.leftChild
def hasBothChildren(self):
return self.rightChild and self.leftChild
def replaceNodeData(self,key,value,lc,rc):
self.key = key
self.leftChild = lc
self.rightChild = rc
if self.hasLeftChild():
self.leftChild.parent = self
if self.hasRightChild():
self.rightChild.parent = self
class BinarySearchTree:
def __init__(self):
self.root = None
self.size = 0
def length(self):
return self.size
def __len__(self):
return self.size
def put(self,key,val):
if self.root:
self._put(key,val,self.root)
else:
self.root = TreeNode(key,val)
self.size = self.size + 1
def _put(self,key,val,currentNode):
if key < currentNode.key:
if currentNode.hasLeftChild():
self._put(key,val,currentNode.leftChild)
else:
currentNode.leftChild = TreeNode(key,val,parent=currentNode)
else:
if currentNode.hasRightChild():
self._put(key,val,currentNode.rightChild)
else:
currentNode.rightChild = TreeNode(key,val,parent=currentNode)
def __setitem__(self,k,v):
self.put(k,v)
def get(self,key):
if self.root:
res = self._get(key,self.root)
if res:
else:
return None
else:
return None
def _get(self,key,currentNode):
if not currentNode:
return None
elif currentNode.key == key:
return currentNode
elif key < currentNode.key:
return self._get(key,currentNode.leftChild)
else:
return self._get(key,currentNode.rightChild)
def __getitem__(self,key):
return self.get(key)
def __contains__(self,key):
if self._get(key,self.root):
return True
else:
return False
def delete(self,key):
if self.size > 1:
nodeToRemove = self._get(key,self.root)
if nodeToRemove:
self.remove(nodeToRemove)
self.size = self.size-1
else:
raise KeyError('Error, key not in tree')
elif self.size == 1 and self.root.key == key:
self.root = None
self.size = self.size - 1
else:
raise KeyError('Error, key not in tree')
def __delitem__(self,key):
self.delete(key)
def spliceOut(self):
if self.isLeaf():
if self.isLeftChild():
self.parent.leftChild = None
else:
self.parent.rightChild = None
elif self.hasAnyChildren():
if self.hasLeftChild():
if self.isLeftChild():
self.parent.leftChild = self.leftChild
else:
self.parent.rightChild = self.leftChild
self.leftChild.parent = self.parent
else:
if self.isLeftChild():
self.parent.leftChild = self.rightChild
else:
self.parent.rightChild = self.rightChild
self.rightChild.parent = self.parent
def findSuccessor(self):
succ = None
if self.hasRightChild():
succ = self.rightChild.findMin()
else:
if self.parent:
if self.isLeftChild():
succ = self.parent
else:
self.parent.rightChild = None
succ = self.parent.findSuccessor()
self.parent.rightChild = self
return succ
def findMin(self):
current = self
while current.hasLeftChild():
current = current.leftChild
return current
def remove(self,currentNode):
if currentNode.isLeaf(): #leaf
if currentNode == currentNode.parent.leftChild:
currentNode.parent.leftChild = None
else:
currentNode.parent.rightChild = None
elif currentNode.hasBothChildren(): #interior
succ = currentNode.findSuccessor()
succ.spliceOut()
currentNode.key = succ.key
else: # this node has one child
if currentNode.hasLeftChild():
if currentNode.isLeftChild():
currentNode.leftChild.parent = currentNode.parent
currentNode.parent.leftChild = currentNode.leftChild
elif currentNode.isRightChild():
currentNode.leftChild.parent = currentNode.parent
currentNode.parent.rightChild = currentNode.leftChild
else:
currentNode.replaceNodeData(currentNode.leftChild.key,
currentNode.leftChild.leftChild,
currentNode.leftChild.rightChild)
else:
if currentNode.isLeftChild():
currentNode.rightChild.parent = currentNode.parent
currentNode.parent.leftChild = currentNode.rightChild
elif currentNode.isRightChild():
currentNode.rightChild.parent = currentNode.parent
currentNode.parent.rightChild = currentNode.rightChild
else:
currentNode.replaceNodeData(currentNode.rightChild.key,
currentNode.rightChild.leftChild,
currentNode.rightChild.rightChild)
mytree = BinarySearchTree()
mytree[3]="red"
mytree[4]="blue"
mytree[6]="yellow"
mytree[2]="at"
print(mytree[6])
print(mytree[2])
yellow
at
### Balanced Binary Search Trees¶
A special kind of binary search tree that automatically makes sure that the tree remains balanced at all times. This tree is called an AVL tree and is named for its inventors: G.M. Adelson-Velskii and E.M. Landis.
An AVL tree implements the Map abstract data type just like a regular binary search tree, the only difference is in how the tree performs. To implement our AVL tree we need to keep track of a balance factor for each node in the tree. We do this by looking at the heights of the left and right subtrees for each node. More formally, we define the balance factor for a node as the difference between the height of the left subtree and the height of the right subtree.
$$balanceFactor=height(leftSubTree)−height(rightSubTree)$$
Using the definition for balance factor given above we say that a subtree is left-heavy if the balance factor is greater than zero. If the balance factor is less than zero then the subtree is right heavy. If the balance factor is zero then the tree is perfectly in balance. For purposes of implementing an AVL tree, and gaining the benefit of having a balanced tree we will define a tree to be in balance if the balance factor is -1, 0, or 1. Once the balance factor of a node in a tree is outside this range we will need to have a procedure to bring the tree back into balance.
In [65]:
def _put(self, key, val, currentNode):
if key < currentNode.key:
if currentNode.hasLeftChild():
self._put(key, val, currentNode.leftChild)
else:
currentNode.leftChild = TreeNode(key, val, parent=currentNode)
self.updateBalance(currentNode.leftChild)
else:
if currentNode.hasRightChild():
self._put(key, val, currentNode.rightChild)
else:
currentNode.rightChild = TreeNode(key, val, parent=currentNode)
self.updateBalance(currentNode.rightChild)
def updateBalance(self, node):
if node.balanceFactor > 1 or node.balanceFactor < -1:
self.rebalance(node)
return
if node.parent != None:
if node.isLeftChild():
node.parent.balanceFactor += 1
elif node.isRightChild():
node.parent.balanceFactor -= 1
if node.parent.balanceFactor != 0:
self.updateBalance(node.parent)
def rotateLeft(self, rotRoot):
newRoot = rotRoot.rightChild
rotRoot.rightChild = newRoot.leftChild
if newRoot.leftChild != None:
newRoot.leftChild.parent = rotRoot
newRoot.parent = rotRoot.parent
if rotRoot.isRoot():
self.root = newRoot
else:
if rotRoot.isLeftChild():
rotRoot.parent.leftChild = newRoot
else:
rotRoot.parent.rightChild = newRoot
newRoot.leftChild = rotRoot
rotRoot.parent = newRoot
rotRoot.balanceFactor = rotRoot.balanceFactor + \
1 - min(newRoot.balanceFactor, 0)
newRoot.balanceFactor = newRoot.balanceFactor + \
1 + max(rotRoot.balanceFactor, 0)
def rebalance(self, node):
if node.balanceFactor < 0:
if node.rightChild.balanceFactor > 0:
self.rotateRight(node.rightChild)
self.rotateLeft(node)
else:
self.rotateLeft(node)
elif node.balanceFactor > 0:
if node.leftChild.balanceFactor < 0:
self.rotateLeft(node.leftChild)
self.rotateRight(node)
else:
self.rotateRight(node)
### Summary of Map ADT Implementations¶
Table 1: Comparing the Performance of Different Map Implementations
operation Sorted List Hash Table Binary Search Tree AVL Tree
put O(n)$O(n)$ O(1)$O(1)$ O(n)$O(n)$ O(log2n)$O(\log_2{n})$
get O(log2n)$O(\log_2{n})$ O(1)$O(1)$ O(n)$O(n)$ O(log2n)$O(\log_2{n})$
in O(log2n)$O(\log_2{n})$ O(1)$O(1)$ O(n)$O(n)$ O(log2n)$O(\log_2{n})$
del O(n))$O(n))$ O(1)$O(1)$ O(n)$O(n)$ O(log2n)$O(\log_2{n})$
|
# Lower bound to $\epsilon$-expansion of a subset of a half-sphere
Below are two known lemmas on a $$d$$-dimensional sphere (related to the isoperimetric inequality). I would like to know: does a similar statement like this holds for a $$d$$-dimensional dome also (i.e. half the sphere)
Lemma 1: Suppose $$s$$ is a subset of the $$d$$-sphere with normalised measure $$\mu(s)=1/2$$. The $$\epsilon$$-expansion of the set $$s$$ is then at least as large as the $$\epsilon$$-expansion of a half sphere, if we use the geodesic metric.
Lemma 2: An $$\epsilon$$-expansion of a half sphere in the geodesic metric has a normalised measure $$\geq 1-\sqrt{\pi/8} \exp(-d \epsilon^2/2)$$
Definition used: $$\epsilon$$-expansion of a set $$s$$ contains all points that are at most $$\epsilon$$ units away from $$s$$ according to a specified metric.
Suppose we don't have a sphere to begin with, but a $$d$$-dimensional dome or half a sphere instead. We can keep using the geodesic metric. Then we take a subset $$s'$$ of this dome with normalised measure $$\mu(s')=0.5$$. What is the normalised measure of the $$\epsilon$$-expansion of this set $$s'$$? Can it be lower-bounded by some function $$f(d, \epsilon)$$ just like lemma 2 for the sphere?
Any leads on this problem would be much appreciated! Thank you in advance!
|
## Duke Mathematical Journal
### Flops and the S-duality conjecture
Yukinobu Toda
#### Abstract
We prove the transformation formula of Donaldson–Thomas (DT) invariants counting two-dimensional torsion sheaves on Calabi–Yau 3-folds under flops. The error term is described by the Dedekind eta function and the Jacobi theta function, and our result gives evidence of a 3-fold version of the Vafa–Witten S-duality conjecture. As an application, we prove a blow-up formula of DT-type invariants on the total spaces of canonical line bundles on smooth projective surfaces. It gives an analogue of the similar blow-up formula in the original S-duality conjecture by Yoshioka, Li and Qin, and Göttsche.
#### Article information
Source
Duke Math. J., Volume 164, Number 12 (2015), 2293-2339.
Dates
Received: 20 December 2013
Revised: 15 October 2014
First available in Project Euclid: 16 September 2015
Permanent link to this document
https://projecteuclid.org/euclid.dmj/1442364464
Digital Object Identifier
doi:10.1215/00127094-3129595
Mathematical Reviews number (MathSciNet)
MR3397387
Zentralblatt MATH identifier
1331.14055
#### Citation
Toda, Yukinobu. Flops and the S-duality conjecture. Duke Math. J. 164 (2015), no. 12, 2293--2339. doi:10.1215/00127094-3129595. https://projecteuclid.org/euclid.dmj/1442364464
#### References
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• [17] A. Gholampour, A. Sheshmani, and R. P. Thomas, Counting curves on surfaces in Calabi-Yau 3-folds, Math. Ann. 360 (2014), 67–78.
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• [19] L. Göttsche, Invariants of moduli spaces and modular forms, Rend. Istit. Mat. Univ. Trieste 41 (2009), 55–76.
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• [21] J. Hu and W. P. Li, The Donaldson-Thomas invariants under blowups and flops, J. Differential Geom. 90 (2012), 391–411.
• [22] D. Huybrechts and M. Lehn, “Geometry of moduli spaces of sheaves” in Aspects in Mathematics, E31, Vieweg, Braunschweig, 1997.
• [23] D. Joyce, Configurations in abelian categories, II: Ringel-Hall algebras, Adv. Math. 210 (2007), 635–706.
• [24] D. Joyce, Configurations in abelian categories, IV: Invariants and changing stability conditions, Adv. in Math 217 (2008), 125–204.
• [25] D. Joyce and Y. Song, A theory of generalized Donaldson-Thomas invariants, Mem. Amer. Math. Soc. 217, Amer. Math. Soc., Providence, 2012.
• [26] S. Katz, Genus zero Gopakumar-Vafa invariants of contractible curves, J. Differential Geom. 79 (2008), 185–195.
• [27] S. Katz and D. R. Morrison, Gorenstein threefold singularities with small resolutions via invariant theory for Weyl groups, J. Algebraic Geom. 1 (1992), 449–530.
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• [29] M. Kontsevich and Y. Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, preprint, arXiv:0811.2435v1 [math.AG].
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• [31] A. Langer, Moduli spaces of sheaves and principal $G$-bundles, Proc. Sympos. Pure Math. 80 (2009), 273–308.
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|
Does anyone know how to change the color scheme and/or bullets used in Beamer's table of contents (without changing the overall structure color or theme)? Unfortunately, the following doesn't work:
\setbeamertheme{enumerate item}[default]
\setbeamercolor{enumerate item}{black}
I would even be fine removing the numbering altogether.
\setbeamerfont{section number projected}{%
|
Free access
Issue A&A Volume 527, March 2011 A22 7 Extragalactic astronomy http://dx.doi.org/10.1051/0004-6361/201015323 20 January 2011
## 1. Introduction
We discuss the situation when the energy release due to accretion into the black hole of an AGN is small, and the energy activity is determined generally by the Blandford-Znajek mechanism. As we see below, the main energy output from a rotating black hole is in a relativistic jet. Thus, we mean LLAGNs with a jet. They are low-luminosity Seyfert galaxies and LINERs, the central source of which is a low luminosity AGN. A LINER is the low-ionization nuclear emission-line region at the center of a bright galaxy. LINERs are characterized by collisionally excited lines of neutral and singly ionized gas (Maoz 2007).
We first describe in Sect. 2 how the Blandford- Znajek mechanism operates. We describe configurations of the magnetic field and electric currents in both a disk and the black hole magnetosphere above a disk. We also show how the energy and angular momentum are transmitted from the black hole rotation to a jet. In Sect. 3, we calculate the synchrotron radiation of fast protons in a disk. The production of high energy γ-rays by relativistic protons is presented in Sect. 4. The observational consequences of the suggested scheme are described in the last section.
## 2. Blandford-Znajek mechanism
The Blandford-Znajek mechanism assumes that energy and angular momentum are extracted from the black hole rotation. The rotation energy Erot stored in the black hole rotation is large, for slow rotation being . Here we introduce the dimensionless parameter of rotation, a = Jc/M2G, where J is the angular momentum of the black hole. For a black hole, a < 1. The value of ΩH is the angular velocity of black hole rotation, ΩH = ac/2rH. The extraction of the rotation energy is possible if there exists a poloidal magnetic field B near the black hole horizon. The black hole in this case works as a dynamo machine, creating the voltage U, U = ΩHfH/2πc (Landau & Lifshitz 1984; Thorne et al. 1986), where fH is the flux of the magnetic field reaching the horizon, . The voltage generates the electric current I = U/(R + RH), which on one side is closed at the black-hole horizon surface of resistivity RH = 4π/c ≈ 377 ohms (Thorne et al. 1986). The resistivity of the outer part of the system is R. Thus, the extracted power is . The power L reaches its maximum value Lm when R = RH, . This value is proportional to the square of the black hole mass, Lm ∝ M2, and for large enough magnetic fields can exceed the Eddington luminosity, B > BEdd = 5.5 × 109a-1(M/M)−1/2 Gauss. For high masses of AGN black holes, the value of the magnetic field BEdd is quite moderate, BEdd ≃ 105 Gauss (a ≃ 1).
We note that to ensure in general that the gravitation energy release is extremely efficient, there must be a high mass accretion rate on to the massive black hole and a high radiative efficiency ηr, whereas the Blandford-Znajek mechanism provides such efficiencies when there is black hole rotation, ΩH ≠ 0, and a strong enough magnetic field B near the black hole horizon. Formally, the Blandford-Znajek mechanism does not need accretion. In addition, the accumulation of a strong magnetic field requires some accretion process, but accretion is not a source of energy.
### 2.1. Magnetic field and electric current configurations
For disk accretion, the magnetic field inside the disk and nearby must have no component that is perpendicular to the disk (Istomin & Sol 2009). In the axisymmetric stationary electromagnetic field, a charged particle conserves the generalized angular momentum, ρpφ + qf/c = const., where ρ is the cylindrical distance from the center, f is the flux of the poloidal magnetic field, and q is the charge of a particle. However, two terms in this relation are practically not commensurable in the case of disk accretion. The first one is proportional to the frequency of rotation of a particle in the disk vφ/ρ, the second one is proportional to the cyclotron frequency, ωc = qBz/mc, of the particle rotation in the perpendicular magnetic field Bz. A charged particle then can move in the radial direction if it is not magnetized, ωc ≃ vφ/ρ, i.e. in a practically zero perpendicular magnetic field. The radial Bρ and the azimuthal Bφ components can be arbitrary. We see that the accretion of matter provides only the radial component of the poloidal magnetic field towards the black hole vicinity. And the magnetic field in the expression for the Blandford-Znajek luminosity is the radial magnetic field B = Bρ produced by the conducting matter of the accretion disk. For there to be a zero component of the magnetic field perpendicular to the disk, an electric current Iρ must flow through the disk, which is just the current I = −Iρ generated by the voltage U. The accretion disk is not only the origin of the flux of the matter on to the black hole, but also conducts the electric current. This current then flows onto the black hole horizon and closes through a jet in outer space. Owing to the spiral motion of charged particles in the disk, there exist not only a radial current jρ but also an azimuthal current jφ. Only the azimuthal current creates the radial magnetic field. The ratio of the radial current to the azimuthal one αj = jρ/jφ can be found by considering that from one side the current I is produced by the black hole rotation, I = U/(R + RH) = (a/16π)BρrHc [ RH/(R + RH) ] , and from another side the Maxwell equation determines the radial magnetic field through the azimuthal current, I = αjrHcBρ. We obtain αj = (a/16π) [ RH/(R + RH) ] ≪ 1, which shows that the toroidal magnetic field Bφ is much less than the radial magnetic field Bρ, Bφ = αjBρ.
### 2.2. Jet power
The radial magnetic field at the black hole horizon allows the transfer of energy and angular momentum from the black hole rotation to particles of the black hole magnetosphere and to particles leaving the accretion disk. The lines of the radial magnetic field begin to rotate with the angular velocity ΩF < ΩH, ΩF = ΩHR/(R + RH) (Thorne et al. 1986). For the optimal condition R = RH, ΩF = ΩH/2 (Blandford & Znajek 1977). Forced by the centrifugal acceleration towards the light cylinder surface (rL = cF = 2a-1rHR/(R + RH)), particles achieve azimuthal velocities, which are close to the speed of light, and significant energies. The main energy is in protons because of their low synchrotron losses in the strong magnetic field. The Lorentz factor γ of protons on the light cylinder surface is γ = (ωcLF)1/2 (Istomin & Sol 2009), where ΩcL = eBL/mpc is the non – relativistic proton cyclotron frequency in the magnetic field BL on the light cylinder surface. Taking into account that the radial magnetic field falls as ρ-1, we obtain γ = (ωcHrH/c)1/2, where ωcH is the proton cyclotron frequency near a black hole. Almost all particle energy is in the azimuthal motion, pφ ≃ mpγc, and only a small part is in the radial one, pρ ≃ mpγ1/2c (Istomin & Sol 2009). Thus, rotating energetic protons slowly flow outside the light cylinder surface, vρ ≃ cγ−1/2, forming the relativistic jet. The energy density of particles on the light cylinder is equal to the density of the electromagnetic energy (Istomin 2010). We can then calculate the jet luminosity . The total Blandford-Znajek luminosity L must, of course, be higher than the jet luminosity LJ. This implies the definite condition for the magnetic field stress near the black hole (1)or for R = RH(2)For AGN black-hole masses of M ≃ 108 M, this condition (B > 3 × 103 Gauss) is not onerous and AGN with such moderate magnetic fields can produce a relativistic jet. Less massive black holes must have higher magnetic fields. At the center of our Galaxy, for example, there must be B ≈ 105 Gauss. However, a magnetic field of the order of 1011 Gauss for micro-quasars seems problematic when producing relativistic jets. However, Karitskaya et al. (2009) measured the disk magnetic field to be 600 Gauss at a distance 2 × 105rH in Cygnus X-1. We see in Eqs. (1) and (2) a strong dependence of the magnitude of the magnetic field near the black hole B on the black hole rotation, B ∝ a-8. This increases the value of magnetic fields for slowly rotating black holes. We note, however, that the rotation cannot be too slow: the light cylinder surface must be inside the jet radius for producing a relativistic jet, rJ > rL, a > 4rH/rJ.
### 2.3. The resistivity R
The effective work done by the Blandford-Znajek mechanism depends on the value of resistivity R. The maximum output is for R = RH. However, the real resistivity of the system can differ from this value. We now estimate the resistivity Rc of the current loop created by the unipolar inductor voltage U. The current resistivity, of course, is determined by the electron motion. We assume the conducting system to be a box with the cross-section S and the length l along the direction of the electric current. The resistivity of this system is then Rc = l/, where σ is the electron conductivity, σ = ne2τe/me, n is the electron density, me and e are its mass and charge, and τe is the relaxation time of electrons. Coulomb collisions of electrons dominate even in a low ionized plasma, and we can write the electron conductivity in the form (3)We introduce the coefficient ησ < 1, which takes into account the possible abnormal electron conductivity due to turbulent or another processes decreasing the electron conductivity. The other parameters are ε the mean electron energy, and Λ the Coulomb logarithm, Λ ≈ 15−20. Thus, the electron resistivity is (4)The electron resistivity strongly depends on the electron energy, decreasing as the energy increases. The energy ε can be estimated from the bolometric luminosity Lb in a continuous spectrum, Lb = S1σSε4, where S1 is the surface of the current system, S1 ≈ 2S + 4S1/2l. The constant σS is the Stefan-Boltzmann constant in energetic units, σS = π2/60ħ3c2 = 1.6 × 1059 erg-3 cm-2 s-1. Substituting the expression ε = (Lb/S1σS)1/4 into Eq. (4), we get (5)(6)Characteristic luminosities L1 and L2 are defined as (7)(8)Equations (7) and (8) show that for the real bolometric luminosities of AGN and LLAGN the resistivity Rc is very small, Rc ≪ RH, and the current system is far from the optimal condition Rc = RH. We note that for relativistic electrons the characteristic luminosity L2 does not depend on the size of the system, and the factor l2S1/S2 is only the geometric factor which is of the order of unity. In contrast, the luminosity L1 for non-relativistic electrons is inversely proportional to the system size l, L1 ∝ l−2/3. According to this estimation, we can conclude that the Blandford-Znajek mechanism for extracting energy from rotating black holes is ineffective for Ohmic heating of outer space. The only possible way to extract power by means of the Blandford-Znajek mechanism is to transform it into relativistic jet luminosity LJ. We can attribute to the jet some value of the resistivity RJ using the relation LJ = RJI2. Neglecting the electron resistivity, we have L = LJ. This implies that the equation for determining the jet resistivity is (9)Denoting the quantity κ = a2(ωcHrH/c)1/4/128π, (κ ≥ 1), we find that (10)Only for κ = 1 do we achieve the optimal efficiency of the central machine, R = RH. For κ > 1, we can assign to the jet two values of resistivity, one greater than RH (the sign + in Eq. (10)), another less than RH (the sign – in Eq. (10)). For both values, the jet luminosity is the same. However, the solution in Eq. (10) with the negative sign is unstable because a fluctuation of the magnetic field δB in this case results in a negative feedback with the power of the central machine, δL/δB < 0, while the value δLJ/δB is always positive since δLJ/δB = 7LJ/4B.
## 3. Acceleration of protons in accretion disk: proton synchrotron radiation
Producing a relativistic jet, a rotating black hole transmits an electric current I of high magnitude through an accretion disk. This current creates the magnetic field not only outside the disk, but also inside. Internal magnetic fields Bρ and Bφ, such that Bφ ≪ Bρ, are of the same order as that above the disk, except that they are equal to zero at the disk equator. Fields are frozen to the disk plasma motion. Therefore, a turbulent motion in the disk induces a turbulent electric field , where is the turbulent plasma velocity. This stochastic electric field accelerates disk particles. This scenario of particle acceleration by large-scale 2D turbulence in a disk was discussed by Istomin & Sol (2009). They found that protons are indeed accelerated by this mechanism up to high energies. The maximum value of the proton Lorentz factor γm ≫ 1. Electrons are almost not accelerated at all because of large synchrotron losses (Istomin & Sol 2009). Thus, for the pure Blandford-Znajek mechanism almost no non-thermal high-energy radiation is produced by disk electrons because of the strong magnetic field and, connected with this, large synchrotron losses in any acceleration process.
In contrast, fast disk protons can radiate the synchrotron emission in strong magnetic fields. The distribution function of fast protons follows a power-law function fp = bγβ. The index β is the ratio of the loss energy rate to the rate of acceleration by stochastic electric field (Istomin & Sol 2009) (11)where n is the proton density, σE is the cross-section of the proton-proton collisions in the disk σE ≈ 10-26 cm2, and the time τc is the correlation time of the turbulence. We consider that relativistic protons have energies in the range 10 < γ < 108 eV/Td, where the pricipal means of energy loss for fast protons is in their collisions with disk protons and Td is the disk temperature in eV units. The correlation time τc is of the order of the time of the azimuthal gyration of the matter in the disk, τc ≃ ρ/uφ ∝ ρ3/2. We consider the velocity uφ to be uφ ∝ ρ−1/2, as in the Keplerian disk. The same law applies to the turbulent motion, , and the disk density is n ∝ ρ-2. We see that the power-law index β increases with the radius ρ as ρ3/2, β = βH(ρ/rH)3/2, where βH is the index value near the black hole. The density of energetic protons is determined by the condition that their energy density is of the order of the energy density of the stochastic electric field . Therefore, the distribution function of fast protons is (12)We see that the fast proton density decreases as ∝ ρ-3. Substituting this function into the well known expression for synchrotron radiation of one particle P1 = 2(e2/mpc2)2cB2γ2/3 and integrating over γ up to γm, we find that the density of the synchrotron power W is (13)The total synchrotron luminosity Ls is the integral of Eq. (13) over the disk volume (14)Here we introduce the dimensionless disk width near the black hole, h = H/rH. Because the index β increases with distance from the black hole, i.e., β = βHx3/2, the main contribution to the total synchrotron luminosity comes from the inner part of the disk, x < x1 = (3/βH)2/3 if βH < 3. For βH > 3, the luminosity is lower. For lnγm ≫ 1, the result of the integration in Eq. (14) with logarithmic accuracy is (15)(16)To generate a relativistic proton jet, an AGN must have a strong magnetic field near the central black hole. This condition is given by Eq. (1), ωcHrH/c ≥ (128π)4a-8. Using that, we obtain (17)(18)For our Galaxy, the estimated synchrotron luminosity of the disk is Ls ≃ 1035 erg/s, which is close to its bolometric luminosity Lb ≃ 1036 erg/s. In any case, the synchrotron luminosity from a turbulent disk given by Eqs. ((15), (16)) is always much less than the total power extracted from a AGN rotating black hole.
The frequencies of radiation are in the range . According to Eq. (1) ωcH ≥ (128π)4a-8c/rH, we estimate that . For γm ≃ 103 − 104, frequencies are in the infrared band for AGNs (M ≃ 108 M). Observed LLAGNs with radiatively inefficient accretion flow indeed show a peak in infrared emission (Maoz 2007). The center also radiates infrared light, and according to observations (Genzel et al. 2003; Nishiyama et al. 2009), this emission comes from the rotating accretion disk. We suggest that this infrared emission is due to the proton synchrotron radiation from the disk. The emitted spectrum of radiation is locally a power law F(ν) ∝ ν−(β−1)/2 because the fast proton distribution function is a power law with index β. However, β changes in the disk since β = βH(ρ/rH)3/2. The integration over the disk gives the following dependence F(ν) ∝ ν−(βH−1)/2/ln(ν), which is almost a power law, but corrected by the logarithmic function. We note that the observed power-law index of the infrared radiation from the Galactic center is −0.6 (Meyer et al. 2009), implying that index of the proton distribution is βH ≃ 2.2.
## 4. Very high energy radiation
In the Blandford-Znajek mechanism, almost all energy is transformed into protons, a jet, or a disk. Thus, it appears to be a barionic scenario for the very high energy (VHE) photon production. VHE photons are measured by Cherenkov telescopes and have energy in the TeV band. Energetic protons collide with the ambient matter and produce pions and then gamma quanta. Sources of the VHE radiation can be in the disk and the jet. We first calculate the VHE radiation from a disk. The spectrum of photons reproduces the spectrum of fast protons in Eq. (12), γ ≫ 1, and is equal to (19)where n is the disk proton density, γph is the photon energy Eph in units of the proton rest energy, γph = Eph/mpc2. The integration in Eq. (19) is over the disk volume. The quantities , and β depend on the radial distance ρ as we have discussed. The result of the integration with logarithmic accuracy, lnγph > 1, is (20)We obtain the total luminosity of VHE radiation from the disk by integrating the spectrum given by Eq. (19) over photon energies mpc2γph(21)We see that the VHE luminosity is proportional to the energy of the magnetic field near the black hole, LVHE ∝ B2M3, and increases with the black hole mass. Substituting the condition in Eq. (2) in to the expression (21), we get (22)which implies that for M ≃ 108 M. We note that the luminosity of VHE photons from the disk increases with the black hole mass in Eq. (22), while the bolometric luminosity of the proton synchrotron radiation in the disk represented by Eqs. (17) and (18) decreases with mass because of the different dependences of luminosities on the magnetic field B, Ls ∝ B4, LVHE ∝ B2. The magnetic field must be stronger for low-mass black holes (see Eq. (2)).
Fast protons of the jet can also produce VHE photons. Their energy density on the light cylinder surface, ρ = rL = cF ≃ 4a-1rH, is equal to the energy density of the electromagnetic field near this surface (Istomin 2010). Thus, the jet proton density on the light cylinder surface is nL = a2B2/64πmpc2γ. Moving further away the light surface, protons diminish in density nJ in line with the jet poloidal magnetic field to which they are frozen, nJ = nL(RL/ρ)2. The total luminosity of the jet in γ-rays is
where n is the density of the interstellar gas inside the jet and V is the jet volume. The integration provides a simple formula analogous to Eq. (21) (23)where l is the jet length and rJ is the outer radius of the jet at its base. Comparing Eqs. (21) and (23), we conclude that they are similar and that both contain the column density of the matter, which for the disk is nHH and the jet is nl. We assume an angular resolution Δφ. In the field of the central source, there is a contribution of the jet emission along its length l = DΔφ, where D is the distance to the source. For a resolution Δφ < (rH/D)(nH/n), this means that the VHE luminosity of the disk dominates over the observed flux of the central source. In contrast, for a resolution Δφ > (rH/D)(nH/n) we will observe only the radiation of the jet. Using the condition given by Eq. (2), the expression in Eq. (23) becomes (24)If the jet length l does not depend on the black-hole scale length rH, as it occurs over the accretion disk width H, then the VHE luminosity of the jet does not depend on the black hole mass and is defined only by the column density nl, .
## 5. Discussion
It seems that LLAGNs are good candidates in which to observe the Blandford-Znajek mechanism. Accreting matter onto a black hole in LLAGNs is a weak source of energy because of either the low accretion mass rate or low radiative efficiency. It may then be possible to observe a black hole operating like a dynamo machine. The accretion disk would play the role of a conductor through which the electric current would flow and the electric current would then follow a jet. For jet creation, a strong magnetic field near the black hole is needed. For AGNs of mass 108 M, this field should not be too high, B ≥ 3 × 103 Gauss. The magnetic field could be accumulated during previous epochs of high accretion rate. The rotating black hole loses its rotation energy and angular momentum, which are both transmitted to the jet. Rotating with the black hole, the radial magnetic field transfers its rotation to the surrounding matter, which leeds to relativistic energies being attained on the light surface. The energy is mainly in protons, which form the relativistic jet. The accretion disk around the black hole can be observed in millimetre and infrared bands. To accrete matter, the disk must be turbulent (abnormal transport coefficients). The turbulent motion in the strong magnetic field generates a turbulent electric field, which accelerates disk ions. Electrons are not accelerated to relativistic energies because of their large synchrotron losses. Disk fast protons radiate synchrotron emission in the infrared range. The high proton energies should correspond to a high disk luminosity in the very high energy (VHE) photon range, and that a correlation exists between the infrared luminosity Ls of the disk and its VHE luminosity . Using Eqs. (15) and (21), and excluding the unknown value of the magnetic field, we find the correlation (25)We can check the validity of this relation for low-luminosity AGNs with known luminosities LVHE and Ls, namely the Galactic center (Sgr A*), M 87, and Centaurus A. The nucleus of Centaurus A has a high bolometric luminosity Lb ≃ 1.3 × 1041 erg/s (Meisenheimer et al. 2009). At this luminosity, a high density of infrared photons of energy εph ≃ 0.1 eV in the central source prevents the free escape of VHE photons due to photon-photon collisions and the production of electron-positron pairs. The simple estimate Lb < εphcd/σT, where σT is the Thomson cross-section and d is the length scale of the central engine d ≃ 102rH = 1.5 × 1015cm for Centaurus A for which M = 5 × 107 M, gives the condition Lb < 1.1 × 1037 erg/s at which the photon-photon annihilation is ineffective. The absorption of VHE quanta and the generation of e + e − pairs result in the re-radiation of VHE emission, as described by Stawarz et al. (2006), for the interaction of VHE radiation with the starlight radiation from stars of the host galaxy. Thus, for Centaurus A the observed luminosity LVHE does not reflect the direct VHE radiation from the black hole vicinity.
For Sgr A*, we have LVHE = 3 × 1034 erg/s (Aharonian et al. 2009a) and Ls = LIR = 1036 erg/s (Yuan et al. 2003), M = 3.6 × 106 M, and for M 87 LVHE = 3 × 1040 erg/s (Aharonian et al. 2006) and Ls = LIR = 1039 erg/s (Perlman et al. 2007), M = 3 × 109 M. Substituting these values in to Eq. (25), we find
for Sgr A*, and
for M 87. These data do not contradict our model. To reach this conclusion, we must be sure that the observed VHE luminosity LVHE comes from the disk, and not the jet. For the angular resolution of current VHE instruments, this requires that nH/n > 10-3(D/rH). For M 87, the argument in favor of a disk origin of the VHE radiation is its short time variability, which excludes the large scale jet of 2 kpc length (Aharonian et al. 2006). On the basis of our estimate of the VHE luminosity of the total jet given by Eq. (21), , the jet is less luminous than the luminosity observed, while from Eq. (22). For the region of Sgr A*, the origin of the VHE emission is not quite clear (HESS collaboration 2010). One argument in favor of a disk origin of the VHE radiation is that the observed spectral index and the cut-off energy of VHE radiation (Aharonian et al. 2009a), β ≃ −2.1 and Ec ≃ 16 TeV, are close to the values that follow from the observation of IR radiation from the disk, β ≃ −2.2, γm ≃ 104, and Em ≃ 10 TeV. The argument of Aharonian et al. (2008) that VHE radiation from Sgr A* is likely from the jet rather than the disk is based on the assumptions that the X-ray emission originates in the disk and there is no time variability in the VHE flux during the X-ray flare. However, we discuss below another possible origin of X-ray radiation in the Blandford-Znajek mechanism, which is not in the disk.
Although Centaurus A is unsuitable for a comparison of the discussed model with observations, substituting its data into Eq. (25) provides values of the VHE luminosity not far from previous estimates, i.e.,
where LVHE = 2.6 × 1039 erg/s (Aharonian 2009b), LIR = 1.3 × 1041 erg/s, and M = 5 × 107 M. This implies that the re-radiation of VHE emission does not strongly affect its power.
We should also consider whether the observed infrared radiation from Sgr A* and M 87 might originate in the disk. For Sgr A*, the short time variability of the NIR emission, which has a period of the order of 20 min, is strong evidence that it originates in the disk (Genzel et al. 2003; Nishiyama et al. 2009). The measured period corresponds to the rotation of a hot spot on the disk around the black hole at distances close to the black hole horizon. We recall that M 87 is a LINER. Apart from the thermal component of its mid-infrared emission, we also observe its power law synchrotron-like emission of similar intensity ≃1039 erg/s (Perlman et al. 2007). The thermal component of temperature ≃50 K is the radiation of the dust around the central energy source. The power-law component is thought to be the radiation from the disk.
LLAGNs also radiate significant power in the X-ray band (Maoz 2007). Because the Thomson cross-section of the scattering of the electromagnetic radiation for protons is (me/mp)2 times less than that for electrons the process of the inverse Compton scattering, which is important to models of standard AGN radiation, can not be applied to explain the X-ray radiation of LLAGNs in the scheme suggested here. However, a second component of fast protons exists in the jet formed by the Blandford-Znajek mechanism. These are protons accelerated in the disk, then ejected into the black hole magnetosphere that obtain additional energy while crossing the light cylinder surface. This two-step mechanism of proton acceleration up to very high energies was suggested by Istomin & Sol (2009). The Lorentz factor of these particles is γ = (γdωcLF)1/2. We recall that ωcL is the non-relativistic proton cyclotron frequency at the light surface, ΩF is the angular frequency of rotation of magnetic field lines, ΩF ≃ ΩH/2, and γd is the Lorentz factor of disk fast protons. The ratio ωcLF is a very large number, ≫ γd, and the energy of these particles is much greater than the energy of fast particles in the disk. They radiate synchrotron emission in the region behind the light surface above the outer part of the disk near the base of the jet. The ratio of frequencies of the synchrotron radiation in both this region and the inner disk is , where is the proton cyclotron frequency at the jet base averaged over the volume from rL to rJ. In the jet behind the light surface, the poloidal magnetic field weakens like ∝ ρ-2, the toroidal field decreases more slowly, ∝ ρ-1, but initially at ρ = rLBφ is small, Bφ ≪ Bρ. Because of this we can consider B(ρ) = BL(rL/ρ)2 out to the outer jet radius rJ and . As a result, we obtain
Using the expression in Eq. (1) for the ratio ωcHrH/c ≃ 3 × 1010 and the estimates γd ≃ 104 and rJ/rL ≃ 10−102, we obtain νJ/νd ≃ 104−105, which corresponds to νJ frequencies in the X-ray band when νd is in the IR band. Unfortunately, we cannot estimate the X-ray luminosity from the jet base because we do not know the fraction of fast protons escaping the disk and being collected by the jet, which depends on the disk model. The variability of this X-ray emission is expected from plasma instabilities in the jet (Istomin 2010).
In conclusion, we can say that LLAGNs, or at least some of them, could be be extracting the energy from the black hole rotation by means of the Blandford-Znajek mechanism. The black hole spends almost all its energy on the jet production and proton acceleration. These LLAGNs are probably sources of high-energy cosmic rays. Their VHE luminosity should reflect the intensive process of proton acceleration. Our scenario predicts that the value of LVHE increases with the black hole mass as M3/2 and with the infrared luminosity of the disk as (see Eq. (25)). The discovery of new bright VHE sources from LLAGNs could confirm our hypotheses.
## Acknowledgments
We acknowledge support from the Observatoire de Paris and the LEA ELGA. This work also was partially supported by the Russian Foundation for Basic Research (grant No. 08-02-00749) and the State Agency for Science and Innovation (state contract No. 02.740.11.0250).
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# ISEE Middle Level Reading : Textual Relationships in Science Passages
## Example Questions
### Example Question #21 : Extrapolating From The Text In Natural Science Passages
Adapted from “Humming-Birds: As Illustrating the Luxuriance of Tropical Nature” in Tropical Nature, and Other Essays by Alfred Russel Wallace (1878)
The food of hummingbirds has been a matter of much controversy. All the early writers down to Buffon believed that they lived solely on the nectar of flowers, but since that time, every close observer of their habits maintains that they feed largely, and in some cases wholly, on insects. Azara observed them on the La Plata in winter taking insects out of the webs of spiders at a time and place where there were no flowers. Bullock, in Mexico, declares that he saw them catch small butterflies, and that he found many kinds of insects in their stomachs. Waterton made a similar statement. Hundreds and perhaps thousands of specimens have since been dissected by collecting naturalists, and in almost every instance their stomachs have been found full of insects, sometimes, but not generally, mixed with a proportion of honey. Many of them in fact may be seen catching gnats and other small insects just like fly-catchers, sitting on a dead twig over water, darting off for a time in the air, and then returning to the twig. Others come out just at dusk, and remain on the wing, now stationary, now darting about with the greatest rapidity, imitating in a limited space the evolutions of the goatsuckers, and evidently for the same end and purpose. Mr. Gosse also remarks, ” All the hummingbirds have more or less the habit, when in flight, of pausing in the air and throwing the body and tail into rapid and odd contortions. This is most observable in the Polytmus, from the effect that such motions have on the long feathers of the tail. That the object of these quick turns is the capture of insects, I am sure, having watched one thus engaged pretty close to me.”
Which of the following inferences does the passage expect its readers to make?
Scientists rarely learn about hummingbirds by dissecting them.
If a hummingbird eats gnats, it will not eat honey.
Fly-catchers are a type of insect.
If a hummingbird consumes flower nectar, this nectar will turn into the honey that can be found in its stomach.
The author is the first scientist to ever have investigated what hummingbirds eat.
If a hummingbird consumes flower nectar, this nectar will turn into the honey that can be found in its stomach.
Explanation:
Let’s consider each of the answer choices to identify the correct one.
“The author is the first scientist to ever have investigated what hummingbirds eat.” - This cannot be true, because the author begins the passage by saying “The food of hummingbirds has been a matter of much controversy. All the early writers down to Buffon believed that they lived solely on the nectar of flowers, but since that time, every close observer of their habits maintains that they feed largely, and in some cases wholly, on insects.” He also cites numerous other scientists’ opinions throughout the passage, so he can’t be the first person to have investigated what hummingbirds eat.
“Fly-catchers are a type of insect.” - The passage mentions fly-catchers in the following sentence: “Many [hummingbirds] in fact may be seen catching gnats and other small insects just like fly-catchers, sitting on a dead twig over water, darting off for a time in the air, and then returning to the twig.” This is a tricky answer choice in that it’s easy to misread the sentence and think that “just like flycatchers” refers to “other small insects” when in fact it refers to the act of “catching.” The sentence is saying that hummingbirds catch insects in the same manner as fly-catchers, not that fly-catchers are a type of insect. Plus, we are being asked to identify an inference readers are expected to make, and if this sentence did mean that fly-catchers were insects, it would be overtly telling us this, and there would be nothing we’d have to infer.
“Scientists rarely learn about hummingbirds by dissecting them.” - This answer choice is proven wrong by the following sentence: “Hundreds and perhaps thousands of specimens have since been dissected by collecting naturalists, and in almost every instance their stomachs have been found full of insects, sometimes, but not generally, mixed with a proportion of honey.”
“If a hummingbird eats gnats, it will not eat honey.” - Given that the questions of whether hummingbirds eat insects or honey and in what proportions is the topic of the passage, it may be easy to choose this answer choice because it seems like the one closest to the passage’s main idea; however, nothing in the passage supports this assertion.
“If a hummingbird consumes flower nectar, this nectar will turn into the honey that can be found in its stomach.” - This is the correct answer! The author initially states that “All the early writers down to Buffon believed that [hummingbirds] lived solely on the nectar of flowers”; however, he later states that “Hundreds and perhaps thousands of specimens have since been dissected by collecting naturalists, and in almost every instance their stomachs have been found full of insects, sometimes, but not generally, mixed with a proportion of honey.” The author does not address the idea that flower nectar and honey could be different substances, and instead expects the reader to treat these as one source of food.
### Example Question #21 : Textual Relationships In Science Passages
Adapted from “Humming-Birds: As Illustrating the Luxuriance of Tropical Nature” in Tropical Nature, and Other Essays by Alfred Russel Wallace (1878)
The food of hummingbirds has been a matter of much controversy. All the early writers down to Buffon believed that they lived solely on the nectar of flowers, but since that time, every close observer of their habits maintains that they feed largely, and in some cases wholly, on insects. Azara observed them on the La Plata in winter taking insects out of the webs of spiders at a time and place where there were no flowers. Bullock, in Mexico, declares that he saw them catch small butterflies, and that he found many kinds of insects in their stomachs. Waterton made a similar statement. Hundreds and perhaps thousands of specimens have since been dissected by collecting naturalists, and in almost every instance their stomachs have been found full of insects, sometimes, but not generally, mixed with a proportion of honey. Many of them in fact may be seen catching gnats and other small insects just like fly-catchers, sitting on a dead twig over water, darting off for a time in the air, and then returning to the twig. Others come out just at dusk, and remain on the wing, now stationary, now darting about with the greatest rapidity, imitating in a limited space the evolutions of the goatsuckers, and evidently for the same end and purpose. Mr. Gosse also remarks, ” All the hummingbirds have more or less the habit, when in flight, of pausing in the air and throwing the body and tail into rapid and odd contortions. This is most observable in the Polytmus, from the effect that such motions have on the long feathers of the tail. That the object of these quick turns is the capture of insects, I am sure, having watched one thus engaged pretty close to me.”
Based on the way the term is used in passage, what is “the Polytmus”?
A type of carnivorous mammal that eats hummingbirds
A type of hummingbird with particularly bright coloring
A type of hummingbird with a long tail
A scientific term for a fledgling hummingbird that cannot yet fly
A species of flower that often attracts hummingbirds
A type of hummingbird with a long tail
Explanation:
Let’s look at the spot in the passage where “the Polytmus” is mentioned:
“Mr. Gosse also remarks, ‘All the hummingbirds have more or less the habit, when in flight, of pausing in the air and throwing the body and tail into rapid and odd contortions. This is most observable in the Polytmus, from the effect that such motions have on the long feathers of the tail.’”
From this context, we can tell that the Polytmus isn’t a carnivorous hummingbird-eating mammal, or a species of flower: it is a hummingbird. It is mentioned in the context of flying, so it can’t refer to a fledgling hummingbird that can’t yet fly. So, is it mentioning a type of hummingbird with particularly bright coloring, or one with a long tail? Mr. Gosse mentions the Polytmus in particular because observers can easily see it contort in midair “from the effect that such motions have on the long feathers of the tail.” So, the Polytmus must be “a type of hummingbird with a long tail.”
### Example Question #1 : Inferences And Predictions In Argumentative Science Passages
Adapted from “Introduced Species That Have Become Pests” in Our Vanishing Wild Life, Its Extermination and Protection by William Temple Hornaday (1913)
The man who successfully transplants or "introduces" into a new habitat any persistent species of living thing assumes a very grave responsibility. Every introduced species is doubtful gravel until panned out. The enormous losses that have been inflicted upon the world through the perpetuation of follies with wild vertebrates and insects would, if added together, be enough to purchase a principality. The most aggravating feature of these follies in transplantation is that never yet have they been made severely punishable. We are just as careless and easygoing on this point as we were about the government of the Yellowstone Park in the days when Howell and other poachers destroyed our first national bison herd, and when caught red-handed—as Howell was, skinning seven Park bison cows—could not be punished for it, because there was no penalty prescribed by any law. Today, there is a way in which any revengeful person could inflict enormous damage on the entire South, at no cost to himself, involve those states in enormous losses and the expenditure of vast sums of money, yet go absolutely unpunished!
The gypsy moth is a case in point. This winged calamity was imported at Maiden, Massachusetts, near Boston, by a French entomologist, Mr. Leopold Trouvelot, in 1868 or 69. History records the fact that the man of science did not purposely set free the pest. He was endeavoring with live specimens to find a moth that would produce a cocoon of commercial value to America, and a sudden gust of wind blew out of his study, through an open window, his living and breeding specimens of the gypsy moth. The moth itself is not bad to look at, but its larvae is a great, overgrown brute with an appetite like a hog. Immediately Mr. Trouvelot sought to recover his specimens, and when he failed to find them all, like a man of real honor, he notified the State authorities of the accident. Every effort was made to recover all the specimens, but enough escaped to produce progeny that soon became a scourge to the trees of Massachusetts. The method of the big, nasty-looking mottled-brown caterpillar was very simple. It devoured the entire foliage of every tree that grew in its sphere of influence.
The gypsy moth spread with alarming rapidity and persistence. In course of time, the state authorities of Massachusetts were forced to begin a relentless war upon it, by poisonous sprays and by fire. It was awful! Up to this date (1912) the New England states and the United States Government service have expended in fighting this pest about $7,680,000! The spread of this pest has been retarded, but the gypsy moth never will be wholly stamped out. Today it exists in Rhode Island, Connecticut, and New Hampshire, and it is due to reach New York at an early date. It is steadily spreading in three directions from Boston, its original point of departure, and when it strikes the State of New York, we, too, will begin to pay dearly for the Trouvelot experiment. Based on the first paragraph, the author would be most likely to support __________. Possible Answers: introducing damaging invasive species to the South granting Howell clemency for his actions keeping bison out of Yellowstone National Park an effort to catalogue the exact amount of money invasive species have cost the United States a law severely punishing those who introduce invasive species that damage the environment Correct answer: a law severely punishing those who introduce invasive species that damage the environment Explanation: One of the author’s main points in the first paragraph is that harsher legal repercussions are needed for those who release damaging invasive species into the United States. This is clear when the author writes, “The most aggravating feature of these follies in transplantation is that never yet have they been made severely punishable.” Thus, we can infer that the author would be most likely to support “a law severely punishing those who introduce invasive species that damage the environment.” Though the author does discuss the potential for someone to introduce invasive species to the South, he is not in favor of this, and he clearly doesn’t want to grant Howell clemency for his actions. (Furthermore, “clemency” somewhat implies that Howell has been charged with a crime, and the author explains that this isn’t the case.) The author does state, “The enormous losses that have been inflicted upon the world through the perpetuation of follies with wild vertebrates and insects would, if added together, be enough to purchase a principality,” and we can therefore assume that he might support cataloguing the amount of money invasive species have cost the United States. However, this inference requires a much larger logical leap than does the one that the author would support harsher legal punishments for those who introduce damaging invasive species, making “a law severely punishing those who introduce invasive species that damage the environment” the best answer. If you’re unsure when picking between answers to an inference question, it’s usually a good idea to see which one is more relevant to the passage’s topic and has the most evidence supporting it. ### Example Question #11 : Inference About The Author dapted from “Introduced Species That Have Become Pests” in Our Vanishing Wild Life, Its Extermination and Protection by William Temple Hornaday (1913) The man who successfully transplants or "introduces" into a new habitat any persistent species of living thing assumes a very grave responsibility. Every introduced species is doubtful gravel until panned out. The enormous losses that have been inflicted upon the world through the perpetuation of follies with wild vertebrates and insects would, if added together, be enough to purchase a principality. The most aggravating feature of these follies in transplantation is that never yet have they been made severely punishable. We are just as careless and easygoing on this point as we were about the government of the Yellowstone Park in the days when Howell and other poachers destroyed our first national bison herd, and when caught red-handed—as Howell was, skinning seven Park bison cows—could not be punished for it, because there was no penalty prescribed by any law. Today, there is a way in which any revengeful person could inflict enormous damage on the entire South, at no cost to himself, involve those states in enormous losses and the expenditure of vast sums of money, yet go absolutely unpunished! The gypsy moth is a case in point. This winged calamity was imported at Maiden, Massachusetts, near Boston, by a French entomologist, Mr. Leopold Trouvelot, in 1868 or 69. History records the fact that the man of science did not purposely set free the pest. He was endeavoring with live specimens to find a moth that would produce a cocoon of commercial value to America, and a sudden gust of wind blew out of his study, through an open window, his living and breeding specimens of the gypsy moth. The moth itself is not bad to look at, but its larvae is a great, overgrown brute with an appetite like a hog. Immediately Mr. Trouvelot sought to recover his specimens, and when he failed to find them all, like a man of real honor, he notified the State authorities of the accident. Every effort was made to recover all the specimens, but enough escaped to produce progeny that soon became a scourge to the trees of Massachusetts. The method of the big, nasty-looking mottled-brown caterpillar was very simple. It devoured the entire foliage of every tree that grew in its sphere of influence. The gypsy moth spread with alarming rapidity and persistence. In course of time, the state authorities of Massachusetts were forced to begin a relentless war upon it, by poisonous sprays and by fire. It was awful! Up to this date (1912) the New England states and the United States Government service have expended in fighting this pest about$7,680,000!
The spread of this pest has been retarded, but the gypsy moth never will be wholly stamped out. Today it exists in Rhode Island, Connecticut, and New Hampshire, and it is due to reach New York at an early date. It is steadily spreading in three directions from Boston, its original point of departure, and when it strikes the State of New York, we, too, will begin to pay dearly for the Trouvelot experiment.
If the author were to learn that the gypsy moth could be efficiently repelled from trees by coating them with a cheap, natural substance, he would likely feel __________.
pessimistic
horrified
exuberant
anxious
unsurprised
exuberant
Explanation:
Throughout the passage, the author makes it apparent that he feels that the gypsy moth is a very damaging invasive species that causes a lot of problems in the United States. He calls it a “winged calamity” and, in the third paragraph, describes how it spread:
“The gypsy moth spread with alarming rapidity and persistence. In course of time, the state authorities of Massachusetts were forced to begin a relentless war upon it, by poisonous sprays and by fire. It was awful! Up to this date (1912) the New England states and the United States Government service have expended in fighting this pest about \$7,680,000!”
From this paragraph, we can tell that if the author were to learn that the gypsy moth could be efficiently stopped from damaging trees, he would be most likely to feel “exuberant,” or excited and happy. Nothing in the passage supports any of the other answers.
### Example Question #21 : Reading Comprehension
Adapted from Volume Four of The Natural History of Animals: The Animal Life of the World in Its Various Aspects and Relations by James Richard Ainsworth Davis (1903)
The examples of protective resemblance so far quoted are mostly permanent adaptations to one particular sort of surrounding. There are, however, numerous animals which possess the power of adjusting their color more or less rapidly so as to harmonize with a changing environment.
Some of the best known of these cases are found among those mammals and birds that inhabit countries more or less covered with snow during a part of the year. A good instance is afforded by the Irish or variable hare, which is chiefly found in Ireland and Scotland. In summer, this looks very much like an ordinary hare, though rather grayer in tint and smaller in size, but in winter it becomes white with the exception of the black tips to the ears. Investigations that have been made on the closely allied American hare seem to show that the phenomenon is due to the growth of new hairs of white hue.
The common stoat is subject to similar color change in the northern parts of its range. In summer it is of a bright reddish brown color with the exception of the under parts, which are yellowish white, and the end of the tail, which is black. But in winter, the entire coat, save only the tip of the tail, becomes white, and in that condition the animal is known as an ermine. A similar example is afforded by the weasel. The seasonal change in the vegetarian Irish hare is purely of protective character, but in such an actively carnivorous creature as a stoat or weasel, it is aggressive as well, rendering the animal inconspicuous to its prey.
What can we infer preceded this paragraph?
Descriptions of animals that have not adapted to their environments
Descriptions of animals that defend themselves by looking like things in a changing environment
Descriptions of animals that hunt other animals efficiently by camouflaging themselves
Descriptions of animals that defend themselves by looking like things in a stable environment
Descriptions of changing environments
Descriptions of animals that defend themselves by looking like things in a stable environment
Explanation:
In order to infer what likely “preceded,” or came before, this passage, we should take at what the passage is talking about right when it starts. The passage’s first sentence says, “The examples of protective resemblance so far quoted are mostly permanent adaptations to one particular sort of surrounding.” The “so far quoted” means so far said or provided and tells us that the writer has been talking about “examples of protective resemblance.” This means that the writer most likely discussed “animals that defend themselves by looking like things in a stable environment” in the part of the book that comes right before the passage.
### Example Question #31 : Science Passages
Adapted from Volume Four of The Natural History of Animals: The Animal Life of the World in Its Various Aspects and Relations by James Richard Ainsworth Davis (1903)
The examples of protective resemblance so far quoted are mostly permanent adaptations to one particular sort of surrounding. There are, however, numerous animals which possess the power of adjusting their color more or less rapidly so as to harmonize with a changing environment.
Some of the best known of these cases are found among those mammals and birds that inhabit countries more or less covered with snow during a part of the year. A good instance is afforded by the Irish or variable hare, which is chiefly found in Ireland and Scotland. In summer, this looks very much like an ordinary hare, though rather grayer in tint and smaller in size, but in winter it becomes white with the exception of the black tips to the ears. Investigations that have been made on the closely allied American hare seem to show that the phenomenon is due to the growth of new hairs of white hue.
The common stoat is subject to similar color change in the northern parts of its range. In summer it is of a bright reddish brown color with the exception of the under parts, which are yellowish white, and the end of the tail, which is black. But in winter, the entire coat, save only the tip of the tail, becomes white, and in that condition the animal is known as an ermine. A similar example is afforded by the weasel. The seasonal change in the vegetarian Irish hare is purely of protective character, but in such an actively carnivorous creature as a stoat or weasel, it is aggressive as well, rendering the animal inconspicuous to its prey.
Based on the passage, what can we infer about the weasel?
Like the stoat, it also lives in burrows.
Like the Irish hare, has been the subject of investigations.
Like the stoat, it also changes its coat color.
Like the stoat, it has claws.
Like the Irish hare, it has grey fur in the summer.
Like the stoat, it also changes its coat color.
Explanation:
The weasel is mentioned in two places in the passage, both in the passage’s last paragraph, both reproduced here:
“But in winter, the entire coat [of the stoat], save only the tip of the tail, becomes white, and in that condition the animal is known as an ermine. A similar example is afforded by the
weasel. The seasonal change in the vegetarian Irish hare is purely of protective character, but in such an actively carnivorous creature as a stoat or weasel, it is aggressive as well, rendering the animal inconspicuous to its prey.”
What does the passage tell us about the weasel? Well, we can infer that it is in some way like the stoat, because the passage says “A similar example is afforded by the weasel” right after describing how the stoat’s fur changes color. We are also told that it is carnivorous, but this is not an inference we have to make, and it doesn’t relate to any of the answer choices. The best answer choice is “Like the stoat, it also changes its coat color.” This captures the specific similarity between the stoat and weasel being discussed when the author writes, “A similar example is afforded by the weasel.”
### Example Question #1 : Making Inferences And Predictions In Science Passages
Adapted from Cassell’s Natural History by Francis Martin Duncan (1913)
The penguins are a group of birds inhabiting the southern ocean, for the most part passing their lives in the icy waters of the Antarctic seas. Like the ratitae, penguins have lost the power of flight, but the wings are modified into swimming organs and the birds lead an aquatic existence and are scarcely seen on land except in the breeding season. They are curious-looking creatures that appear to have no legs, as the limbs are encased in the skin of the body and the large flat feet are set so far back that the birds waddle along on land in an upright position in a very ridiculous manner, carrying their long narrow flippers held out as if they were arms. When swimming, penguins use their wings as paddles while the feet are used for steering.
Penguins are usually gregarious—in the sea, they swim together in schools, and on land, assemble in great numbers in their rookeries. They are very methodical in their ways, and on leaving the water, the birds always follow well-defined tracks leading to the rookeries, marching with much solemnity one behind the other in soldierly order.
The largest species of penguins are the king penguin and the emperor penguin, the former being found in Kerguelen Land, the Falklands, and other southern islands, and the latter in Victoria Land and on the pack ice of the Antarctic seas. As they are unaccustomed from the isolation of their haunts to being hunted and persecuted by man, emperor penguins are remarkably fearless, and Antarctic explorers invading their territory have found themselves objects of curiosity rather than fear to the strange birds who followed them about as if they were much astonished at their appearance.
The emperor penguin lays but a single egg and breeds during the intense cold and darkness of the Antarctic winter. To prevent contact with the frozen snow, the bird places its egg upon its flat webbed feet and crouches down upon it so that it is well covered with the feathers. In spite of this precaution, many eggs do not hatch and the mortality amongst the young chicks is very great.
We can infer from the passage that penguins probably eat __________ because the passage tells us that __________.
polar bears . . . penguins are unafraid of explorers
kelp and other sea plants . . . penguins are slow in the water
fish . . . penguins live most of their lives in the water
microscopic sea creatures . . . the emperor penguin only lays one egg at a time
seals . . . penguins are carnivorous and hunt in groups
fish . . . penguins live most of their lives in the water
Explanation:
We can narrow down our answer choices by considering which of the options for the second blank the passage actually tells us. This allows us to eliminate the answer choices containing “penguins are carnivorous and hunt in groups” and “penguins are slow in the water.” Deciding between the remaining three answer choices, it doesn’t make sense that penguins would eat “polar bears” because they “are unafraid of explorers”—the second part of the answer may suggest that penguins are bold, and from this we might infer that penguins might try to fight and eat a polar bear, but it’s unlikely that would end well; the reverse is true. It is also doesn’t make sense that penguins would eat “microscopic sea creatures” because “the emperor penguin only lays one egg at a time”—these ideas are unrelated. This leaves us with the correct answer, the idea that penguins probably eat “fish” because “penguins live most of their lives in the water.” This makes sense; if penguins spend most of their lives in the water, they probably eat something that is found in the water, and fish are found in the water.
### Example Question #72 : Narrative Science Passages
Adapted from Cassell’s Natural History by Francis Martin Duncan (1913)
The penguins are a group of birds inhabiting the southern ocean, for the most part passing their lives in the icy waters of the Antarctic seas. Like the ratitae, penguins have lost the power of flight, but the wings are modified into swimming organs and the birds lead an aquatic existence and are scarcely seen on land except in the breeding season. They are curious-looking creatures that appear to have no legs, as the limbs are encased in the skin of the body and the large flat feet are set so far back that the birds waddle along on land in an upright position in a very ridiculous manner, carrying their long narrow flippers held out as if they were arms. When swimming, penguins use their wings as paddles while the feet are used for steering.
Penguins are usually gregarious—in the sea, they swim together in schools, and on land, assemble in great numbers in their rookeries. They are very methodical in their ways, and on leaving the water, the birds always follow well-defined tracks leading to the rookeries, marching with much solemnity one behind the other in soldierly order.
The largest species of penguins are the king penguin and the emperor penguin, the former being found in Kerguelen Land, the Falklands, and other southern islands, and the latter in Victoria Land and on the pack ice of the Antarctic seas. As they are unaccustomed from the isolation of their haunts to being hunted and persecuted by man, emperor penguins are remarkably fearless, and Antarctic explorers invading their territory have found themselves objects of curiosity rather than fear to the strange birds who followed them about as if they were much astonished at their appearance.
The emperor penguin lays but a single egg and breeds during the intense cold and darkness of the Antarctic winter. To prevent contact with the frozen snow, the bird places its egg upon its flat webbed feet and crouches down upon it so that it is well covered with the feathers. In spite of this precaution, many eggs do not hatch and the mortality amongst the young chicks is very great.
What can you infer is “the ratitae," underlined in the first paragraph?
A type of flightless bird
A bird that lives in Antarctica
A type of fish
A bone found in most birds’ skeletons
A type of flightless bird
Explanation:
The ratitae is mentioned in the paragraph’s first passage, when the author writes, “Like the ratitae, penguins have lost the power of flight, but the wings are modified into swimming organs and the birds lead an aquatic existence and are scarcely seen on land except in the breeding season.” In analyzing this, it’s important to ask: to what part of the sentence is “Like the ratitae” specifically referring? It is referring to the fact that “penguins have lost the power of flight.” Thus, the comparison being made here is that both the ratitae and the penguins “have lost the power of flight.” Therefore, we can infer that “the ratitae” is “a type of flightless bird.”
You may have picked out “a bird that lives in Antarctica” based on the information that precedes the comparison or “a bird that swims” based on the information that follows it. However, neither of these comparisons are being made in the passage: it is specifically the fact that both the ratitae and penguins “have lost the power of flight” that is being stated. Nothing in the passage suggests that “the ratitae” is “a bone found in most birds’ skeletons” or “a type of fish.”
### Example Question #101 : Content Of Natural Science Passages
Adapted from “Feathers of Sea Birds and Wild Fowl for Bedding” from The Utility of Birds by Edward Forbush (ed. 1922)
In the colder countries of the world, the feathers and down of waterfowl have been in great demand for centuries as filling for beds and pillows. Such feathers are perfect non-conductors of heat, and beds, pillows, or coverlets filled with them represent the acme of comfort and durability. The early settlers of New England saved for such purposes the feathers and down from the thousands of wild-fowl which they killed, but as the population increased in numbers, the quantity thus furnished was insufficient, and the people sought a larger supply in the vast colonies of ducks and geese along the Labrador coast.
The manner in which the feathers and down were obtained, unlike the method practiced in Iceland, did not tend to conserve and protect the source of supply. In Iceland, the people have continued to receive for many years a considerable income by collecting eider down, but there they do not “kill the goose that lays the golden eggs.” Ducks line their nests with down plucked from their own breasts and that of the eider is particularly valuable for bedding. In Iceland, these birds are so carefully protected that they have become as tame and unsuspicious as domestic fowls In North America. Where they are constantly hunted they often conceal their nests in the midst of weeds or bushes, but in Iceland, they make their nests and deposit their eggs in holes dug for them in the sod. A supply of the ducks is maintained so that the people derive from them an annual income.
In North America, quite a different policy was pursued. The demand for feathers became so great in the New England colonies about the middle of the eighteenth century that vessels were fitted out there for the coast of Labrador for the express purpose of securing the feathers and down of wild fowl. Eider down having become valuable and these ducks being in the habit of congregating by thousands on barren islands of the Labrador coast, the birds became the victims of the ships’ crews. As the ducks molt all their primary feathers at once in July or August and are then quite incapable of flight and the young birds are unable to fly until well grown, the hunters were able to surround the helpless birds, drive them together, and kill them with clubs. Otis says that millions of wildfowl were thus destroyed and that in a few years their haunts were so broken up by this wholesale slaughter and their numbers were so diminished that feather voyages became unprofitable and were given up.
This practice, followed by the almost continual egging, clubbing, shooting, etc. by Labrador fishermen, may have been a chief factor in the extinction of the Labrador duck, that species of supposed restricted breeding range. No doubt had the eider duck been restricted in its breeding range to the islands of Labrador, it also would have been exterminated long ago.
Which of the following would you LEAST expect to be discussed elsewhere in the book from which this passage was taken?
The raising of chickens for their eggs
The practice of sending messages by carrier pigeon
The use of tropical birds’ feathers as hat decorations
Falconry
The types of birds encountered by the first Antarctic explorers
The types of birds encountered by the first Antarctic explorers
Explanation:
The passage describes how humans use the eider down produced by eider ducks as a commodity for its insulating properties. Given this focus, along with the title of the book from which the passage is taken, The Utility of Birds, we can assume that other topics discussed in the books would deal with ways in which birds are useful to humans. “The use of tropical birds’ feathers as hat decorations,” “the raising of chickens for their eggs,” “falconry,” and “the practice of sending messages by carrier pigeon” all deal with ways in which birds are useful to humans, but “The types of birds encountered by the first Antarctic explorers” does not relate to how birds are useful to humans, so it would be least likely to be discussed elsewhere in a book called The Utility of Birds and is the correct answer.
### Example Question #101 : Content Of Natural Science Passages
Adapted from “Feathers of Sea Birds and Wild Fowl for Bedding” from The Utility of Birds by Edward Forbush (ed. 1922)
In the colder countries of the world, the feathers and down of waterfowl have been in great demand for centuries as filling for beds and pillows. Such feathers are perfect non-conductors of heat, and beds, pillows, or coverlets filled with them represent the acme of comfort and durability. The early settlers of New England saved for such purposes the feathers and down from the thousands of wild-fowl which they killed, but as the population increased in numbers, the quantity thus furnished was insufficient, and the people sought a larger supply in the vast colonies of ducks and geese along the Labrador coast.
The manner in which the feathers and down were obtained, unlike the method practiced in Iceland, did not tend to conserve and protect the source of supply. In Iceland, the people have continued to receive for many years a considerable income by collecting eider down, but there they do not “kill the goose that lays the golden eggs.” Ducks line their nests with down plucked from their own breasts and that of the eider is particularly valuable for bedding. In Iceland, these birds are so carefully protected that they have become as tame and unsuspicious as domestic fowls In North America. Where they are constantly hunted they often conceal their nests in the midst of weeds or bushes, but in Iceland, they make their nests and deposit their eggs in holes dug for them in the sod. A supply of the ducks is maintained so that the people derive from them an annual income.
In North America, quite a different policy was pursued. The demand for feathers became so great in the New England colonies about the middle of the eighteenth century that vessels were fitted out there for the coast of Labrador for the express purpose of securing the feathers and down of wild fowl. Eider down having become valuable and these ducks being in the habit of congregating by thousands on barren islands of the Labrador coast, the birds became the victims of the ships’ crews. As the ducks molt all their primary feathers at once in July or August and are then quite incapable of flight and the young birds are unable to fly until well grown, the hunters were able to surround the helpless birds, drive them together, and kill them with clubs. Otis says that millions of wildfowl were thus destroyed and that in a few years their haunts were so broken up by this wholesale slaughter and their numbers were so diminished that feather voyages became unprofitable and were given up.
This practice, followed by the almost continual egging, clubbing, shooting, etc. by Labrador fishermen, may have been a chief factor in the extinction of the Labrador duck, that species of supposed restricted breeding range. No doubt had the eider duck been restricted in its breeding range to the islands of Labrador, it also would have been exterminated long ago.
Which of the following most likely happened after the Labrador feather voyages were no longer organized?
A population of the Labrador duck was reestablished.
Eider down began to be used for other purposes in North America.
The quality of bedding in North America became preferable to that found in Iceland.
North Americans imported eider down from Iceland.
The price of eider down in North America plummeted.
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# User:LuckyM11
Hello I´m a pupil of the gisela gym, munich I have mr Oswald as teacher in physics he forced me to write this i´m 16 years old and I love physics ^^
a)
x(t) = x0 + v * $\delta {t}$
Bussard:x(t) = 14 km + 95 $\frac {km}{h}$ * $\delta {t}$
Falke:x(t) = 0km + 153 $\frac {km}{h}$ * $\delta {t}$
Gleichgesetzt:14 km + 95 $\frac {km}{h}$ * $\delta {t}$ =$153 \frac {km}{h}$ * $\delta {t}$
$\delta {t}$ = 0,24 h
Gerundet:$\delta {t}$ = 14 min
Eingesetzt:x(t) = $153\frac {km}{h}$ * $0,24h$
x(t) = $36,72 km$ Gerundet:$x(t)=37 km$
2.9)
a)
t = $\frac {v}{a}$
t = 2,1 * 10-4
b)
x(t) = $\frac{1}{2}$ * a * t 2
t= $\frac {v}{a}$
x(t)=$\frac {1}{2}$ $\frac{v^2}{a}$
a= $\frac {v^2}{2*x(t)}$
a=3,6 *10 4 $\frac {m}{s^2}$
3.20 / 4.1
a)
v²=2ax
$\frac {v^2}{2x}$ = $\frac {192.90 \frac {m^2}{s^2}}{400m}$=0.48$\frac {m}{s^2}$
b) Fmotor=FHangabtrieb+FReibung+FBeschleunigung
FR = µ * FG = 0.15 * (35000Kg * 9.81 $\frac{N}{Kg}$) = 51.5 KN
α=tan -1=5.71°
FH=FG* sin α = 343350n * 0.099 =51.5kN
FB=$m * a = 35000kG * 0.48 \frac{m}{s^2}$= 102.2 kN
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1. Aug 25, 2004
in a maths book i have there is an example of solving a quadratic. part of the process reads:
y^2 - 12y + 32 = 0
implies that
(y-8) (y-4) = 0
i don't understand how this second equation was reached based on the first one.
could someone add in the steps that gets me from the first equation to the implication.
thanks.
2. Aug 25, 2004
### jcsd
expand:
(y + a)(y + b)
and you get:
y^2 + (a+b)y + ab
so from your equation above you know that:
a + b = -12
and
ab = 32
as
-8 + -4 = -12
and
-8*-4 = 32
a and b must be -8 and -4.
3. Aug 25, 2004
### xt
i don't think you can, it is obvious.
if you actually want the logics of thinking its like this:
we want to express the quadratic as (y-a)(y-b),
now a*b = 32
and a + b = 12
so try all the integers you can think of, you'll find 8 and 4 does.
4. Aug 25, 2004
### arildno
A simple way uses the idea of "completing the square:
$$y^{2}-12y+32=0$$
Now, regarding -12=2*(-6), we add 0 to our equation in this manner:
$$y^{2}-12y+32+(-6)^{2}-(-6)^{2}=0$$
Or, equivalently, for the Left Hand Side:
$$y^{2}-12y+32+(-6)^{2}-(-6)^{2}=(y^{2}-2*6y+6^{2})+(32-6^{2})=(y-6)^{2}-4$$
Furthermore, since $$4=2^{2}$$ we have:
$$(y-6)^{2}-4=(y-6)^{2}-2^{2}=(y-6+2)(y-6-2)=(y-4)(y-8)$$
Finally, by setting this expression (which is equivalent to our original left hand side) equal to our originil right hand side (that is,0) we gain:
$$(y-4)(y-8)=0$$
as required.
5. Aug 25, 2004
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# Review of 2017
Year 2018 comes soon, at the tail of 2017, I would like to review the whole year,sum up both professional gains and self-learning gains. This year is a turningpoint in my life: I finished my study ...
Year 2018 comes soon, at the tail of 2017, I would like to review the whole year, sum up both professional gains and self-learning gains. This year is a turning point in my life: I finished my study in school and entered the workplace. Thanks to the rich education of Toulouse School of Economics and the trust of my lead, I did my modest part to contribute franprix as a Data Scientist.
In franprix, I play a role as a “full stack” data scientist, which means when I put my hands to a new project, I do ETL(Extract-Transform-Load), data cleaning, data visualisation, create model, apply algorithm, analyse and give a presentation to requesters all by myself. Not easy for a junior, but I really enjoy it and learnt lots of new things in technology, I know how to communicate with my colleagues and organise time when I have a new project. In the following, I’ll go into detail.
## Technology
### Projects at franprix
In this year, I studied many interesting subjects. In order to find the impact of weather on sales, I created a linear regression, which can specify the impact of weather on each subfamily.
In order to help merchandise team to organise the shelves, I used apriori algorithm analysed consumers’ preference, determined the shelves that often sold together, and found the association between each two shelves.
Furthermore, we also helped HR Department. As enterprise growing, there are more and more shops spread all over France, so are its employees. Some employees take such a long duration for working and going home. To facilitate the way between one’s home and workplace(shop), CEO proposed that 2 employees can exchange their workplaces, if it saves time for both. This project is for finding the optimal solution for all employees. We combined R and Google Maps API, created a tool for visualising and simplifying the employee-switching. Voilà, like this graph, we can enter an employee ID and click “choice 1”, then his domicile and workplace position will display on the map, the best switch for him as well; it will also display the distance and duration for arriving at the new workplace.
Here I just show you the most interesting projects of this year, we also achieved many other projects like automating creation of report and sending e-mail by Talend, visualising data by Microstrategy, analysing stores’ performance, analysing tourist-area stores, etc.
### Self-learning
As a data scientist, we should never stop learning new things(languages, technologies, algorithms). In this year, I participated Microstrategy’s formation and learnt how to make a clear and simple visual report with its tools, and continue to participate online courses. Thanks again to datacamp, I started to learn basic knowledge of Python. I also learnt linear regression, ridge regression, lasso and logistic regression in books “Introduction to Machine Learning with Python” and “Python Machine Learning Cookbook”.
Besides python, I learnt a new framework, git, on youtube. Git is a version control system for tracking changes in computer files and coordinating work on those files among multiple people. It is primarily used for source code management in software development, but it can be used to keep track of changes in any set of files[1]. Thanks to the videos, I learnt that merge means from the chosen branch switch to the current HEAD branch, when there are merge conflicts, you can UNDO a merge(abort) and start over; rebase is like an alternative to merge, we should use rebase only locally on unpublished commits, and never rebase commits that have already been pushed to a remote repository. If you’re interested for the details, you will find more things on youtube and Git documentation.
Moreover, I keep learning R this year, not only on datacamp, but also reading a book named “R IN ACTION”. By reading it, I strengthened basic knowledge of R, like data structures, simple visualisation, also enrich my knowledge reserve of machine learning. I learnt more about OLS regression models, using regression diagnostics to assess the data’s fit to statistical assumptions, and methods for modifying the data to meet these assumptions more closely; I looked at how to create time series in R, assess trends, examine seasonal effects, and considered two of the most popular approaches to forecasting: exponential models and ARIMA models; I understand the logistic regression is a type of generalized linear model to predict a binary outcome from a set of numeric variables; I know that decision trees involve creating a set of binary splits on the predictor variables in order to created a tree that can be used to classify new observations into one of two groups. All these algorithms helped our team on different projects.
Apart from participating online courses and reading books, I also joined technical meetups and exchanged ideas and technologies with others.
## Communication
Besides technical progress, I got some tips on communicating with my colleagues. During our first discussion on a new project, I used to asking their objective, the problems they met, project’s deadline and all elements for the project. Then thinking about its feasibility, possible duration and give requesters the first response on project planning, so that we can assure the deadline according to our ability.
## Time planning
Since there are lots of projects to analyse, we need to well organise our time for being efficient. Everyday before leaving for home, I write a todo list for tomorrow on my agenda, it can remind me the tasks immediately when I arrive at my office the next day. How to make myself to be efficient? Besides listing tasks on my agenda, I also organise the duration for each task. For the latter, I follow Pomodoro Technique with a tool named Pomotodo, I assign time-consuming of each task to several pomodoro, and keep focused during each pomodoro. It works pretty well for me.
## Self-examination
Sometimes we can’t avoid taking a roundabout course in our work. Under these circumstances, I always asked myself the reason, why I misguided? What leads it? What should I notice in the following projects? Persisting in so doing, this kind of self-examination helps me to go forward faster.
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# Question on finding work done on charge
Tags:
1. Jun 8, 2016
### RoboNerd
1. The problem statement, all variables and given/known data
2. Relevant equations
coulomb's law
3. The attempt at a solution
Hi everyone. I understand their approach with the integration to find the amount of work that "a person" would have to do to bring the charge q3 from infinity to its current position.
I understand that the force that this person would have to do is the opposite of the coulombic forces between the q3 and q1 and 2.
My question: why did they not account for the y-components of the coulombic forces between q1 and 2 to a3? The x-components of the two forces would cancel, so the only coulombic forces acting on q3 would be the y-components.
Thus I did exactly what they did, but I added a sine term to account for the y-component and then re-wrote the sine in terms of the vertical distance and "s/2" with pythagorean theorem and then did the integration.
Could anyone please weight in on why their approach is legit and they did not do what I planned to do with the sine term?
Thank you very much in advance.
2. Jun 9, 2016
### ehild
If you correctly do the integration of the y component of the electric force you get the same result.
The solution shown uses the potential to get the potential energy of q3, which is q3U. The potential of a point charge is kq/r where r is the distance from the charge. The potential function is scalar and additive, so the net potential at a point is the sum of the potentials from all charges.
3. Jun 9, 2016
### Delta²
I think what is needed to be emphasized is that the work is independent of the path that the q3 follows as is brought from the infinite to the desired place, it depends only on the end points of the path (one point at infinite and one point at the third vertex of the triangle). This holds only if the field is static so that $\nabla \times E=0$, $E=\nabla\phi$ where $\phi$ is the electrostatic potential (scalar).
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## 2010/05/09
### Microsoft Troubles - IX, the story unfolds with Apple closing in on Microsoft size.
Three pieces in the trade press showing how things are unfolding.
Om Malik points out that Intel and Microsoft fortunes are closely intertwined.
Jean-Louis Gassée suggests that "Personal Computing" (on those pesky Personal Computers) is downsizing and changing.
Joe Wilcox analyses Microsoft latest results and contrasts a little with Apple.
## 2010/05/03
### Everything Old is New Again: Cray's CPU design
I found myself writing, during a commentary on the evolution of SSD's in servers, that large-slow-memory like Seymour Cray used (not cache), would affect the design of Operating Systems. The new scheduling paradigm:
Allocate a thread to a core, let it run until it finishes and waits for (network) input, or it needs to read/write to the network.
This leads into how Seymour Cray dealt with Multi-Processing, he used multi-level CPU's:
• There were Application processors, many bits, many complex features like Floating Point and other fancy stuff, but had no kernel mode features or access to protected regions of hardware or memory, and
• Peripheral Processors (PP's), really a single very simple, very high-speed processor, multiplexed to look like 10 small, slower processors that performed all kernel functions and controlled the operation of the Application Processors (AP's)
Not only did this organisation result in very fast systems (Cray's designs were the fastest in the world for around 2 decades), but very robust and secure ones as well: the NSA and other TLA's used them extensively.
The common received wisdom is that interrupt-handling is the definitive way to interface unpredictable hardware events with the O/S and rest of the system. That polling devices, the old-way, is inefficient and expensive.
Creating a fixed overhead scheme is more expensive in compute cycles than an on-demand, or queuing, system, until the utilisation rate is very high. Then the cost of all the flexibility (or Variety in W. Ross Ashby's Cybernetics term) comes home to roost.
Piers Lauder of Sydney University and Bell Labs improved total system throughput of a VAX-11/780 running Unix V8 under continuous full (student/teaching) load by 30% by changing the serial-line device driver from 'interrupt handling' to polling.
All those expensive context-switches went away, to be replaced by a predictable, fixed overhead.
Yes, when the system was idle or low-load, it spent a little more time polling, but marginal.
And if the system isn't flat-out, what's the meaning of an efficiency metric?
Dr Neil J Gunther has written about this effect extensively with his Universal Scaling Law and other articles showing the equivalence of the seemingly disparate approaches of Vector Processing and SMP systems in the limit of their performance.
My comment about big, slow memory changing Operating System scheduling can be combined with the Cray PP/AP organisation.
In the modern world of CMOS, micro-electronics and multi-core chips, we are still facing the same Engineering problem Seymour Cray was attempting to address/find an optimal solution to:
For a given technology, how do you balance maximum performance with the Power/Heat Wall?
More power gives you more speed, this creates more Heat, which results in self-destruction, the "Halt and Catch Fire" problem. Silicon junctions/transistors are subject to thermal run-away, as they get hotter, they consume more power and get hotter still. At some point that becomes a viscous cycle (positive feedback loop) and its game over. Good chip/system designs balance on just the right side of this knife edge.
How could the Cray PP/AP organisation be applied to current multi-core chip designs?
1. Separate the CPU designs for kernel-mode and Application Processors.
A single chip needs only have a single kernel-mode CPU controlling a number of Application CPU's. With its constant overhead cost already "paid for", scaling of Application performance is going to be very close to linear right up until the limit.
2. Application CPU's don't have forced context switches. They roar along as fast as they can for as long as they can, or the kernel scheduler decides they've had their fair share.
3. System Performance and Security both improve by using different instruction sets and processor architectures for different applications. While a virus/malware might be able to compromise an Application, it can't migrate into the kernel unless it's buggy. The Security Boundary and Partitioning Model is very strong.
4. There doesn't have to be competition between the kernel-mode CPU and the AP's for cache memory 'lines'. In fact, the same memory cell designs/organisations used for L1/L2 cache can be provided as small (1-2MB) amounts of very fast direct access memory. The modern equivalent of "all register" memory.
5. Because the kernel-mode CPU and AP's don't contend for cache lines, each will benefit hugely in raw performance.
Another, more subtle, benefit is the kernel can avoid both the 'snoopy cache' (shared between all CPU's) and VM systems. It means a much simpler, much faster and smaller (= cooler) design.
6. The instruction set for the kernel-mode CPU will be optimised for speed, simplicity and minimal transistor count. You can forget about speculative execution and other really heavy-weight solutions necessary in the AP world.
7. The AP instruction set must be fixed and well-know, while the kernel-mode CPU instruction set can be tweaked or entirely changed for each hardware/fabrication iteration. The kernel-mode CPU runs what we'd now call either a hypervisor or a micro-kernel. Very small, very fast and with just enough capability. A side effect is that the chip manufacturers can do what they do best - fiddle with the internals - and provide a standard hypervisor for other O/S vendors to build upon.
Cheaper, Faster, Cooler, more robust and Secure and able to scale better.
What's not to like in this organisation?
### A Good Question: When will Computer Design 'stabilise'?
The other night I was talking to my non-Geek friend about computers and he formulated what I thought was A Good Question:
When will they stop changing??
This was in reaction to me talking about my experience in suggesting a Network Appliance, a high-end Enterprise Storage device, as shared storage for a website used by a small research group.
It comes with a 5 year warranty, which leads to the obvious question:
will it be useful, relevant or 'what we usually do' in 5 years?
I think most of the elements in current systems are here to stay, at least for the evolution of Silicon/Magnetic recording. We are staring at 'the final countdown', i.e. hitting physical limits of these technologies, not necessarily their design limits. Engineers can be very clever.
The server market has already fractioned into "budget", "value" and "premium" species.
The desktop/laptop market continues to redefine itself - and more 'other' devices arise. The 100M+ iPhones, in particular, already out there demonstrate this.
There's a new major step in server evolution just breaking:
Flash memory for large-volume working and/or persistent storage.
What now may be called internal or local disk.
This implies a major re-organisation of even low-end server installations:
Fast local storage and large slow network storage - shared and reliable.
When the working set of Application data in databases and/or files will fit on (affordable) local flash memory, response times improve dramatically because all that latency is removed. By definition, data outside the working set isn't a rate limiting step, so its latency only slightly affects system response time. However, throughput, the other side of the Performance Coin, has to match or beat that of the local storage, or it will become the system bottleneck.
An interesting side question:
How will Near-Zero-Latency local storage impact system 'performance', both response times (a.k.a. latency) and throughput.
I conjecture that both system latency and throughput will improve markedly, possibly super-linearly, because one of the bug-bears of Operating Systems, the context switch, will be removed. Systems have to expend significant effort/overhead in 'saving their place', deciding what to do next, then when the data is finally ready/available, to stop what they were doing and start again where they left off.
The new processing model, especially for multi-core CPU's, will be:
Allocate a thread to a core, let it run until it finishes and waits for (network) input, or it needs to read/write to the network.
Near zero-latency storage removes the need for complex scheduling algorithms and associated queuing. It improves both latency and throughput by removing a bottleneck.
It would seem that Operating Systems might benefit from significant redesign to exploit this effect, in much the same way that RAM is now large and cheap enough that system 'swap space' is now either an anachronism or unused.
The evolution of USB flash drives saw prices/Gb halving every year. I've recently seen 4Gb SDHC cards at the supermarket for ~$15, whereas in 2008, I paid ~$60 for USB 4Gb.
Rough server pricing for RAM in 2010 is A$65/Gb ±$15.
List prices by Tier 1/2 vendors for 64Gb SSD is $750-$1000 (around 2-4 times cheaper from 'white box' suppliers).
I've seen this firmware limited to 50Gb to improve performance and reliability comparable to current production HDD specs.
This is $12-$20/Gb, depending on what base size and prices used.
Disk drives are ~A$125 for 7200rpm SATA and$275-$450 for 15K SAS drives. With 2.5" drives priced in-between. Ie.$0.125/Gb for 'big slow' disks and $1 per GB for fast SAS disks. Roll forward 5 years to 2015 and 'SSD' might've doubled in size three times, plus seen the unit price drop. Hard disks will likely follow the same trend of 2-3 doublings. Say SSD 400Gb for$300: $1.25/Gb 2.5" drives might be up to 2-4Tb in 2015 (from 500Gb in 2010) and cost$200: $0.05-0.10/Gb RAM might be down to$15-$30/Gb. A caveat with disk storage pricing: 10 years ago RAID 5 became necessary for production servers to avoid permanent data loss. We've now passed another event horizon: Dual-parity, as a minimum, is required on production RAID sets. On production servers, price of storage has to factor in the multiple overheads of building high-reliability storage (redundant {disks, controllers, connections}, parity and hot-swap disks and even fully mirrored RAID volumes plus software, licenses and their Operations, Admin and Maintenance) from unreliable parts. A problem solved by electronics engineers 50+ years ago with N+1 redundancy. Multiple Parity is now needed because in the time taken to recreate a failed drive, there's a significant chance of a second drive failure and total data loss. [Something NetApp has been pointing out and addressing for some years.] The reason for this is simple: the time to read/write a whole drive has steadily increased since ~1980. Recording density (bits per inch) times areal density (tracks per inch) have increased faster than read/write speeds, roughly multiplying recording density times rotational speed. Which makes running triple-mirrors a much easier entry point, or some bright spark has to invent a cheap-and-cheerful N-way data replication system. Like a general use Google File System. Another issue is that current SSD offerings don't impress me. They make great local disk or non-volatile buffers in storage array, but are not yet, in my opinion, quite ready for 'prime time'. I'd like to see 2 things changed: • RAID-3 organisation with field-replaceable mini-drives. hot-swap preferred. • PCI, not SAS or SATA connection. I.e. they appear as directly addressable memory. This way the hardware can access flash as large, slow memory and the Operating System can fabricate that into a filesystem if it chooses - plus if it has some knowledge of the on-chip flash memory controller, it can work much better with it. It saves multiple sets of interfaces and protocol conversions. Direct access flash memory will be always be cheaper and faster than SATA or SAS pseudo-drives. We would then see following hierarchy of memory in servers: • Internal to server • L1/2/3 cache on-chip • RAM • Flash persistent storage • optional local disk (RAID-dual parity or triple mirrored) • External and site-local • network connected storage array, optimised for size, reliability, streaming IO rate and price not IO/sec. Hot swap disks and in-place/live expansion with extra controllers or shelves are taken as a given. • network connected near-line archival storage (MAID - Massive Array of Idle Disks) • External and off-site • off-site snapshots, backups and archives. Which implies a new type of business similar to Amazon's Storage Cloud. The local network/LAN is going to be ethernet (1Gbps or 10Gbps Ethernet, a.k.a 10GE), or Infiniband if 10GE remains very expensive. Infiniband delivers 3-6Gbps over short distances on copper, external SAS currently uses the "multi-lane" connector to deliver four channels per cable. This is exactly right for use in a single rack. I can't see a role for Fibre Channel outside storage arrays, and these will go if Infiniband speed and pricing continues to drop. Storage Arrays have used SCSI/SAS drives with internal copper wiring and external Fibre interfaces for a decade or more. Already the premium network vendors, like CISCO, are selling "Fibre Channel over Ethernet" switches (FCoE using 10GE). Nary a tape to be seen. (Hooray!) Servers should tend to be 1RU either full-width or half-width, though there will still be 3-4 styles of servers: • budget: mostly 1-chip • value: 1 and 2-chip systems • lower power value systems: 65W/CPU-chip, not 80-90W. • premium SMP: fast CPU's, large RAM and many CPU's (90-130W ea) If you want removable backups, stick 3+ drives in a RAID enclosure and choose between USB, firewire/IEEE 1394, e-SATA or SAS. Being normally powered down, you'd expect extended lifetimes for disks and electronics. But they'll need regular (3-6-12 months) read/check/rewrite cycling or the data will degrade and be permanently lost. Random 'bit-flipping' due to thermal activity, cosmic rays/particles and stray magnetic fields is the price we pay for very high density on magnetic media. Which is easy to do if they are kept in a remote access device, not unlike "tape robots" of old. Keeping archival storage "on a shelf" implies manual processes for data checking/refresh, and that is problematic to say the least. 3-5 2.5" drives will make a nice 'brick' for these removable backup packs. Hopefully commodity vendors like Vantec will start selling multiple-interface RAID devices in the near future. Using current commodity interfaces should ensure they are readable at least a decade into the future. I'm not a fan of hardware RAID controllers in this application because if it breaks, you need to find a replacement - which may be impossible at a future date. (fails 'single point of failure' test). Which presents another question using a software RAID and filesystem layout: Will it still be available in your O/S of the future? You're keeping copies of your applications, O/S, licences and hardware to recover/access archived data, aren't you? So this won't be a question... If you don't intend to keep the environment and infrastructure necessary to access archived data, you need to rethink what you're doing. These enclosures won't be expensive, but shan't be cheap and cheerful: Just what is your data worth to you? If it has little value, then why are you spending money on keeping it? If it is a valuable asset, potentially irreplaceable, then you must be prepared to pay for its upkeep in time, space and dollars. Just like packing old files into archive boxes and shipping them to a safe off-site facility cost money, it isn't over once they are out of your sight. Electronic storage is mostly cheaper than paper, but it isn't free and comes with its own limits and problems. Summary: • SSD's are best suited and positioned as local or internal 'disks', not in storage arrays. • Flash memory is better presented to an Operating System as directly accessible memory. • Like disk arrays and RAM, flash memory needs to seamlessly cater for failure of bits and whole devices. • Hard disks have evolved to need multiple parity drives to keep the risk of total data loss acceptably low in production environments. • Throughput of storage arrays, not latency, will become their defining performance metric. New 'figures of merit' will be: • Volumetric: Gb per cubic-inch • Power: Watts per Gb • Throughput: Gb per second per read/write-stream • Bandwidth: Total Gb per second • Connections: Number simultaneous connections. • Price:$ per Gb available and \$ per Gb/sec per server and total
• Reliability: probability of 1 byte lost per year per Gb
• Archive and Recovery features: snapshots, backups, archives and Mean-Time-to-Restore
• Expansion and Scalability: maximum size (Gb, controllers, units, I/O rate) and incremental pricing
• Off-site and removable storage: RAID-5 disk-packs with multiple interfaces are needed.
• Near Zero-latency storage implies reorganising and simplifying Operating Systems and their scheduling/multi-processing algorithms. Special CPU support may be needed, like for Virtualisation.
• Separating networks {external access, storage/database, admin, backups} becomes mandatory for performance, reliability, scaling and security.
• Pushing large-scale persistent storage onto the network requires a commodity network faster than 1Gbps ethernet. This will either be 10Gbps ethernet or multi-lane 3-6Gbps Infiniband.
What might Desktops look like in 5 years?
For a definitive theoretical treatment of aspects of storage hierarchies, Dr. Neil J Gunther, ex-Xerox PARC, now Performance Dynamics, has been writing about "The Virtualization Spectrum" for some time.
Footnote 1:
Is this idea of multi-speed memory (small/fast and big/slow) new or original?
No: Seymour Cray, the designer of the world's fastest computers for ~2 decades, based his designs on it. It appears to me to be a old idea whose time has come again.
From a 1995 interview with the Smithsonian:
SC: Memory was the dominant consideration. How to use new memory parts as they appeared at that point in time. There were, as there are today large dynamic memory parts and relatively slow and much faster smaller static parts. The compromise between using those types of memory remains the challenge today to equipment designers. There's a factor of four in terms of memory size between the slower part and the faster part. Its not at all obvious which is the better choice until one talks about specific applications. As you design a machine you're generally not able to talk about specific applications because you don't know enough about how the machine will be used to do that.
There is also a great PPT presentation on Seymour Cray by Gordon Bell entitled "A Seymour Cray Perspective", probably written as a tribute after Cray's untimely death in an auto accident.
Footnote 2:
The notion of "all files on the network" and invisible multi-level caches was built in 1990 at Bell Labs in their Unix successor, "Plan 9" (named for one of the worst movies of all time).
Wikipedia has a useful intro/commentary, though the original on-line docs are pretty accessible.
Ken Thompson and co built Plan 9 around 3 elements:
• A single protocol (9P) of around 14 elements (read, write, seek, close, clone, cd, ...)
• The Network connects everything.
• Four types of device: terminals, CPU servers, Storage servers and the Authentication server.
Ken's original storage server had 3 levels of transparent storage (in sizes unheard of at the time):
• 1Gb of RAM (more?)
• 100Gb of disk (in an age where 1Gb drives where very large and exotic)
• 1Tb of WORM storage (write-once optical disk. Unheard of in a single device)
The usual comment was, "you can go away for the weekend and all your files are still in either memory or disk cache".
They also pioneered permanent point-in-time archives on disk in something appearing to the user as similar to NetApp's 'snapshots' (though they didn't replicate inode tables and super-blocks).
My observations in this piece can be paraphrased as:
• re-embrace Cray's multiple-memory model, and
• embrace commercially the Plan 9 "network storage" model.
### Promises and Appraising Work Capability and Proficiency
Max Wideman, PMI Distinguished Contributor and Person of the Year and Canadian author of several Project Management books plus a slew of published papers, not only responded to, and published, some comments and conversations of between us, he then edited up some more emails into a Guest Article of his site.
Many thanks to you Max for all your fine work and for seeing something useful in what I penned.
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Question
A trail is
9
10
of a mile long. Abdul hiked
2
5
of the length of the trail. Which steps could be used to determine how far he hiked, in simplest terms? Select three options.
9
10
to
2
5
.
Add , 9 tenths, to , 2 fifths, .
Find the product as follows:
9
10
×
2
5
=
11
50
.
Find the product as follows: , 9 tenths times 2 fifths is equal to 11 fiftieths, .
Find the product as follows:
9
10
×
2
5
=
18
50
.
Find the product as follows: , 9 tenths times 2 fifths is equal to 18 fiftieths, .
Find a common factor for the numerator and denominator.
Find a common factor for the numerator and denominator.
Simplify the product as
9
25
mile.
Simplify the product as , 9 twenty-fifths, mile.
1. bonexptip
The third and fifth ones
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Register for: AustMS2017
# Abstracts for AustMS2017
250 abstracts submitted.
Authors Title Session Timetabled Abstract
Aaron J Kaw, Assoc Prof Adelle Coster Delivery and diffusion in membranes Math Biol 2017-12-12 14:00:00 Lateral diffusion of proteins on cell membranes ...
Ainsley Pullen Concrete mathematical incompleteness and the Finite Upper Shift Kernel theorem Alg 2017-12-12 16:30:00 Incompleteness was famously investigated by Gö\-del ...
Alan Haynes, Tanja Schindler Trimmed sums for observables on the doubling map Dyn Erg 2017-12-13 14:00:00 For measure preserving ergodic dynamical systems there ...
Aleksey Ber, Jinghao Huang, Galina Levitina, Fedor Sukochev Derivations into ideals of semifinite von Neumann algebras Func An 2017-12-12 15:00:00 The derivation problem introduced by Barry Johnson ...
Ali Eshragh, Hadi Charkhgard A new approach to select the best subset of predictors in linear regression modeling Math Opt 2017-12-13 15:00:00 We study the problem of selecting the ...
Amie Albrecht, Phil Howlett, Geetika Verma The fundamental equations for the generalized resolvent of an elementary pencil in a unital Banach algebra Func An 2017-12-14 16:00:00 We show that the generalized resolvent of ...
Amie Albrecht, Phil Howlett, Peter Pudney, Xuan Vu, Peng Zhou The two-train separation problem on non-level track App Ind 2017-12-12 14:00:00 When two trains travel along the same ...
Assoc Prof Anthony Licata Linear braids Rep Th 2017-12-14 16:00:00 The Artin groups associated to Coxeter groups ...
Assoc Prof Catherine Greenhill, Daniel Altman, Peter Ayre, Amin Coja-Oghlan, Mikhail Isaev, Reshma Ramadurai Two threshold problems for random graphs and hypergraphs Plen 2017-12-15 10:00:00 Probabilistic combinatorics is the study of random ...
Assoc Prof David Brander Cauchy problems for surfaces related to harmonic maps Compl Geom 2017-12-14 12:00:00 It has been known since ...
Assoc Prof Deborah King (UoM), Prof Cristina Varsavsky (Monash), Dr Kelly Matthews (UQ), Assoc Prof Shaun Belward (JCU) Investigating students' perceptions of graduate learning outcomes in mathematics Edu 2017-12-14 12:00:00 In this talk we will report on ...
Assoc Prof Dzmitry Badziahin $p$-adic Littlewood conjecture: what can potential counter-examples look like? Num Th 2017-12-12 16:00:00 In 2004 de~Mathan and Teulie proposed the ...
Assoc Prof Gerd Schmalz Homogeneous tube domains in higher dimensions Compl Geom 2017-12-13 15:00:00 The study of homogeneous domains goes back ...
Assoc Prof Kais Hamza General bootstrap random walks Prob Stoch 2017-12-13 15:00:00 Given the increments of a simple symmetric ...
Assoc Prof Lesley Ward Using the Schottky--Klein prime function to compute the harmonic measure distribution function of a doubly connected planar domain Compl Geom 2017-12-12 14:00:00 Consider releasing a Brownian particle from a ...
Assoc Prof Mark Nelson Analysis of nitrogen removal in the activated sludge process App Ind 2017-12-12 16:00:00 The activated sludge process is the most ...
Assoc Prof Mark Nelson Modelling the spread of smoking as an infectious disease Math Biol 2017-12-13 16:30:00 SIR models were originally developed to analyse ...
Assoc Prof Michael Batanin $E_3$-algebra structure on the Davydov--Yetter deformation complex Top 2017-12-13 16:00:00 We show that the Davydov--Yetter deformation complex ...
Assoc Prof Peter Kim Modelling evolution of post-menopausal human longevity: the Grandmother Hypothesis Math Biol 2017-12-15 14:00:00 Human post-menopausal longevity makes us unique among ...
Assoc Prof Yan Dolinsky Duality and convergence for binomial markets with friction Prob Stoch 2017-12-15 15:00:00 We prove limit theorems for the super-replication ...
Assoc Prof Yvonne Stokes Can we make that fibre? Plen 2017-12-14 16:30:00 The development of microstructured optical fibres, containing ...
Audrey J. Markowskei, Paul D. Smith Quantifying the change in the far-field pattern induced by rounding the corners of a scatterer illuminated by a plane-wave electromagnetic field App Ind 2017-12-13 14:30:00 When a perfectly electrically conducting two-dimen\-s\-io\-nal scatterer, ...
Ben Webster The representation theory of symplectic singularities Rep Th 2017-12-13 14:00:00 There are a lot of non-commutative algebras ...
Christopher C. Green, Marie A. Snipes, Lesley A. Ward Using the Schottky--Klein prime function to compute harmonic measure distribution functions of a class of multiply connected planar domains Compl Geom 2017-12-12 14:30:00 We will show how to construct explicit ...
Daniel Daners, Juli\'an L\'opez-G\'omez Global dynamics of generalized logistic equations PDEs 2017-12-12 14:00:00 We consider a parameter dependent parabolic population ...
Daniel Murfet A-infinity categories of matrix factorisations Cat K Associated to any hypersurface singularity is a ...
David Pask, Aidan Sims, Adam Sierakowski Unbounded quasitraces, stable finiteness and pure infiniteness Func An 2017-12-15 15:00:00 We prove that if $A$ is a ...
Diarmuid Crowley The topological period–index conjecture for almost complex 6-manifolds Top 2017-12-12 14:00:00 The period–index conjecture for functions fields is ...
Diarmuid Crowley Functors to categories of manifolds Cat K 2017-12-12 16:00:00 In this talk I will describe how ...
Dr Adam Sikora Riesz transform and harmonic functions Harm An 2017-12-12 14:00:00 Let $(X,d,\mu)$ be a doubling metric measure ...
Dr Alessandro Ottazzi Lie groups contacto-morphic to nilpotent Lie groups Compl Geom 2017-12-14 15:30:00 Consider a $3$-dimensional connected and simply connected ...
Dr Alessandro Ottazzi Spectral multipliers for sub-Laplacians on $NA$ groups Harm An 2017-12-12 14:30:00 Let $G=NA$, where $N$ is a stratified ...
Dr Alexander Campbell Enriched algebraic weak factorisation systems Cat K 2017-12-15 16:30:00 It is known that if $\mathcal{V}$ is ...
Dr Alexandr Garbali Lattice integrable stochastic processes Math Phys 2017-12-12 16:30:00 Using a certain twisting procedure $XXZ$-type integrable ...
Dr Andrea Collevecchio The branching-ruin number and the critical parameter of once-reinforced random walk on trees Prob Stoch 2017-12-14 15:30:00 The motivation for this paper is the ...
Dr Anh Bui On the flows associated to self-adjoint operators on metric measure spaces Harm An 2017-12-14 11:30:00 Let $X$ be a metric space with ...
Dr Arik Wilbert Two-block Springer fibers and Springer representations in type D Rep Th 2017-12-14 11:30:00 We explain how to construct an explicit ...
Dr Artem Pulemotov Metrics with prescribed curvature on homogeneous spaces with intermediate subgroups Geom An 2017-12-13 14:00:00 We will discuss the problem of recovering ...
Dr Artem Pulemotov Metrics with prescribed curvature on generalised flag manifolds Top 2017-12-14 12:00:00 We will discuss the problem of recovering ...
Dr Azam Asanjarani, Dr Yoni Nazarathy Bursty Markovian arrival processes Prob Stoch 2017-12-13 16:30:00 We call a Markovian arrival process (MAP) ...
Dr Bishnu Lamichhane A new minimisation principle for the Poisson equation leading to a flexible finite-element approach Comp Math 2017-12-12 15:00:00 We introduce a new minimisation ...
Dr Changwei Xiong Convexity of non-negatively curved hypersurfaces with free boundary on a sphere Geom An 2017-12-14 15:30:00 When is an immersed hypersurface in Euclidean ...
Dr Chi-Kwong Fok Real K-theory of compact Lie groups Cat K 2017-12-13 16:30:00 Let $G$ be a compact connected Lie ...
Dr Chris Tisdell No, Professor: you don't need to reverse the order of integration! Edu 2017-12-12 16:00:00 Calculus books are full of worked examples ...
Dr Chuong Thai Doan Optimality conditions for non-smooth, multi-objective, bilevel optimization problems Math Opt 2017-12-12 16:00:00 This talk is devoted to the study ...
Dr Collin Grant Phillips Employing cultural plasticity in STEM workshops for prospective indigenous Engineering and Information Technology students Edu 2017-12-13 14:00:00 Indigenous school students from the ACT, NSW, ...
Dr Daniel Hauer A strong maximum principle on cones PDEs 2017-12-13 16:30:00 In this talk, I present a new ...
Dr Daniel Hauer A generalised Gagliardo--Nirenberg type inequality with application to the $p(x)$-Laplacian Harm An 2017-12-13 14:30:00 In this talk, we present a generalised ...
Dr Daniel Mathews Knot invariants and cluster algebras Top 2017-12-12 16:00:00 Cluster algebras are a type of commutative ...
Dr Danya Rose Who gets the girl? On the operational sex ratio as an index for male strategy Math Biol 2017-12-14 15:30:00 The operational sex ratio (OSR) is defined ...
Dr Deborah Jackson Impact and sustainability of a cross-disciplinary mathematics support program Edu 2017-12-12 14:30:00 Mathematics support' materializes in many forms. A ...
Dr Dmitry Demskoi Automated assessment by means of computer algebra, \LaTeX\ and PDF forms \LaTeX 2017-12-14 15:30:00 I will explain a method of creating ...
Dr Duy Ho On the classification of toroidal circle planes Rep Th 2017-12-13 16:00:00 We consider the problem of classifying toroidal ...
Dr Emma Carberry Toroidal soap bubbles: constant mean curvature tori in $\mathbb{S}^3$ and $\mathbb{R}^3$ Compl Geom 2017-12-14 11:30:00 Constant mean curvature (CMC) tori in $\mathbb{S}^3$, ...
Dr Fu Ken Ly An embedding result for Hermite distribution spaces Harm An 2017-12-14 12:00:00 We present certain Sobolev-type embeddings for weighted ...
Dr Gizem Intepe Examining students' interaction with mathematics consultation using Text Mining Edu 2017-12-14 15:30:00 There has always been a requirement for ...
Dr Guillermo Pineda-Villavicencio On the reconstruction of polytopes Math Opt 2017-12-12 16:30:00 Blind and Mani, and later Kalai, showed ...
Dr Guo Chuan Thiang Hyperbolic and crystalline topological matter via Baum--Connes isomorphisms Func An 2017-12-14 12:00:00 Much of the literature on topological phases ...
Dr Guo Chuan Thiang Duality methods for topological matter Math Phys 2017-12-14 15:30:00 Band theory has been used to predict ...
Dr Guoyin Li Computing radius of robust feasibility of uncertain linear conic programs via semidefinite programs Math Opt 2017-12-14 12:00:00 The radius of robust feasibility provides a ...
Dr Hang Wang Twisted Donaldson invariants Top 2017-12-14 15:30:00 Donaldson's polynomial invariants are topological invariants for ...
Dr Harkirat Dhindsa, AP Leanne Rylands Tertiary students' perceptions of assessments of, and attitudes to, Mathematics Edu 2017-12-13 16:00:00 The aim of this study was to ...
Dr Holger Dullin Stability of doubly periodic shear flow of the Euler equations Dyn Erg 2017-12-13 14:30:00 The Euler fluid equations allow for explicit ...
Dr Huanhuan li Graded Steinberg algebras and their representations Cat K 2017-12-15 15:00:00 We study the category of left unital ...
Dr Ilknur Tulunay ST-metric method in Finance App Ind 2017-12-15 14:30:00 This method uses an information theoretic concept, ...
Dr James East Integer polygons with given perimeter Alg 2017-12-12 14:00:00 A classical result of Honsberger states that ...
Dr Jan Rozendaal Stability theory for semigroups using $(L^{p},L^{q})$ Fourier multipliers PDEs 2017-12-15 14:30:00 In this talk I will discuss some ...
Dr Jerome Droniou Designing high-order schemes for diffusion problems on generic polytopal cells Comp Math 2017-12-12 14:00:00 Dealing with meshes presenting complex geometries has ...
Dr Jesse Burke Transferring $A$-infinity structures along quasi-isomorphisms Cat K 2017-12-15 14:00:00 It has been long understood how to ...
Dr Jessica Fintzen On the Moy--Prasad filtration and super-cuspidal representations Rep Th 2017-12-12 15:00:00 The Moy--Prasad filtrations of $p$-adic groups play ...
Dr Jessica Fintzen Families of $p$-adic automorphic forms on unitary groups Num Th 2017-12-14 15:30:00 We will start with an introduction to ...
Dr Jiakun Liu Bergman--Toeplitz operators on weakly pseudoconvex domains PDEs 2017-12-15 14:00:00 In this talk, we prove that for ...
Dr Joan Licata Contact manifolds with boundary Top 2017-12-13 14:00:00 Contact geometry\,---\,the odd-dimensional analogue of symplectic geometry\,---\,equips ...
Dr John Banks The {\it Register!} conference registration system \LaTeX 2017-12-14 12:00:00 {\it Register!} was originally created to handle ...
Dr John Bourke Braidings for skew monoidal categories Cat K 2017-12-14 16:00:00 Skew monoidal categories are a certain laxified ...
Dr Judy-anne Osborn Communities of practice across pre-undergraduate and undergraduate mathematics Edu 2017-12-15 14:30:00 I report on research at the University ...
Dr Justin Wishart Homogeneous wavelet expansions of some fractional Gaussian fields Stoch App 2017-12-13 15:00:00 Homogeneous, and ideally non-homogeneous, wavelet expansions of ...
Dr Kevin Coulembier Auslander--Dlab--Ringel algebras and Ringel duality Rep Th 2017-12-12 14:00:00 I will introduce a new class of ...
Dr Kieran Clancy Extending a linear programming formulation for TSP Math Opt 2017-12-13 16:30:00 The travelling salesman problem (TSP) is a ...
Dr Laurence Field Spatial decomposition for Brownian motion and SLE curves Prob Stoch 2017-12-14 16:00:00 Decomposition of a stochastic process in spatial ...
Dr Liangyi Zhao Mean-value results of Hecke $L$-functions with fixed-order characters Num Th 2017-12-12 16:30:00 We will present some mean-value results for ...
Dr Maolin Zhou A principal eigenvalue problem with large degenerate advection Harm An 2017-12-13 14:00:00 In this article, we study the asymptotic ...
Dr Marcy Robertson Presheaf models for infinity modular operads Cat K 2017-12-13 14:30:00 Getzler and Kapronov introduced the notion of ...
Dr Marcy Robertson An action of the Grothendieck--Teichm\"uller group Top 2017-12-14 11:30:00 The Grothendieck--Teichm\"uller group is an explicitly defined ...
Dr Martin Ehler Optimal Monte Carlo integration Comp Math 2017-12-12 16:00:00 The decay rate of Monte Carlo integration ...
Dr Masoud Kamgarpour Examples of mesopelagic Langlands correspondence Rep Th 2017-12-14 15:30:00 The aim of the geometric Langlands program ...
Dr Matthew Randall Flat $(2,3,5)$-distributions and Chazy's equations Compl Geom 2017-12-12 16:00:00 In the theory of generic $2$-plane fields ...
Dr Michael Greg Watson, Prof Mary R. Myerscough Multi-phase modelling of early fibrous cap formation in \textit{atherosclerosis} Math Biol 2017-12-14 16:00:00 Advanced \textit{atherosclerotic} plaques are characterised by the ...
Dr Michael Stewart More sensitive mixture detection using the empirical moment-generating function Stoch App 2017-12-13 14:30:00 The higher criticism' method of mixture detection, ...
Dr Michal Ferov Isomorphism problem for virtually-free groups Alg 2017-12-12 16:00:00 The Isomorphism problem is one of the ...
Dr Milena Radnovic Short trajectories of integrable billiards Dyn Erg 2017-12-14 11:30:00 We classify short periodic trajectories of billiards ...
Dr Min Sha On the irregular primes with respect to Euler polynomials Num Th 2017-12-14 16:00:00 An odd prime $p$ is called irregular ...
Dr Minh N. Dao On the generalized Douglas--Rachford algorithm for feasibility problems Math Opt 2017-12-13 17:00:00 In this work, we study the generalized ...
Dr Mircea Voineagu Computations in Bredon motivic cohomology Cat K 2017-12-12 15:00:00 We construct a category of $\mathbb{Z}/2$-equivariant motives ...
Dr Mumtaz Hussain The Hausdorff measure version of Gallagher's theorem\,---\,closing the gap and beyond Num Th 2017-12-12 14:30:00 I will present my recent results in ...
Dr Nathan Brownlowe On Baumslag--Solitar monoids and their $C^*$-algebras Func An 2017-12-15 14:30:00 At the 2016 AustMS meeting, Zahra Afsar ...
Dr Peter Hochs K-theory and characters Func An 2017-12-14 11:30:00 Let $G$ be a semisimple Lie group. ...
Dr Peter Hochs Blattner's conjecture as an index theorem Rep Th 2017-12-12 16:00:00 Let $G$ be a semisimple Lie group ...
Dr Qirui Li The planar dual Minkowski problem Geom An 2017-12-13 17:30:00 Various geometric measures have been discovered and ...
Dr Qirui Li A class of optimal transportation problems on the sphere PDEs 2017-12-13 16:00:00 Optimal transportation problem has attracted much attention ...
Dr Quoc Thong Le Gia Sparse isotropic regularisation for spherical harmonic representations of random fields on the sphere Comp Math 2017-12-12 16:30:00 We discuss sparse isotropic regularisation for a ...
Dr Richard Garner Ultrafilters Cat K 2017-12-15 14:30:00 Ultrafilters are important structures in areas of ...
Dr Robert McDougall On base radical operators Part 2: classes of finite Puczylowski algebras Alg 2017-12-12 15:00:00 In this presentation, we use the methods ...
Dr Robyn Patrice Araujo The simple complexity of robust networks Math Biol 2017-12-14 11:30:00 While mathematics has long been considered an ...
Dr Roland Dodd, Antony Dekkers, Prof William Guo Bridging the gap for inclusive transition Edu 2017-12-15 16:30:00 The mathematics capability of students, entering CQUs ...
Dr Ross Moore Authoring Tagged PDF' documents with \LaTeX \LaTeX 2017-12-14 11:30:00 Tagged PDF' is at the basis of ...
Dr Ross Ogilvie The space of harmonic tori in the $3$-sphere Geom An 2017-12-14 11:30:00 It is known that harmonic tori in ...
Dr Shengguo Zhu Recent progress on classical solutions for compressible isentropic Navier--Stokes equations with degenerate viscosities and vacuum PDEs 2017-12-12 16:00:00 In this talk, I will present ...
Dr Stephen Meagher Chebotarev's density theorem over finite fields Num Th 2017-12-13 16:30:00 Chebotarev's density theorem is well known over ...
Dr Thomas Quella Protection of topological phases in quantum spin systems by quantum deformed symmetries Math Phys 2017-12-14 16:00:00 We show that topological phases of quantum ...
Dr Thomas Taimre, Patrick Laub Exploiting asymptotic structure for efficient rare-event estimation for sums of random variables Prob Stoch 2017-12-13 16:00:00 We consider the problem of estimating the ...
Dr Thomas Wong \LaTeX\ + First Year Calculus = ??? Edu 2017-12-15 15:00:00 Assignments form a major component of continuous ...
Dr Tim Stokes Generalised domain and $E$-inverse semigroups Alg 2017-12-14 16:00:00 Various generalisations of Green's relations on semigroups ...
Dr Timothy Trudgian Primes and squares\,---\,in less than two pages! Num Th 2017-12-13 16:00:00 There are more sums of squares than ...
Dr Vera Roshchina Multipoint Voronoi cells Math Opt 2017-12-14 11:30:00 Given a finite set $S$ in a ...
Dr Vladimir Mangazeev Integrable structure of products of complex random matrices Math Phys 2017-12-12 14:00:00 We consider the squared singular values of ...
Dr Wolfgang Globke Affinely flat algebraic groups and a conjecture of Popov Compl Geom 2017-12-14 16:00:00 A Lie group $G$ admits a flat ...
Dr Xin Zhang Quantum integrable models and the off-diagonal Bethe ansatz method Math Phys 2017-12-12 14:30:00 The quantum integrable models, which are defined ...
Dr Yinan Zhang Computing $p$-adic regulators Num Th 2017-12-14 12:00:00 I will give an update on joint ...
Dr Yingying Sun The Sylvester equation and the elliptic Korteweg--de Vries system PDEs 2017-12-12 14:30:00 In this talk, I will introduce the ...
Dr Yong Wei Volume preserving flow by powers of $k$-th mean curvature PDEs We consider the flow of closed convex ...
Dr Yong Wei Volume preserving flow in hyperbolic space Geom An 2017-12-14 16:00:00 We consider the volume preserving flow of ...
Dr Yucang Wang, Prof William Guo, Dr Roland Dodd Linking mathematical theories to computation and modelling for engineering applications Edu 2017-12-12 14:00:00 Most engineering and scientific problems are too ...
Dr Zahra Afsar KMS states on the $C^*$-algebras of Fell bundles over groupoids Func An 2017-12-15 16:00:00 We consider fibre-wise singly generated Fell bundles ...
Dr Zdravko Botev Sampling via regenerative chain Monte Carlo Prob Stoch 2017-12-12 16:00:00 Markov chain Monte Carlo (MCMC) is a ...
Dr Zlatko Jovanoski A stochastic differential equation approach to modelling the growth phase of fire spread App Ind 2017-12-13 14:00:00 From a point source, landscape fires accelerate ...
Dr Zongzheng Zhou Green's function of a random length random walk on the torus Prob Stoch 2017-12-15 14:30:00 It was generally believed that above upper ...
Dr Zongzheng Zhou Unified correlation function behaviours on high-dimensional tori Math Phys 2017-12-14 11:30:00 Above the upper critical dimensions, Ising model, ...
Elena D. Vinogradova Regularization of the first-kind surface integral equations arising in the wave diffraction on 2D arbitrary cavities with longitudinal slit App Ind 2017-12-15 15:00:00 Diffraction of acoustic and electromagnetic waves from ...
Erchuan Zhang, Lyle Noakes Riemannian cubics in the manifold $\operatorname{SPD}(n)$ of all $n\times n$ symmetric positive-definite matrices Geom An 2017-12-13 17:00:00 The manifold $\operatorname{SPD}(n)$ of all $n\times n$ ...
Frank Valckenborgh, Dilshara Hill Assessment: a multi-pronged tool to motivate and engage Edu 2017-12-14 11:30:00 Engagement and motivation of students studying mathematics ...
Gerd Schmalz, Masoud Ganji A criterion for the embedding of a $3$-dimensional CR structure Compl Geom 2017-12-13 16:00:00 For any $3$-dimensional CR structure $M$ we ...
Hanz Martin Cheng, Jerome Droniou, Kim-Ngan Le Convergence analysis of a family of ELLAM schemes for a fully coupled model of miscible displacement in porous media Comp Math 2017-12-12 14:30:00 We analyse the convergence of numerical schemes ...
Heather Lonsdale, Matthew Allen, Christel Ernest, Kai Striega Third-year undergraduate projects in mathematics education: analysing student attitudes, student reflections, and predicting student performance Edu 2017-12-14 16:00:00 In this talk I will present a ...
Ian Sloan, Ivan Graham (Univ. Bath), Frances Kuo (UNSW), Dirk Nuyens (KU Leuven), Robert Sceichl (Univ. Bath) On the generation of random fields Comp Math 2017-12-13 15:00:00 The generation of Gaussian random fields is ...
Ivan Mirkovic, Yaping Yang, Gufang Zhao Towards a construction of higher dimensional loop Grassmannians Rep Th 2017-12-12 16:30:00 I will talk about a construction of ...
Jacqui Ramagge C*-algebras from self-similar actions and their states Func An 2017-12-13 14:00:00 We will consider self-similar actions of groupoids ...
Jan Rozendaal Operator-valued $(L^{p},L^{q})$ Fourier multipliers Harm An 2017-12-12 15:00:00 In this talk I will discuss some ...
John P. Wormell Spectral Galerkin methods for transfer operators in uniformly expanding dynamics Dyn Erg 2017-12-13 16:00:00 We present a spectral method for numerically ...
Kevin Aguyar Brix Investigating symbolic dynamics using C*-algebras Func An 2017-12-13 16:30:00 The symbiotic relationship between symbolic dynamics and ...
Marc Levine, Yaping Yang, Gufang Zhao Algebraic elliptic cohomology and flops Cat K 2017-12-13 17:30:00 I will talk about the algebraic elliptic ...
Martin Sagradian Potential theory problems for arbitrary rotationally symmetric double-connected conductors: rigorous approach App Ind 2017-12-12 16:30:00 This paper considers potential theory problems for ...
Michael Assis Exactly-solved origami statistical mechanics Math Phys 2017-12-13 17:30:00 Using the methods of exactly-solved models in ...
Michael Assis Systematic analysis of OEIS generating functions Comp Math 2017-12-13 16:30:00 Given a sequence of integers, one would ...
Michael Hallam, Mathai Varghese End-periodic $K$-homology and positive scalar curvature Cat K 2017-12-13 14:00:00 In this talk I will introduce a ...
Miss Becky Armstrong Twisted $C^*$-algebras of topological higher-rank graphs: keeping things simple! Func An 2017-12-13 16:00:00 Since their introduction twenty years ago, $C^*$-algebras ...
Miss Lauren Thornton On base radical operators Part 1: classes of finite associative rings Alg 2017-12-12 14:30:00 Class operators are used to give a ...
Miss Lijing Ma, Dr Georgy Sofronov Multiple change-point detection in an AR(1) Process: comparison of different methods Stoch App 2017-12-13 17:00:00 The multiple change-point detection problem in time ...
Miss Meng Shi Bootstrap random walk Prob Stoch 2017-12-13 14:00:00 Consider a one-dimensional simple random walk $X$. ...
Miss Rebecca Smith; Dr Mumtaz Hussain Engagement-focused learning in large service-level courses Edu 2017-12-12 15:00:00 We will explore students' attitudes towards Mathematics ...
Miss Sogol Mohammadian Investigating Hamilton cycles through extreme points of a certain polytope Math Opt 2017-12-13 16:00:00 In this talk we study a certain ...
Miss Sophie Ham Geometric triangulations of knot complements Top 2017-12-12 16:30:00 To date, it remains an open conjecture ...
Mr Abrahim Steve Nasrawi Lifted worm process for the Ising model Math Phys 2017-12-14 12:00:00 We construct a lifted worm process for ...
Mr Adarsh Kumbhari Modelling the impact of T-cell avidity on cancer vaccines Math Biol 2017-12-13 14:30:00 Therapeutic cancer vaccines treat cancers that have ...
Mr Alex Casella Representations of fibered $3$-manifolds using flags Top 2017-12-13 15:00:00 In 2007, Fock and Goncharov use ideal ...
Mr Alex Parkinson Averaging of discrete-time singularly perturbed optimal control problems Math Opt 2017-12-13 14:00:00 Current methods for solutions of singularly perturbed ...
Mr Alexander Majchrowski Neck detection for the fully nonlinear flow $G$ Geom An 2017-12-13 15:00:00 I will discuss neck detection and a ...
Mr Bojan Crnkovic Lattice structure detection and refinement DMD algorithm Dyn Erg 2017-12-14 12:00:00 Dynamic Mode Decomposition (DMD) is a class ...
Mr Charles Walker Universal properties of polynomials via doctrinal Yoneda structures Cat K 2017-12-12 14:30:00 Given a category $\mathcal{E}$ with pullbacks, one ...
Mr Csaba Nagy Classifying $8$-dimensional E-manifolds Top 2017-12-12 15:00:00 A manifold $M$ is called an E-manifold ...
Mr Csaba Nagy A functorial approach to classifying manifolds Cat K 2017-12-12 16:30:00 The aim of this talk is to ...
Mr Daniel Lin Presheaves over join restriction categories Cat K 2017-12-13 17:00:00 Restriction categories were first introduced in the ...
Mr Dominic Tate Higher Teichm\"uller theory on closed and finite-area surfaces using techniques of Fock and Goncharov Top 2017-12-13 14:30:00 In 2007 Fock and Goncharov devised an ...
Mr Edoardo Lanari $\infty$-groupoids and the Homotopy Hypothesis Cat K 2017-12-13 15:00:00 In this talk I will introduce a ...
Mr Fadi Antown Optimal linear response for Markov chains Dyn Erg 2017-12-13 17:00:00 The linear response of a dynamical system ...
Mr Florian Martin Laurent De Leger Contractibility of nerve of classifiers and application to the Turchin--Dwyer--Hess theorem (with Michael Batanin) Top 2017-12-13 16:30:00 A result of Dwyer--Hess and Turchin asserts ...
Mr Hafiz Khusyairi Unexpected new formula for Grothendieck duality Alg 2017-12-14 12:00:00 A few years ago, a surprising new ...
Mr Hao Guo, Elder Prof Varghese Mathai, Dr Hang Wang Positive scalar curvature for proper actions Func An 2017-12-15 14:00:00 In this talk, we study equivariant index ...
Mr Harry Crimmins Stability of Statistical Properties for some Dynamical Systems Dyn Erg 2017-12-13 16:30:00 For sufficiently chaotic dynamical systems the ...
Mr Jean-Jerome Casanova Fluid structure system with boundary conditions involving the pressure PDEs 2017-12-12 16:30:00 We study a coupled fluid-structure system involving ...
Mr Jeffrey Lay An explicit bound for the divisor function Num Th 2017-12-12 15:00:00 Landreau proved in 1988 a class of ...
Mr Jon Xu Chevalley groups and finite geometry Rep Th 2017-12-13 16:30:00 In this talk, I will describe a ...
Mr Jonathan Julian Zhu Min--max theory for constant mean curvature hypersurfaces Geom An 2017-12-13 14:30:00 We describe the construction of closed constant ...
Mr Kam Hung Yau Distribution of $\alpha n+ \beta$ modulo 1 over some arithmetic set Num Th 2017-12-13 15:00:00 For any sufficiently small real number $\varepsilon ... Mr Liam S. Hodgkinson The long-term behaviour of an occupancy process Prob Stoch 2017-12-12 16:30:00 Treating a complex network as a large ... Mr Matthew James Spong The$K$-theory of loop spaces and elliptic cohomology Cat K 2017-12-13 16:00:00 Let$T$be a torus. In 1994 ... Mr Michael Arthur Mampusti Mauldin--Williams graphs and their KMS states Func An 2017-12-12 16:30:00 In this talk, we will look at ... Mr Michael Hendriksen Non-binary unrooted tree-based networks Math Biol 2017-12-12 15:00:00 There is contemporary debate in biology as ... Mr Oscar Peralta, Dr Bo Friis Nielsen, Dr Mogens Bladt On a class of bivariate phase-type distributions and its applications in risk theory Prob Stoch 2017-12-14 12:00:00 A multivariate phase-type (MPH*) distributed vector can ... Mr Petru A. Cioica-Licht Stochastic integration in quasi-Banach spaces Prob Stoch 2017-12-12 15:00:00 The goal of this talk is to ... Mr Scott Lindstrom Strong convergence for relaxed iterated approximate projection methods for convex feasibility problems Math Opt 2017-12-13 14:30:00 We investigate iterative schemes for solving convex ... Mr Simon Macourt Visible points on exponential curves Num Th 2017-12-14 11:30:00 We provide an introduction to the problem ... Mr Thanakorn Nitithumbundit Modelling multivariate financial time series with variance gamma innovations Stoch App 2017-12-13 16:00:00 Modelling multivariate financial time series returns data ... Mr Thomas Pedersen On the$C^*$-algebras of a graph of groups Func An 2017-12-13 17:00:00 Graphs of groups, consisting of an undirected ... Mr Timothy Siu Combinatorial model for the dynamics of birational maps over finite fields Dyn Erg 2017-12-13 17:30:00 Consider a random permutation of$N$points ... Mr Turker Topal, Elena Vinogradova, Yu. A. Tuchkin Accurate calculation of complex eigenvalues for TM-modes in 2D arbitrary cavities with longitudinal slit App Ind 2017-12-13 15:00:00 The rigorous Method of Regularization ([1]--[3]) is ... Mr Yuki Maehara Mahavier limits Cat K 2017-12-14 11:30:00 Sequential limits have long been valued by ... Mr Yunxuan Liu Invariance principle for biased bootstrap random walks Prob Stoch 2017-12-13 14:30:00 Our main goal is to study a ... Mr Zihua Guo Generalized Strichartz estimates for Schr\"odinger equation PDEs In this talk I will talk about ... Mrs Shaymaa Shawkat Kadhim Al-shakarchi Isomorphisms of$\mathit{AC}(\sigma)$spaces Func An 2017-12-12 16:00:00 The classical Banach--Stone Theorem for spaces$C(K)$... Ms Adrianne Jenner Modelling heterogeneity in biology: how do cancer-killing viruses interact with tumour cells? Math Biol 2017-12-13 15:00:00 For some time now, the use ... Ms Catheryn Gray Akt translocation as a harmonic oscillator Math Biol 2017-12-12 14:30:00 Akt is a key signalling protein of ... Ms Dorothee Frey Sparse dominations and sharp weighted estimates for singular integral operators Harm An 2017-12-12 16:30:00 We shall discuss the concept of sparse ... Ms Elizabeth Bradford Recursive algorithms for inversion of linear operator pencils Func An 2017-12-13 17:30:00 There are numerous examples of systems that ... Ms Jie Yen Fan Measure-valued population processes and their asymptotics Prob Stoch 2017-12-14 11:30:00 Population process in a general setting, where ... Ms Lyn Armstrong, Mr Donald Shearman From where do our students come? Edu 2017-12-15 16:00:00 It is well known that the mathematics ... Ms Pantea Pooladvand Do T-cells compete for antigen? Math Biol 2017-12-13 14:00:00 When antigen is presented to helper T ... Ms Sinead Wilson Stabilisers of eigenvectors in complex reflection groups Rep Th 2017-12-14 12:00:00 Complex reflection groups are finite subgroups of ... Ms Stephanie Mills The connection between vanishing reverse-H\"older weights and functions of vanishing mean oscillation Harm An 2017-12-12 16:00:00 One way to quantify how far a ... Ms Zeying Chen Duality in mASEP and tKZ equation Math Phys 2017-12-12 16:00:00 In stochastic processes, duality is defined by ... Murray Elder, Yoong Kuan Goh Permutations sorted by a finite and an infinite stack in series Alg 2017-12-14 15:30:00 An \emph{antichain} is a subset of a ... Mx Andrew Grant Comparing multivariate spectral densities Stoch App 2017-12-13 16:30:00 In this talk we present a test ... Mx Bregje Pauwels Gerstenhaber structure of a class of special biserial algebras Rep Th 2017-12-12 14:30:00 I will talk about a recent computation ... 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Prof Andrei Okounkov Enumerative geometry and geometric representation theory Plen 2017-12-12 11:30:00 Certain problems in enumerative geometry turn out ... Prof Andrew Eberhard Radius theorems for monotone mappings Math Opt 2017-12-12 15:00:00 For a Hilbert space$X$and a ... Prof Birgit Loch Teaching wirelessly with a pen-enabled tablet Edu 2017-12-13 14:30:00 In this presentation, I will discuss the ... Prof Dominic Verity, Assoc Prof Emily Riehl Generator notions in$\infty$-cosmology Cat K 2017-12-15 16:00:00$\infty$-Cosmoi provide a framework ... Prof Fima Klebaner Random initial conditions in differential equations Prob Stoch 2017-12-12 14:00:00 We give a result for approximation of ... Prof Geoffrey Prince Variationality of PDEs PDEs 2017-12-14 16:00:00 When can a system of PDEs be ... Prof Georgia Benkart Walking on graphs to Invariant Theory Plen 2017-12-13 11:30:00 Molien's 1897 formula for the Poincar\'e series ... 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Encyclopedia > Electronvolt
The electronvolt (symbol eV) is a unit of energy. In theoretical physics, where distinctions between mass and energy are not concrete, it is often used also as a unit of mass (AAAS Science journal, 2006). It is the amount of kinetic energy gained by a single unbound electron when it passes through an electrostatic potential difference of one volt, in vacuo. In other words, it is equal to one volt (1 volt = 1 joule per coulomb) times the (unsigned) charge of a single electron. The one-word spelling is the modern recommendation[1], although the use of the earlier electron volt still exists. The kinetic energy of an object is the extra energy which it possesses due to its motion. ... e- redirects here. ... A student demonstrating the effects of electrostatics. ... Josephson junction array chip developed by NIST as a standard volt. ... Look up Vacuum in Wiktionary, the free dictionary. ... Josephson junction array chip developed by NIST as a standard volt. ... The joule (IPA pronunciation: or ) (symbol: J) is the SI unit of energy. ... The coulomb (symbol: C) is the SI unit of electric charge. ... The elementary charge (symbol e or sometimes q) is the electric charge carried by a single proton, or equivalently, the negative of the electric charge carried by a single electron. ...
One electronvolt is a very small amount of energy:
1 eV = 1.602 176 53(14)×10−19 J. [2] (or approximately 0.160 aJ)
The unit electronvolt is accepted (but not encouraged) for use with SI. It is widely used in solid state, atomic, nuclear, and particle physics, often with prefixes m, k, M, G or T. In a recorded lecture from 1961 Richard Feynman apologized to his students for this failure by atomic physicists to use the appropriate SI unit (which would be the attojoule): To help compare different orders of magnitude we list here energies between 10−19 joules and 10−18 joules (0. ... The joule (IPA pronunciation: or ) (symbol: J) is the SI unit of energy. ... Atto- (symbol a) is an SI prefix to a unit and means that it is 10-18 times this unit. ... Look up si, Si, SI in Wiktionary, the free dictionary. ... Solid-state physics, the largest branch of condensed matter physics, is the study of rigid matter, or solids. ... Atomic physics (or atom physics) is the field of physics that studies atoms as isolated systems comprised of electrons and an atomic nucleus. ... Nuclear physics is the branch of physics concerned with the nucleus of the atom. ... Thousands of particles explode from the collision point of two relativistic (100 GeV per ion) gold ions in the STAR detector of the Relativistic Heavy Ion Collider. ... An SI prefix (also known as a metric prefix) is a name or associated symbol that precedes a unit of measure (or its symbol) to form a decimal multiple or submultiple. ... Richard Phillips Feynman (May 11, 1918 – February 15, 1988; IPA: ) was an American physicist known for expanding the theory of quantum electrodynamics, the physics of the superfluidity of supercooled liquid helium, and particle theory. ... Atto- (symbol a) is an SI prefix to a unit and means that it is 10-18 times this unit. ...
"A single atom is such a small thing that to talk about its energy in joules would be inconvenient. But instead of taking a definite unit in the same system, like 10−20 J, [physicists] have unfortunately chosen, arbitrarily, a funny unit called an electronvolt (eV) ... I am sorry that we do that, but that's the way it is for the physicists." [3]
In chemistry, it is often useful to have the molar equivalent, that is the kinetic energy that would be gained by a mole of electrons passing through a potential difference of one volt. This quantity is equal to 96.48538(2) kJ/mol. Ionization energies and other atomic properties are often quoted in electronvolts, especially in older texts. Chemistry - the study of interactions of chemical substances with one another and energy based on the structure of atoms, molecules and other kinds of aggregrates Chemistry (from Egyptian kēme (chem), meaning earth[1]) is the science concerned with the composition, structure, and properties of matter, as well as the... The mole (symbol: mol) is the SI base unit that measures an amount of substance. ... To help compare different orders of magnitude we list here energies between 10,000 joules and 100,000 joules. ... The joule per mole (symbol: J·mol-1) is an SI derived unit of energy per amount of material. ... The ionization energy (IE) of an atom or of a molecule is the energy required to strip it of an electron. ...
## Using electronvolts to measure mass GA_googleFillSlot("encyclopedia_square");
Albert Einstein reasoned that energy is equivalent to mass, as famously expressed in the mass-energy equivalence formula E = mc² (1.0000 kg = 89.876 PJ). It is thus common in particle physics, where mass and energy are often interchanged, to use eV/c² or even simply eV as a unit of mass. “Einstein†redirects here. ... This article or section is in need of attention from an expert on the subject. ... 15ft sculpture of Einsteins 1905 E = mc² formula at the 2006 Walk of Ideas, Germany In physics, mass-energy equivalence is the concept that all mass has an energy equivalence, and all energy has a mass equivalence. ... The kilogram or kilogramme (symbol: kg) is the SI base unit of mass. ... The joule (symbol J, also called newton metre, or coulomb volt) is the SI unit of energy and work. ...
For example, an electron and a positron, each with a mass of 0.511 MeV/c², can annihilate to yield 1.022 MeV of energy. The proton has a mass of 0.938 GeV/c², making GeV a very convenient unit of mass for particle physics. The first detection of the positron in 1932 by Carl D. Anderson The positron is the antiparticle or the antimatter counterpart of the electron. ... In physics, the proton (Greek proton = first) is a subatomic particle with an electric charge of one positive fundamental unit (1. ... Thousands of particles explode from the collision point of two relativistic (100 GeV per ion) gold ions in the STAR detector of the Relativistic Heavy Ion Collider. ...
1 eV/c² = 1.783×10−36 kg
1 keV/c² = 1.783×10−33 kg
1 MeV/c² = 1.783×10−30 kg
1 GeV/c² = 1.783×10−27 kg
1 TeV/c² = 1.783×10−24 kg
1 PeV/c² = 1.783×10−21 kg
1 EeV/c² = 1.783×10−18 kg
See: Orders of magnitude (mass) Category: ...
In some older documents, and in the name Bevatron, the symbol "BeV" is used, which stands for "billion-electron-volt"; it is equivalent to the GeV (gigaelectronvolt). [[Category:]] Edwin McMillan and Edward Lofgren on the shielding of the Bevatron. ...
## Electronvolts and energy
For comparison:
• 3.2×10−11 joule or 200 MeV - total energy released in nuclear fission of one U-235 atom (on average; depends on the precise break up)
• 3.5×10−11 joule or 210 MeV - total energy released in fission of one Pu-239 atom (also on average)
• Molecular bond energies are on the order of an electronvolt per molecule.
• The typical atmospheric molecule has a kinetic energy of about 1/40 eV. This corresponds to room temperature.
The joule (IPA pronunciation: or ) (symbol: J) is the SI unit of energy. ... An induced nuclear fission event. ... In chemistry, bond energy (E) is a measure of bond strength in a chemical bond. ... To help compare different orders of magnitude we list here energies between 10−21 joule and 10−20 joule (0. ... Room temperature describes a certain temperature within enclosed space that is uses for various purposes by human beings. ...
## Electronvolts and photon properties
The energy E, frequency f, and wavelength λ of a photon are related by
$E=hf=frac{hc}{lambda}=frac{1240~rm{nm~eV}}{lambda}$
where h is Planck's constant and c is the speed of light. For example, the spectrum of visible light consists of wavelengths ranging from 400 nm to 700 nm. Photons of visible light therefore have energies ranging from A commemoration plaque for Max Planck on his discovery of Plancks constant, in front of Humboldt University, Berlin. ... A line showing the speed of light on a scale model of Earth and the Moon, about 1. ...
$E_{min} = frac{1240~rm{nm~eV}}{700~rm{nm}} = 1.77~rm{eV}$
to
$E_{max} = frac{1240~rm{nm~eV}}{400~rm{nm}} = 3.10~rm{eV}$.
An electronvolt is also the energy of an infrared photon with a wavelength of approximately 1240nm. 10eV would correspond to ultraviolet of 124nm, etc.
## Using electronvolts to measure time and distance
In particle physics, distances and times are sometimes expressed in inverse electronvolts via the conversion factors[4] Thousands of particles explode from the collision point of two relativistic (100 GeV per ion) gold ions in the STAR detector of the Relativistic Heavy Ion Collider. ...
• $hbar$ = 6.582 118 89(26) x 10-16 eV s
• $hbar c$ = 197.326 960 2(77) eV nm
In these units, the mean lifetime τ of an unstable particle can be reexpressed in terms of its decay width Γ (in eV) via $Gamma = hbar/tau$. For example, the B0 meson has a mean lifetime of 1.542(16) picoseconds, or a decay width of 4.269(44) x 10-4 eV, and its mean decay length is cτ = 462 μm. Given an assembly of elements, the number of which decreases ultimately to zero, the lifetime (also called the mean lifetime) is a certain number that characterizes the rate of reduction (decay) of the assembly. ... A list of mesons. ... A picosecond is an SI unit of time equal to 10-12 of a second. ...
## Electronvolts and temperature
In certain fields, such as plasma physics, it is convenient to use the electronvolt as a unit of temperature. The conversion to kelvins (symbol: uppercase K) is defined by using kB, the Boltzmann constant: A Plasma lamp In physics and chemistry, a plasma is an ionized gas, and is usually considered to be a distinct phase of matter. ... The kelvin (symbol: K) is a unit increment of temperature and is one of the seven SI base units. ... Ludwig Boltzmann The Boltzmann constant (k or kB) is the physical constant relating temperature to energy. ...
${1 mbox{ eV} over k_B} = {1.60217653(14) times 10^{-19} mbox{J} over 1.3806505(24) times 10^{-23} mbox{J/K}} = 11604.505(20) mbox{ kelvins}$
For example, a typical magnetic confinement fusion plasma is 15 keV, or 174 megakelvins. Magnetic confinement fusion is an approach to fusion energy that uses magnetic fields to confine the fusion fuel in the form of a plasma. ...
## References
1. ^ NIST: Units outside the SI
2. ^ Peter J. Mohr and Barry N. Taylor (January 2005). "CODATA recommended values of the fundamental physical constants: 2002" (PDF). Reviews of Modern Physics 77: 1–107. Retrieved on 2006-07-01. An in-depth discussion of how the CODATA constants were selected and determined.
3. ^ Transcript of part of a 1961 lecture by Richard Feynman
4. ^ K. Hagiwara et al, Review of Particle Physics, Phys. Rev. D66, 010001 (2002)
Year 2006 (MMVI) was a common year starting on Sunday (link displays full 2006 calendar) of the Gregorian calendar. ... is the 182nd day of the year (183rd in leap years) in the Gregorian calendar. ...
This article or section does not cite any references or sources. ... Thermodynamics (from the Greek θεÏμη, therme, meaning heat and δυναμις, dunamis, meaning power) is a branch of physics that studies the effects of changes in temperature, pressure, and volume on physical systems at the macroscopic scale by analyzing the collective motion of their particles using statistics. ...
Results from FactBites:
Electronvolt - Wikipedia, the free encyclopedia (378 words) The electronvolt (symbol eV, or, rarely and incorrectly, ev) is a unit of energy. It is the amount of kinetic energy gained by a single unbound electron when it passes through an electrostatic potential difference of one volt, in vacuum. In certain fields, such as plasma physics, it is convenient to use the electronvolt as a unit of temperature.
More results at FactBites »
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Probability of queuing
0liver
New Member
We are students and we developing the project of a music festival that takes place in 3 venues. We have identified some risks; one of those is related to the capacity of the venues, the tickets sold and how people will behave during the festival.
The total capacity of the venues is 6000, however we just sell 5800 tickets to have some spare capacity (200). The tickets allow the audience to access the three venues if there is free space on them.
Venue A Venue B Venue C
Capacity (people) 3000 2000 1000
In order to reduce people queuing outside because the venues are full the following measures have been taking:
The concerts in the three venues start at the same time; this is 18.00h, 19.00h and so forth and last 45minutes each.
The line up has been done in such way that bands playing at the same time have a similar fan-base on the social media.
The venues are located in the same area but walking time between them takes approx 5 minutes between venues is required.
The question are:
What is the probability at any given moment of time to have people queuing outside the venues A, B, C?
Probability of having queues in more than one venues simultaneously (e.g A and C)
It is possible to know an estimate, using statistics, of the queuing time? What time of data would be required to carry such analysis?
We appreciate your time and help.
Best,
Oliver
BGM
TS Contributor
Your problems seems quite interesting but there are quite a number of follow-up questions need to be asked.
1. Do you mean there are a total of 5800 people (inside the venue + queuing outside) at any given moment of time?
2. If yes, "have a similar fan-base on the social media" means that you want to model the number of people supporting the band in three different venues follows
$$\text{Trinomial}\left(5800; \frac {1} {3}, \frac {1} {3}, \frac {1} {3}\right)$$
3. Why do people queuing outside? Do you mean the people buying the ticket will only watch their most supported band without considering the other two?
4. You have mentioned the show started on time 1800 1900 etc. Do you mean there are different people in each hour? Or people enter freely at anytime? Also do you mean the band will rotate the venue at each time?
5. Not sure how the walking time between each venue will be related to the estimates (unless you are taking somehow complicated model considerations that the people are moved freely between the venues based on several factors). So that's why I am not very sure how "queuing time" plays the role here. As mentioned above unless you consider the people to be very dynamic, can moving around freely throughout the 45 minutes show time; but this backs to the question 3 why do people queue outside? why do not they just go to the empty seat?
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# Plotting SHARP keywords and images with python¶
In this notebook, we will be plotting keywords and images, from data taken by the Helioseismic and Magnetic Imager (HMI) instrument on NASA's Solar Dynamics Observatory (SDO) satellite. SDO takes about a terabyte and a half of data a day, which is more data than any other satellite in the NASA Heliophysics Division.
Data from the HMI and Atmospheric Imaging Assembly (AIA) instruments aboard SDO are stored at Stanford University. The metadata are stored in a pSQL database called the Data Record Management System, or DRMS. The image data are stored separately, in storage units called the Storage Unit Management System, or SUMS. Data are merged together, upon export from both systems, as FITS files. DRMS and SUMS together constitute the Joint Science Operations Center, or JSOC.
The easiest way to access SDO HMI and AIA data is via the python drms module, available at PyPI. In addition to the numerous tutorials on both the Read the Docs and Github, all the examples below utilize the drms module. First we'll import the module, and some others:
In [1]:
import drms
import json, numpy as np, matplotlib.pylab as plt, matplotlib.ticker as mtick
from datetime import datetime as dt_obj
import urllib
from astropy.io import fits
from sunpy.visualization.colormaps import color_tables as ct
from matplotlib.dates import *
import matplotlib.image as mpimg
import sunpy.map
import sunpy.io
from IPython.display import Image
%matplotlib inline
%config InlineBackend.figure_format='retina'
The first step in querying for SDO HMI and AIA data is to establish a connection to JSOC. This can be done with the Client() class.
In [2]:
import drms
c = drms.Client()
The Client() class allows one to access both metadata and image data simultaneously via a data series. A data series contains all of particular type of data — e.g. there is a series for continuum intensity data, another for magnetic field data, and so forth. Read Section 4 of the SDO Analysis Guide for more information about how to build a data series query. For example, to find all the SHARP data series, execute the following regular expression query:
In [3]:
c.series(r'hmi\.sharp_')
Out[3]:
['hmi.sharp_720s',
'hmi.sharp_720s_nrt',
'hmi.sharp_cea_720s',
'hmi.sharp_cea_720s_nrt']
In [4]:
# Set a series
si = c.info('hmi.sharp_cea_720s')
In [5]:
si.keywords
Out[5]:
type recscope defval units note linkinfo is_time is_integer is_real is_numeric
name
cparms_sg000 string variable compress Rice none None False False False False
magnetogram_bzero double variable 0 none None False False True True
magnetogram_bscale double variable 0.1 none None False False True True
cparms_sg001 string variable none None False False False False
bitmap_bzero double variable 0 none None False False True True
... ... ... ... ... ... ... ... ... ... ...
ERRMSHA float variable nan Degrees Error in Mean shear angle for B_total None False False True True
ERRUSI float variable nan Amperes Error in Total unsigned vertical current None False False True True
DOFFSET int variable -2147483648 Gauss Constant value added to the noise mask for dis... None False True False True
ERRTPOT float variable nan Ergs per cubic centimeter Error in Total photospheric magnetic energy de... None False False True True
ERRJHT float variable nan Amperes Sum of the Absolute Value of the Net Currents ... None False False True True
211 rows × 10 columns
To find more information about the FITS image data, or segments, that belong to any given series, we can use the following command:
In [6]:
# To see all the segments associated with the series hmi.sharp_cea_720s:
si.segments
Out[6]:
type units protocol dims note
name
magnetogram int Gauss fits VARxVAR Line-of-sight magnetogram in CEA projection
bitmap char Enumerated fits VARxVAR Mask for the patch in CEA coordinates
Dopplergram int m/s fits VARxVAR Dopplergram in CEA projection
continuum int DN/s fits VARxVAR Intensitygram in CEA projection
Bp int Gauss fits VARxVAR B_phi, positive westward
Bt int Gauss fits VARxVAR B_theta, positive southward
Br int Gauss fits VARxVAR B_r, positive up
Bp_err int Gauss fits VARxVAR Standard deviation of B_phi
Bt_err int Gauss fits VARxVAR Standard deviation of B_theta
Br_err int Gauss fits VARxVAR Standard deviation of B_r
conf_disambig char none fits VARxVAR confidence of disambiguation result
The query below retrieves both metadata and image data for active region 11158, which produced an X2.2-class flare on February 15, 2011 at 1:56 UT, from the SHARP data series. The SHARP data series include patches of vector magnetic field data taken by the HMI instrument. These patches encapsulate automatically-detected active regions that are tracked throughout the entirety of their disk passage. The c.query() method takes three arguments:
1. The first argument is the data series. In the example below, the data series is called hmi.sharp_cea_720s. This series is appended with two prime keys: the HARP number (in this case, 377) and the time range (in this case, 2011.02.14_15:00:00/12h). These two prime keys appear in the first two brackets. The HARP number refers to the active region number (see here for a mapping between HARP numbers and NOAA active region numbers). A prime key, or set of prime keys, is a unique identifier. The third bracket, with the argument [? (QUALITY<65536) ?], filters out data where the value of the QUALITY keyword is greater than 65536. (See here for the definition of the QUALITY keyword). While this third bracket is not necessary, it can be a powerful tool for filtering data based on keyword values.
2. The second argument in the search query is a list of keywords. In this example, we will query for the keywords T_REC, USFLUX, and ERRVF.
3. The third argument in the search query is a list of segments. In this example, we will query for the segment Br, or the radial component of the photospheric magnetic field.
In [7]:
keys, segments = c.query('hmi.sharp_cea_720s[377][2011.02.14_15:00:00/12h][? (QUALITY<65536) ?]', key='T_REC, USFLUX, ERRVF', seg='Br')
To convert the keyword T_REC into a datetime object, we can use the function below.
In [8]:
def parse_tai_string(tstr,datetime=True):
year = int(tstr[:4])
month = int(tstr[5:7])
day = int(tstr[8:10])
hour = int(tstr[11:13])
minute = int(tstr[14:16])
if datetime: return dt_obj(year,month,day,hour,minute)
else: return year,month,day,hour,minute
In [9]:
t_rec = np.array([parse_tai_string(keys.T_REC[i],datetime=True) for i in range(keys.T_REC.size)])
Now for some plotting! matplotlib.pyplot generates two objects: a figure and axes. The data are ascribed to the axes. The time axes in particular requires some formatting; in order to free it of clutter, we'll plot tick marks every three hours and label them accordingly.
In [10]:
fig, ax = plt.subplots(figsize=(8,7)) # define the size of the figure
orangered = (1.0,0.27,0,1.0) # create an orange-red color
# define some style elements
marker_style = dict(linestyle='', markersize=8, fillstyle='full',color=orangered,markeredgecolor=orangered)
text_style = dict(fontsize=16, fontdict={'family': 'monospace'})
ax.tick_params(labelsize=14)
ax.yaxis.set_major_formatter(mtick.FormatStrFormatter('%.2f'))
# ascribe the data to the axes
ax.plot(t_rec, (keys.USFLUX)/(1e22),'o',**marker_style)
ax.errorbar(t_rec, (keys.USFLUX)/(1e22), yerr=(keys.ERRVF)/(1e22), linestyle='',color=orangered)
# format the x-axis with universal time
locator = AutoDateLocator()
locator.intervald[HOURLY] = [3] # only show every 3 hours
formatter = DateFormatter('%H')
ax.xaxis.set_major_locator(locator)
ax.xaxis.set_major_formatter(formatter)
# set yrange
ax.set_ylim([2.4,2.8])
# label the axes and the plot
ax.set_xlabel('time in UT',**text_style)
ax.set_ylabel('maxwells x 1e22',**text_style)
ax.set_title('total unsigned flux starting at '+str(t_rec[0])+' UT',**text_style) # annotate the plot with a start time
Out[10]:
Text(0.5, 1.0, 'total unsigned flux starting at 2011-02-14 15:00:00 UT')
## Querying the data¶
The example above shows a simple query for 12 hours of data from one HARPNUM. But we can also perform more complex queries to identify active regions of interest. Here are a few examples.
Example 1. Suppose we want to create a magnetic field model of an active region and we need observations of a strong-field active region near disk center. This query identifies strong-field regions near disk center during a two year period. We define strong active regions as those with a total unsigned flux (USFLUX) greater than $4x10^{22}$ Mx and near disk center as those with a Carrington Longitude (CRLN_OBS) less than $1^{\circ}$. The two year period spans between January 2014 and January 2016.
In [11]:
keys = c.query('hmi.sharp_cea_720s[][2014.01.01 - 2016.01.01][? (CRLN_OBS < 1) AND (USFLUX > 4e22) ?]', key='T_REC, HARPNUM, USFLUX, CRLT_OBS, CRLN_OBS, AREA_ACR')
In [12]:
keys
Out[12]:
T_REC HARPNUM USFLUX CRLT_OBS CRLN_OBS AREA_ACR
0 2014.05.04_14:48:00_TAI 4071 4.007906e+22 -3.818258 0.045204 1297.351685
1 2014.11.11_06:00:00_TAI 4800 4.004017e+22 3.297165 0.676486 1207.938232
2 2014.11.11_06:12:00_TAI 4800 4.023450e+22 3.295724 0.567313 1194.579590
3 2014.11.11_06:24:00_TAI 4800 4.050183e+22 3.294308 0.458156 1198.369873
4 2014.11.11_06:36:00_TAI 4800 4.070347e+22 3.292919 0.349014 1204.132935
5 2014.11.11_06:48:00_TAI 4800 4.081056e+22 3.291558 0.239885 1208.038818
6 2014.11.11_07:00:00_TAI 4800 4.094752e+22 3.290226 0.130767 1199.290039
7 2014.11.11_07:12:00_TAI 4800 4.113331e+22 3.288925 0.021658 1222.337891
8 2015.02.01_05:00:00_TAI 5127 4.801844e+22 -6.020030 0.985052 2273.071289
9 2015.02.01_05:12:00_TAI 5127 4.817782e+22 -6.020372 0.875986 2271.228516
10 2015.02.01_05:24:00_TAI 5127 4.778004e+22 -6.020685 0.766931 2270.977295
11 2015.02.01_05:36:00_TAI 5127 4.801163e+22 -6.020968 0.657885 2278.230469
12 2015.02.01_05:48:00_TAI 5127 4.780960e+22 -6.021223 0.548846 2281.161621
13 2015.02.01_06:00:00_TAI 5127 4.724961e+22 -6.021451 0.439813 2275.408203
14 2015.02.01_06:12:00_TAI 5127 4.682469e+22 -6.021652 0.330784 2282.527344
15 2015.02.01_06:24:00_TAI 5127 4.597347e+22 -6.021828 0.221756 2280.517334
16 2015.02.01_06:36:00_TAI 5127 4.505237e+22 -6.021980 0.112728 2281.434570
17 2015.02.01_06:48:00_TAI 5127 4.448969e+22 -6.022110 0.003698 2298.415039
18 2015.06.17_13:48:00_TAI 5692 4.593700e+22 1.266837 0.923733 1495.361328
19 2015.06.17_14:00:00_TAI 5692 4.592959e+22 1.267220 0.813259 1503.894775
20 2015.06.17_14:12:00_TAI 5692 4.593827e+22 1.267613 0.702742 1489.501465
21 2015.06.17_14:24:00_TAI 5692 4.584389e+22 1.268017 0.592184 1494.039429
22 2015.06.17_14:36:00_TAI 5692 4.636860e+22 1.268434 0.481585 1491.178833
23 2015.06.17_14:48:00_TAI 5692 4.616645e+22 1.268866 0.370945 1495.119507
24 2015.06.17_15:00:00_TAI 5692 4.629078e+22 1.269314 0.260266 1505.565796
25 2015.06.17_15:12:00_TAI 5692 4.652655e+22 1.269780 0.149549 1496.576538
26 2015.06.17_15:24:00_TAI 5692 4.681429e+22 1.270264 0.038794 1479.642456
27 2015.10.31_19:12:00_TAI 6063 5.331063e+22 4.459106 0.909237 1804.466309
28 2015.10.31_19:24:00_TAI 6063 5.314220e+22 4.458659 0.798615 1805.412476
29 2015.10.31_19:36:00_TAI 6063 5.258935e+22 4.458181 0.687989 1811.795410
30 2015.10.31_19:48:00_TAI 6063 5.197402e+22 4.457672 0.577359 1807.541748
31 2015.10.31_20:00:00_TAI 6063 5.231150e+22 4.457130 0.466728 1802.004395
32 2015.10.31_20:12:00_TAI 6063 5.238996e+22 4.456556 0.356098 1813.600952
33 2015.10.31_20:24:00_TAI 6063 5.218206e+22 4.455947 0.245472 1822.455444
34 2015.10.31_20:36:00_TAI 6063 5.241477e+22 4.455305 0.134850 1833.126953
35 2015.10.31_20:48:00_TAI 6063 5.301954e+22 4.454628 0.024236 1834.722900
Example 2. Suppose we are doing a study on flux emergence and we want to identify active regions that live for a long time. This query identifies long-lived active regions within a six year period. We define long-lived active regions as those with a minimum number of observations (N_PATCH1) equal to 1800 (1800 observations with a 12 minute gap between observations means a minimum observation time of 15 days). The six year period spans between January 2012 and January 2018. The T_FRST1=T_REC clause identifies the first observation time in the sequence.
In [13]:
keys = c.query('hmi.sharp_cea_720s[][2012.01.01 - 2018.01.01][? (N_PATCH1 > 1800) AND (T_FRST1=T_REC) ?]', key='T_REC, HARPNUM, NOAA_ARS, N_PATCH1, AREA_ACR')
In [14]:
keys
Out[14]:
T_REC HARPNUM NOAA_ARS N_PATCH1 AREA_ACR
0 2012.07.04_03:24:00_TAI 1834 11519,11520,11521 1820 28.082201
1 2012.07.22_21:12:00_TAI 1879 11529,11530,11532,11533,11536 1883 24.621628
2 2012.07.30_22:24:00_TAI 1907 11538,11539,11540,11541,11544,11545 1809 18.218309
3 2012.09.17_23:24:00_TAI 2040 11575,11577,11583 1986 124.659348
4 2013.04.26_15:36:00_TAI 2696 11732,11734 1846 18.187544
5 2013.06.12_17:24:00_TAI 2852 11769,11770,11771,11772,11774,11775 1864 3.143462
6 2013.12.30_21:00:00_TAI 3563 11943,11944 1822 2.208999
7 2014.01.17_09:00:00_TAI 3647 11958,11959,11960,11963,11964 1813 18.883802
8 2014.02.21_04:36:00_TAI 3784 11987,11989,11993,11994,12001 2018 30.495930
9 2014.03.14_21:24:00_TAI 3856 12008,12010,12012,12015,12019,12023 1838 21.218081
10 2014.04.10_23:00:00_TAI 4000 12035,12038,12043,12046 1878 14.411080
11 2014.07.27_01:00:00_TAI 4396 12127,12128,12130,12131,12132 1927 18.025908
12 2014.12.09_05:12:00_TAI 4920 12235,12237,12238,12242 1841 21.272736
Example 3. Suppose we are doing a study on flare prediction. Schrijver (2007) derived a parameter, called $R$, which quantifies the flux near an active region neutral line. The study concluded that an active region will flare if the log of R is greater than 5. Bobra & Couvidat (2015) also identified a large total unsigned helicity as a relevant characteristic of flaring active regions. This query identifies active regions with a log of R (R_VALUE) greater than 5.5 or a total unsigned helicity (TOTUSJH) greater than 8900 $\frac{G^{2}}{m}$ during a two year period between January 2012 and January 2014.
In [15]:
keys = c.query('hmi.sharp_cea_720s[][2012.01.01 - 2014.01.01][? (R_VALUE > 5.5 AND R_VALUE < 6.0) OR (TOTUSJH >= 8900)?]', key='T_REC,HARPNUM,R_VALUE,TOTUSJH')
In [16]:
keys
Out[16]:
T_REC HARPNUM R_VALUE TOTUSJH
0 2012.07.09_15:12:00_TAI 1834 5.249 8920.152
1 2012.07.09_15:36:00_TAI 1834 5.239 8954.800
2 2012.07.09_15:48:00_TAI 1834 5.253 8923.751
3 2012.07.09_16:00:00_TAI 1834 5.252 8902.096
4 2012.07.09_16:12:00_TAI 1834 5.251 8918.384
5 2012.07.09_16:24:00_TAI 1834 5.256 8901.315
6 2012.07.09_16:36:00_TAI 1834 5.261 8928.521
7 2012.07.09_21:12:00_TAI 1834 5.264 8907.350
8 2012.07.09_21:48:00_TAI 1834 5.271 8910.375
9 2012.07.09_22:12:00_TAI 1834 5.272 8931.835
10 2012.07.10_05:36:00_TAI 1834 5.255 8907.319
11 2012.07.10_05:48:00_TAI 1834 5.285 8903.378
12 2013.11.25_10:00:00_TAI 3376 5.530 55.039
## Plotting the image data¶
We can also query for and plot image data. There are two ways to do this.
1. We can download the image data, as unmerged FITS file, and header data separately. An unmerged FITS file contains the image data, but almost no header metadata (except for a few keywords). This is the quickest and easiest option as the drms.Client() class can query the header and image data at the same time and store the keyword metadata and URLs to the image data in a Pandas dataframe. This eliminates the need to store FITS files locally. This method is also faster, as there is no need to wait for the exportdata system to generate FITS files. We can then download and open the unmerged FITS files via the astropy package for FITS file handling.
2. We can download the merged FITS file, which merges the header metadata and the image data together, and use SunPy Map to plot the image data. This is the easiest way to put the image data into a coordinate system, as the SunPy Map object will automatically use the WCS keyword data to plot the image data in the correct coordinate system. We can also read merged FITS via the astropy package for FITS file handling.
We go through each option below using an image of the radial component of the photospheric magnetic field as an example.
Query the image data and header metadata using drms, then download and open the unmerged FITS file with astropy:
In [17]:
hmi_query_string = 'hmi.sharp_cea_720s[377][2011.02.15_02:12:00_TAI]'
keys, segments = c.query(hmi_query_string, key='T_REC, USFLUX, ERRVF', seg='Br')
url = 'http://jsoc.stanford.edu' + segments.Br[0] # add the jsoc.stanford.edu suffix to the segment name
photosphere_image = fits.open(url) # download and open the unmerged FITS file via astropy
Plot the image data with matplotlib:
In [18]:
hmimag = plt.get_cmap('hmimag')
plt.imshow(photosphere_image[1].data,cmap=hmimag,origin='lower',vmin=-3000,vmax=3000)
print('The dimensions of this image are',photosphere_image[1].data.shape[0],'by',photosphere_image[1].data.shape[1],'.')
The dimensions of this image are 377 by 744 .
There are only a few keywords associated with the unmerged FITS file:
In [19]:
photosphere_image[1].header
Out[19]:
SIMPLE = T / file does conform to FITS standard
BITPIX = -64 / data type of original image
NAXIS = 2 / dimension of original image
NAXIS1 = 744 / length of original image axis
NAXIS2 = 377 / length of original image axis
PCOUNT = 0 / size of special data area
GCOUNT = 1 / one data group (required keyword)
XTENSION= 'BINTABLE' / binary table extension
BLANK = -2147483648
CHECKSUM= 'VCJiX9GfVAGfV9Gf' / HDU checksum updated 2018-05-10T01:45:55
DATASUM = '1982616782' / data unit checksum updated 2018-05-10T01:45:55
But we can get all the header metadata information we like from the drms query:
In [20]:
keys
Out[20]:
T_REC USFLUX ERRVF
0 2011.02.15_02:12:00_TAI 2.653720e+22 6.506040e+18
#### Option 2: Download the merged FITS file and use SunPy Map to plot the image data¶
n.b. The code below will only work with a valid e-mail address. In order to obtain one, users must register on the JSOC exportdata website.
In [21]:
email = '[email protected]'
In [22]:
c = drms.Client(email=email, verbose=True)
In [23]:
# Export the magnetogram as a FITS image
r = c.export(hmi_query_string+'{Br}', protocol='fits', email=email)
fits_url_hmi = r.urls['url'][0]
hmi_map = sunpy.map.Map(fits_url_hmi)
Export request pending. [id=JSOC_20210406_1196, status=2]
Waiting for 5 seconds...
In [24]:
fig = plt.figure()
hmi_map.plot(cmap=hmimag, vmin=-3000,vmax=3000)
Out[24]:
<matplotlib.image.AxesImage at 0x15a6f5f40>
The image is now in the correct coordinate system! We can also inspect the map like this:
In [25]:
hmi_map
Out[25]:
<sunpy.map.sources.sdo.HMIMap object at 0x157cecb20>
Observatory SDO HMI SIDE1 HMI hmi 6173.0 Angstrom 2011-02-15 02:10:12 0.000000 s [744. 377.] pix heliographic_carrington [0.03 0.03] deg / pix [371.5 188. ] pix [ 34.88091278 -21.07690048] deg
Image colormap uses histogram equalization
Click on the image to toggle between units
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# ORB¶
This gem provides a feature detector and descriptor extractor.
Features are used in applications such as:
• 3D reconstruction using structure-from-motion
• Visual odometry (motion tracking) & SLAM
• Content-based image retrieval
• Image alignment & panorama stitching
“ORB” stands for “Oriented FAST and rotated BRIEF”. This shows that ORB is based on FAST, a feature detector, and BRIEF, a binary descriptor.
The original publication by Rublee, et al., titled “ORB: An efficient alternative to SIFT or SURF”, can be found here: http://www.willowgarage.com/sites/default/files/orb_final.pdf
ORB has the following key qualities compared to other feature types:
• Resistant to image noise
• Rotation invariant
• Multi-scale
The Isaac SDK ORB gem follows the original publication closely. In addition to that, it also:
• Implements most of the pipeline in CUDA for higher performance by running on the GPU
• Adds a spatial regularization (“grid filtering”) step, which fixes a fundamental flaw of other implementations (e.g., OpenCV’s) by distributing keypoints over the image more evenly. This has significant benefits for applications such as motion tracking.
The algorithm steps are as follows:
1. Downsample input image to different scale levels
2. Extract FAST features on all levels
3. Apply grid filtering
4. Extract feature orientation
5. Extract descriptors
## Types Provided by the Gem¶
### Keypoints¶
A Keypoint has the following properties:
• x, y: Keypoint coordinates in the image it was extracted from. Note: Since there are multiple scale levels, use the utility function GetCoordsAtTopLevel to get the coordinates at the topmost (initial) image level.
• response: “Strength” of a feature. A higher score means a higher local contrast.
• angle: Orientation angle (in radians)
• scale: Radius of the support region (in pixels)
• level: Scale level from which the feature was extracted. Starts at zero.
The Keypoints type is a vector of keypoints.
### Descriptors¶
A Descriptor has the following properties:
• id: An ID that allows connecting a descriptor back to the keypoint from which it originated
• elements: The binary descriptor data, packed into a small buffer of elements.
The Descriptors type is a vector of descriptors.
## How to Use the Gem (Interface)¶
The gem provides a single function, ExtractOrbFeatures, which is used to extract ORB features and descriptors from an image.
As an input parameter, an image has to be passed.
As an output parameter, Keypoints and Descriptors objects have to be passed.
These are the additional parameters that can be set:
• Max. features (default 1500): The desired number of best features to retain.
• FAST threshold (default 30): A threshold for the minimum local contrast a pixel has to exhibit in order to be considered a feature
• Grid number of cells linear (default 8): How many bins to use for spatial regularization in horizontal / vertical direction. The total number of bins is the square of the parameter. A higher number means spreading features more evenly. Setting to 1 disables the regularization.
• Downsampling factor (default 0.7): The size of the subsequent scale level, as a ratio of the current level. Values between 0.5 (inclusive) and 1.0 (exclusive) are allowed.
• Max. levels (default 4): How many scale levels to use.
## Building the package¶
Run the following command to build the gem, the codelet, and the sample application:
### Embedded Jetson Device¶
1. Deploy //packages/orb/apps/orb_demo:orb_demo-pkg to the robot as explained in Deploying and Running on Jetson.
2. SSH into the device with the following command:
$ssh nvidia@ROBOTIP 1. Run the demo application on the device with the following command, after changing to the deployed directory: $ cd orb_demo-pkg/
\$ ./packages/orb/apps/orb_demo/orb_demo
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The Mathematics Department holds regular seminars on a variety of topics. Please see below for further details.
Monday at 4-5 PM, MSB 110
Organizers: Rankeya Datta and Hema Srinivasan.
Wednesday 5-6 PM, MSB 110
Organizer: Arun Suresh.
Tuesday 2-3 PM, MSB 111
Organizers: Peter Pivovarov and Petros Valettas
Tuesday 1-2 PM, Zoom
Please see the schedule here.
Monday 5-6 PM, Zoom (Contact organizer for link)
Organizer: Stephen Landsittel.
Thursday 1-2 PM, MSB 110
Organizer: Samuel Walsh.
### Fall 2022
Date Speaker Title
9/15
Samuel Walsh (University of Missouri)
Rigidity of three-dimensional internal waves with constant vorticity
In this talk, we will discuss some recent results on the structural implications of constant vorticity for steady three-dimensional internal water waves. It is known that in many physical regimes, water waves beneath vacuum that have constant vorticity are necessarily two dimensional. The situation is more subtle for internal waves that traveling along the interface between two immiscible fluids. When the layers have the same density, there is a large class of explicit steady waves with constant vorticity that are three-dimensional in that the velocity field and pressure depend on one horizontal variable while the interface is an arbitrary function of the other.
Our main theorem states that every three-dimensional traveling internal wave with bounded velocity for which the vorticity in the lower layer $$\boldsymbol{\omega}_1$$ and vorticity in the upper layer $$\boldsymbol{\omega}_2$$ are nonzero, constant, and parallel must belong to this family. If the densities in each layer are distinct, then in fact the flow is fully two dimensional. This result is obtained using a novel but fairly elementary argument based on unique continuation, the maximum principle, and an analysis of streamline patterns.
This is joint work with R. M. Chen, L. Fan, and M. H. Wheeler.
9/29 Tanya Christiansen (University of Missouri) The semiclassical structure of the scattering matrix for a manifold with infinite cylindrical end
We study the microlocal properties of the scattering matrix associated to the semiclassical Schrödinger operator $$P=h^2\Delta_X+V$$ on a Riemannian manifold with an infinite cylindrical end. We will show that under suitable hypotheses the scattering matrix "quantizes" the scattering map. The scattering map $$\kappa$$ and its domain are determined by the Hamilton flow of $$|\xi|^2+V\upharpoonright_{h=0}$$, the principal symbol of $$P$$. As an application we prove that, under additional hypotheses on the scattering map, the eigenvalues of the associated unitary scattering matrix are equidistributed on the unit circle.
The goal of this talk will be to introduce the audience to the big picture: the setting, the objects of the interest, general questions, and some of the challenges involved, rather than giving proofs.
This talk is based on joint work with A. Uribe.
10/13 Mathew Johnson (University of Kansas) Subharmonic dynamics of periodic Lugiato–Lefever waves
In this talk, we will consider the liner and nonlinear dynamics of perturbations of spectrally stable periodic stationary solutions of the Lugiato–Lefever equation (LLE), a damped nonlinear Schrödinger equation with forcing that arises in optics. It is known that spectrally stable $$T$$-periodic solutions are nonlinearly stable to subharmonic perturbations, i.e. to $$NT$$-periodic perturbations for some integer $$N$$, with exponential decay rates. However, both the exponential rates of decay and the allowable size of initial perturbations both tend to zero as $$N \to \infty$$, and hence such subharmonic stability results are not uniform in $$N$$ and are, in fact, empty in the limit $$N=\infty$$.
The primary goal of this talk is to introduce a methodology, in the context of the LLE, by which a uniform in $$N$$ stability result for subharmonic perturbations may be achieved (at least at the linear level). The obtained uniform decay rates are shown to agree precisely with the polynomial decay rates of localized, i.e. integrable on the line, perturbations of such spectrally stable periodic solutions of LLE. If time permits, I will also discuss recent progress towards extending these results for the LLE to the nonlinear level.
This is joint with Mariana Haragus (Univ. Bourgogne Franche-Comté), Wesley Perkins (KU) and Bjorn de-Rijk (Stuttgart)
10/27 Carmen Chicone (University of Missouri) Modeling oscillating heat pipes
I will present background on oscillating heat pipes (OHPs), published results on an ODE model, and give a progress report on a new PDE model. In short, an OHP is a serpentine closed tube (toroidal geometry) partially filled with a liquid. When part of the boundary is heated and part is cooled, it is possible that the liquid separates into a two-phase flow consisting of vapor plugs separating fluid slugs that is set into oscillatory motion and serves as a device to efficiently transfer heat from the hot to the cold zone with no moving mechanical parts except for the fluid motion within the tube. At one level of modeling (usually with ODEs) fluid slugs are tracked and artificial means are used (if at all) to model nucleation and merging of fluid cells.
A more sophisticated approach, and the main focus of the talk, seeks to model the two-phase flow as a phenomenon that arises naturally from the underlying physics and eliminates the need to track slugs. This latter approach, called phase field modeling, is based on the ideas of Allen, Cahn, and Hilliard that lead to the Allen–Cahn and the Cahn–Hilliard equations, which form the basis of all such models. Roughly speaking, the dependent variable in these equations is a so-called order parameter, which is akin to a smoothed indicator function evolving in space and time, which gives the locations of the two phases of the flow. The underlying physics is thermodynamics; in particular, the minimization of Gibbs energy at equilibrium. An overview of this methodology, which has far reaching applications, will be discussed. Its application to OHPs is the motivation for a proposed PDE model whose predictive power has not yet been fully explored.
Most of the content of the talk is joint work with Frank Feng, Steve Lombardo, and Dave Retzloff all colleagues in the College of Engineering.
11/3 Stephen Montgomery-Smith (University of Missouri) The exponential and logarithm of dual quaternions
Dual quaternions are finding increasing use in the robotics and graphics industry, as a method to represent pose (position and orientation) of one frame with respect to another. There is a natural Lie-group/Lie algebra structure, and from this arises a need to compute the exponential and logarithm of a dual quaternion. Quite a few other authors have done this, but their formulas are either wrong, complicated, or hard to use.
In this talk we describe a general functional calculus for dual quaternions. The methods are quite elementary, and it nostalgically brings back an earlier time in our lives when mathematics was essentially simple, and about cute formulas.
11/10 Ming Chen (University of Pittsburgh) Kato's condition for vanishing viscosity near Onsager's critical regularity
In 1984, T. Kato showed that for sufficiently regular solutions, the vanishing viscosity limit is equivalent to having vanishing viscous dissipation in a boundary layer of width proportional to the viscosity. We prove that Kato's criterion holds for Hölder continuous weak solutions with the regularity index arbitrarily close to the Onsager's critical exponent through a new boundary layer foliation and a global mollification technique.
This is a joint work with Zhilei Liang and Dehua Wang.
12/1 Hugo Panzo (Saint Louis University) Improved upper bounds for the Hot Spots constant of Lipschitz domains
The Hot Spots constant was recently introduced by Steinerberger as a means to control the global extrema of the first nontrivial eigenfunction of the Neumann Laplacian by its boundary extrema. We use a probabilistic technique to derive a general formula for a dimension-dependent upper bound that can be tailored to any specific class of bounded Lipschitz domains. This formula is then used to compute upper bounds for the Hot Spots constant of the class of all bounded Lipschitz domains in $$\mathbb{R}^d$$ for both small $$d$$ and for asymptotically large $$d$$ that significantly improve upon the existing results.
This is joint work with Phanuel Mariano and Jing Wang.
Thursday 2-3 PM, Online (Contact organizer for link)
Organizer: Zhenbo Qin.
Friday 4-5 PM, MSB 110
Organizers: Arun Suresh and Luis Flores.
Wedneday 4-5 PM, MSB 12
Organizer: Rankeya Datta.
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# model binding
18691 1 10 129
Orignial message at:
https://sourceforge.net/forum/message.php?msg_id=3468721
By: nobody
Hi. How should I reflect my changes to the view back to the model?
Here is my hello.zul file. I have a controller called testWindow. My model is an attribute of my controller. The model is a simple Person bean with sets and gets for name. The textbox shows the name correctly. However I would like changes to the value in the textbox to be reflected back to the model. How can this be done?
<window id="testWindow" title="Window" border="normal" width="600px"
use="bonzi.TestWindow">
<hbox>
Name:
<textbox id="name" value="${testWindow.person.name}"> </textbox> </hbox> <button label="Hello"><attribute name="onClick">testWindow.onHello();</attribute></button> </window> Andrew delete retag edit ## 1 Reply Sort by » oldest newest answered 2005-12-12 01:26:04 +0800 admin 18691 1 10 129 Orignial message at: https://sourceforge.net/forum/message.php?msg_id=3470454 By: henrichen Hi! Andrew, The ZK's programming model is event-driven. i.e. the ZK engine would send(post) events triggered by end users (at browser) to event handlers (at server) written by programmers. In your case, you should write an event handler to listen to event "onChange" of the textbox with id == "name". <textbox id="name" value="${testWindow.person.name}" onChange="testWindow.person.name = self.value"> </textbox>
the "self" in the handler means the "textbox" component itself.
There are at least three ways to register an event handler to an event. Please take a look of the userguide (http://zk1.sourceforge.net/wp/ZK-userguide.pdf).
And we have setup a wikibooks about ZK's How-Tos (http://en.wikibooks.org/wiki/ZK:_How-Tos). Should there were other questions, please check it first. Though it is kind of slim right now :-); we invite you the users to contribute and share your ZK How-Tos.
Henri
[hide preview]
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# What is the meaning of $k$ in this paragraph from “The Application of Linear Programming to Team Decision Problems” by Radner?
The attached paragraph is from "The Application of Linear Programming to Team Decision Problems."
I do not understand how the profit depends on the capital limit $$k$$ (whose definition is also confusing). Is $$k$$ the maximum capital the company can access?
This is my understanding currently:
• cost for production (alloted) = $$a$$
• cost for promotion (alloted) = $$b$$
• amount produced (can only produce based on unit cost for production) = $$xa$$
• amount that can be sold (effectively how well did promotion work?) = $$yb$$
• additional capital cost. This is the amount raised but at an immediate cost = $$(1+f)$$
• how much profit (this is effectively sold subtracted by money spent) = $$\min(xa+yb)-a-b$$
1. Now this profit can be used for more production if it's postive and above a certain threshold $$k$$. Then the profit is: $$\min(xa+yb)-a-b$$
2. If negative, capital has to be raised. Let profit be less than $$k$$. So, $$a+b. The cost of this capital raise will be $$(1+f)(a+b-k)$$. The profit is then: $$\min(xa+yb)-a-b-(1+f)(a+b-k) = \min(xa+yb)-a-b-a-b+k-f(a+b-k)$$
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+1.617.933.5480
+1.866.649.0192
# Q: finance 4
Southern Healthcare and BestWell are for-profit HMOs that operate in Florida and Georgia. Currently,
both are identical in every respect except that Southern is unleveraged while BestWell has $10 million of 5 percent bonds. Both HMOs report an EBIT of$2 million and pay corporate tax at a rate of 40 p...
0
0
Mark B
0
0
a) Computing the total value of Southern Healthcare:Vu = [EBIT(1- Tc)] /Ruwhere Tc is the tax rate Ru is the cost of capital. Vu is the unleveredVu = [$2,000,000 (1-0.40)] / 0.10 =$1,200,000 / 0
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Article | Open | Published:
# Bottom-up effects on herbivore-induced plant defences: a case study based on compositional patterns of rhizosphere microbial communities
## Abstract
Below-ground soil microorganisms can modulate above-ground plant-insect interactions. It still needs to be determined whether this is a direct effect of single species or an indirect effect of shifts in soil microbial community assemblages. Evaluation of the soil microbiome as a whole is critical for understanding multi-trophic interactions, including those mediated by volatiles involving plants, herbivorous insects, predators/parasitoids and microorganisms. We implemented a regulated system comprising Nerium oleander plants grown in soil initially containing a sterile/non sterile inoculum, herbivore Aphis nerii and predator Chrysoperla carnea. After aphid attack, plants emitted a characteristic blend of volatiles derived from two biosynthetic classes: fatty acid catabolites and aromatic-derived products. Three aliphatic compounds were mainly detected in plants grown in the inoculated microbial soil, a blend which was preferentially chosen by C. carnea adult females. The contrasting effect of the initial inocula was attributed to the different microbial consortia developed in each treatment. We argue that differences in the relative abundance of the active microbial communities in the rhizosphere correlate with those in the emission of selected volatile compounds by attacked plants. The mechanisms involved in how the functional soil microbiome modulates inducible indirect defence of plants are discussed.
## Introduction
As part of the evolutionary adaptation process, phytophagous insect-plant interactions occur according to the behavioural choices of insects and the development of physical and chemical plant defences1. In addition, volatile plant secondary metabolites possibly function as signals in communications with other organisms in the environment. In this context, the plant-mediated effects on predator-prey and host-parasitoid interactions in tri-trophic systems of herbivore-induced plant volatiles (HIPVs) have been well-documented2. The release of HIPVs, generally a mixture of green-leaf volatiles, terpenes and aromatic compounds, among others3, may signal the presence of potential prey or hosts and therefore can be exploited by natural enemies to locate the prey organism. Pioneering plant studies show how chewers, sap feeders and herbivore egg deposition induce the production of volatiles attractive to entomophagous arthropods4,5,6,7.
In addition to indirect interactions, plants act as a link between above- and below-ground communities. It is well known that soil-borne non-pathogenic microbes can modulate plant-insect above-ground interactions via plant growth promotion or induced systemic resistance by triggering biochemical changes in the primary plant metabolism8. From a multi-trophic perspective, the tri-trophic role of plant secondary chemistry has been shown to be central to an understanding of various aspects of trophic phenomena, including top-down and bottom-up regulation of herbivores. In recent years, several studies have investigated the effects of below-ground microbes on the third trophic level organisms via changes in HIPV emission9. However, most studies focusing on modifications in plant volatile emission through interactions with soil microorganisms mainly address plant interactions with single species of non-pathogenic microbes, which have a neutral, synergistic or antagonistic effect on plant secondary chemistry10, 11. Given the significant impact of microbial community diversity and richness on plant signalling pathways, the root microbiome as a whole needs to be considered in relation to many aspects of plant immunity12, 13 such as induced indirect defence and insect population dynamics. Nevertheless, to date, no overall trend has emerged in relation to the effects of increased microbial complexity on microbe-plant-insect interactions, with some evidence showing a significant, limited or zero impact on above-ground herbivores14.
In this study, we aim to evaluate the role of the rhizosphere microbiome in HIPV production and its impact on above-ground community interactions. In a case study, we designed a system-based model with three trophic levels, with Nerium oleander as host plant, Aphis nerii as phloem-sap feeder and Chrysoperla carnea as predator (Fig. 1). The plants were grown for 3 months in sterilized soil with a microbial inoculant and were then infested with A. nerii. The HIPVs produced were sampled using solid-phase micro-extraction (SPME) and were measured by GC-MS. A Y-tube olfactometer was used to investigate the orientational response of C. carnea to the plant volatiles. Using a metagenomic approach, we analyzed the composition and diversity of entire and active microbial communities.
## Results
### Chrysoperla carnea behaviour
Model selection indicated that the most parsimonious model only takes account of fixed factor treatment without considering block and assay variables (AICc = 167.21, ΔAICc = 2, R2 = 0.25). Selected model estimated 2.55 C. carnea individuals chose the branch connected with the sterilised vermicompost treatment, whereas the non-sterilized vermicompost treatment was chosen by a predicted total average of 4.32 C. carnea individuals. This means that 37.12% of individuals chose the Control treatment, while 62.88% selected the Vermicompost treatment (Fig. 2). The results indicate a clear preference of female C. carnea adults for the volatile compounds produced under vermicompost soil treatment conditions.
### Volatiles
The capacity of N. oleander to emit volatiles has been previously described15, 16. However, the volatile blend emitted by N. oleander plants under non-attack conditions was quantitatively insufficient to be detected by SPME-GC in our experiment. Depending on the potting soil media used, N. oleander plants damaged by A. nerii under laboratory conditions emitted a different variety of volatile organic compounds 3 days after infestation (Fig. 3). The 2-decanone, 2-dodecanone and tetradecane compounds were mostly detected when plants were grown in the microbial inoculated soil (Vermicompost treatment).
### Nerium oleander dry weight and chemical characteristics of soil samples
After harvest, no differences in the dry weight of N. oleander plants between the two treatments were detected (Control = 62.91 g, Vermicompost = 64.57 g; F = 0.213, P = 0.656). Detailed information on potting soil characteristics can be found as Supplementary Table S1. As expected, fractional sterilization had no significant effect on the intrinsic chemical properties of the vermicompost17. No significant differences were detected between the two treatments, with values for pH, C:N ratio and macro and micronutrients found to be comparable in Control and Vermicompost soils at the end of the experiment.
### Composition of active and whole microbial communities
After 3 months of incubation, the two microbial communities (from a sterile and natural vermicompost) showed different taxonomical structures when inoculated in the same sterile soil. We retrieved over 160,000 bacterial and fungal operational taxonomic units (OTUs) from potting soils previously surveyed by pyrosequencing rRNA gene and gene transcript amplicons.
### Bacterial communities
Eight dominant phyla (abundance > 1%) were present in the rhizosphere soil samples, accounting for more than 99% of all bacterial sequences (Fig. 4). The total bacterial community was dominated by Firmicutes (Control 79.49%, Vermicompost 46.07%), Actinobacteria (Control 12.14%, Vermicompost 16.52%) and Proteobacteria (Control 4.18%, Vermicompost 20.79%). Its composition differed significantly between treatments, with, most notably, a single phylum, Firmicutes, accounting for almost 80% of the entire bacterial population in the Control treatment. On closer analysis, no particular differences between treatments in terms of genus or species were detected, although relative abundance within bacterial classes did vary across treatments (Supplementary Fig. S1). For example, we observed differences between relative abundances of orders Bifidobacteriales (97% Control, 54% Vermicompost) and Actinomycetales (6% Control, 35% Vermicompost) in Actinobacteria, genera Lactobacillus (94% Control, 76% Vermicompost) and Clostridium (72% Control, 48% Vermicompost) in Bacilli and Clostridia, respectively, and between relative abundances of orders Pseudomonadales (62% Control, 28% Vermicompost) and Xanthomonadales (28% Control, 46% Vermicompost) in Gammaproteobacteria.
With respect to the active bacterial community, the differences in relative abundance at class level were less marked between treatments than those detected in the total bacterial community (Fig. 4). The phylum Verrucomicrobia, which was detected at the DNA level with insufficient relative abundance (<1%), was identified at the RNA level with some differences observed in relative abundance at the order level (Supplementary Figs S2, S3). We also detected differences between relative abundances of orders Actinomycetales in Actinobacteria and order Caulobacteriales in Alfaproteobacteria (21% Control, 9% Vermicompost) as examples. The richness indices showed a similar comparative trend in terms of predicting the number of OTUs in all soil samples, displaying those from the Vermicompost treatment the highest alpha diversity -Chao1 and Faith’s Phylogenetic Diversity- values at both the DNA and RNA levels (Supplementary Fig. S4). Beta diversity patterns in the four datasets (Vermicompost DNA, Control DNA, Vermicompost RNA, Control RNA) were examined using qualitative and quantitative similarity indices at the putative species level (Table 1). The Sorensen-Dice distance and Bray-Curtis dissimilarity indices, based on presence/absence and abundance data, respectively, showed a high degree of differentiation among bacterial communities, particularly between total and active populations in treatment Control. At the RNA level, the Bray-Curtis dissimilarity index also displayed a certain degree of similarity between treatments at the RNA level. UniFrac distances, incorporating information on relative relatedness of community members through the inclusion of the phylogenetic distances between organisms, reflected a similar trend. Both the weighted (quantitative) and unweighted (qualitative) variants of UniFrac point to a medium-to-high degree of dissimilarity in bacterial population structure both in and between treatments at the DNA and RNA levels. It is important to note that the number of denoised sequences (62.160 and 49.730 per sample of DNA and RNA, respectively) is considered an enough sequencing depth to describe a trend in both alpha and beta diversity18.
### Fungal communities
Three dominant phyla accounting for over 99% of all fungal sequences were present in the rhizosphere soil samples. The number of assigned reads was 152.643, 136.583, 128.412 and 137.465 for Control-DNA, -RNA, and Vermicompost-DNA and -RNA sets, respectively. The total and active fungal communities were dominated by two phyla, Ascomycota and Chytridiomycota, with remarkable differences between the two treatments (Fig. 5). While neither total nor active communities ascribed to Ascomycota changed significantly following each treatment (Control DNA 91.52%, Vermicompost DNA 67.93%; Control RNA 97.75%, Vermicompost RNA 74.07%), Chytridiomycota varied to some extent in terms of relative abundance in treatment Control at the RNA level (Control DNA 7.52%, Vermicompost DNA 30.30%; Control RNA not detected, Vermicompost RNA 22.45%). Fungal species belonging to phylum Basiodiomycota were only detected at the DNA level. In class terms, the main differences between treatments at the DNA level were found with respect to phylum Ascomycota, particularly in classes Sordaromycetes (Control 8.37%, Vermicompost 44.39%) and Pezizomycetes (Control 80.2%, Vermicompost 17.2%). The latter was also the predominant active fungal class in both treatments, particularly in rhizosphere soil Control (relative abundance 92%). Class Chytridiomycetes accounted for all Chytridiomycota fungi at the DNA and RNA level.
### Microbial data analysis
Analysis of similarity (ANOSIM) was used to statistically determine the effects of the initial inoculum on final rhizosphere microbial community structure after 3 months. This analysis generated an R value of 1, indicating total separation between soils at either the DNA (P = 0.098) or RNA (P = 0.094) level. Similarity percentage (SIMPER) analyses show the major peaks corresponding to the differences between the two treatments. With respect to total bacterial and fungal communities (Table 2), the OTUs assigned to 18 soil microbial classes accounted for 90% of total dissimilarity (Bray-Curtis dissimilarity = 26.5%). The members of classes Deltaproteobacteria, Betaproteobacteria and Acidobacteria accounted for roughly 25% of community variation between the two soils. With regard to active microbial communities (Bray-Curtis dissimilarity = 21.7%), 11 soil microbial classes accounted for 90% of total variation across different treatments, with Pedosphaerae (Verrucomicrobiota) and Acidobacteria, contributing to 25% of the dissimilarity between the two groups, as the most discriminant species (Table 3). To determine whether community composition affects volatile blend composition, canonical correspondence analysis (CCA) was performed. To test our hypothesis in this multitrophic system, volatiles were regarded as dependent analytical factors. CCA analysis carried out to examine the effect of total and active community composition on volatile emission resulted in a species-volatiles correlation of 0.999 and significant axes in both cases (trace = 0.234, F = 60.185, P = 0.0020), with CCA axis 1and axis 2 accounting for over 98% of variance. The results show two distinct groups corresponding to the two soils, each positively correlated with specific microbial classes at either DNA or RNA level (Fig. 6).
## Discussion
This study is based on the “integration” approach which uses previous evidence concerning the role of the below-ground rhizosphere microbiome at above-ground third trophic level. We tested N. oleander plants grown in a potting soil initially containing a sterile/non-sterile olive waste vermicompost as inoculum. When attacked by A. nerii after 3 months of growth, SPME fiber analysis revealed the presence in both treatments of a characteristic blend of plant volatiles derived from two biosynthetic classes: fatty acid catabolites and aromatic-derived products. The composition of the aromatic blend was, in qualitative and quantitative terms, generally similar in both treatments, with comparable relative amounts of 4-methoxy-benzaldehyde, benzothiazole, benzyl alcohol, dichlorobenzoic acid, alkylbenzenes and 1,2,4-trimethylbenzene (Supplementary Fig. S5). Many volatile compounds containing an aromatic ring produced by the shikimate pathway have been described in a wide range of plant species regarded as HIPVs. For instance, 4-methoxy-benzaldehyde has been described in herbivore-infested Arabidopsis thaliana plants19; in aphid species, perception of this plant-specific volatile component assists olfactory discrimination between host and non-host plants20. The release of insect-induced benzothiazole in rice and sunflower plants damaged by Tibraca limbativentris and Euschistus heros has also been described21. Benzyl alcohol has been identified in Camellia sinensis and Coffea canephora under attack from different insect feeding guilds22 and even under mechanical damage conditions23. Dichlorobenzoic acid has been described as a functional analogue of the defence hormone salicylic acid, which acts as a bioactive plant defence-inducing compound24. Many different alkyl benzenes have also been retrieved from infested plants, especially in the presence of high densities of fungi25. Finally, for example, 1,2,4-trimethyl benzene has been characterized as a Brassica oleracea HIPV when infested with Pieris rapae larvae26. The volatile blends of fatty acid derivatives contained four aliphatic compounds: the methyl ketones 2-decanone, 2-undecanone and 2-dodecanone, as well as the alkane tetradecane, all low-molecular-weight substances previously classified as HIPVs. The results obtained by Lozano et al.27 indicate that 2-decanone is involved in attracting the parasitoids Dendrosoter protuberans and Cheiropachus quadrum to their host Phloeotribus scarabaeoides. Some studies have revealed the role played by 2-undecanone in an integrated pest management strategy for tomato28 and solanaceous crops, the latter due to its capacity to alter Bactericera cockerelli behavior29. However, other studies have also shown that 2-undecanone emissions do not differ between Brassica rapa plants exposed/not exposed by Pieris brassicae to leaf herbivory30. Hervibore-induced 2-dodecanone has been detected as a tomato volatile blend component31. The alkane tetradecane, a known semiochemical for many arthropods32, has also been identified as a volatile biomarker indicating damaged flower head tissue33. The most remarkable finding of our study was that three aliphatic compounds were mainly detected after an aphid attack in plants grown in the inoculated microbial soil. Neither 2-decanone nor 2-dodecanone was detected in any of the plants of the Control treatment, at least above the detection limit of the method and in the experimental conditions described above. The other key volatile, tetradecane, was emitted by all the infected plants, either grown in Control or Vermicompost conditions, being the emission significantly higher in the last.
The biosynthesis and regulation of plant volatiles have been widely studied34. Herbivore damage usually elicits phytohormone-mediated changes in the expression of genes involved in biosynthetic pathways for the production of plant volatiles. Variables influencing plant volatile emissions are multiple, including abiotic and biotic characteristics such as plant species, variety, phenology, physiology and nutritional quality, environmental conditions such as light, temperature and moisture status, herbivore density and population growth, as well as biotic and abiotic soil stresses10. In our experiment, carried out under controlled environmental conditions, any possible impact of these stresses was limited. Plants of the same variety were grown under similar environmental conditions and under the same physico-chemical soil characteristics and no differences in plant dry biomass were detected between the two treatments. Also, even though the pre-reproductive period of a mature A. nerii can be less than 2 days35, no visual differences in aphid population size among treatments were observed similarly to that noted by other authors in comparable experiments36, notwithstanding that aphid number was not actually quantified.
Under all the standardized conditions, the contrasting effect of the initial inocula on shifts in the emission of the fatty acid-derived volatile blends could be mainly attributed to the different microbial consortia developed in each treatment. In this context, the main cause of the synthesis of fatty acid-derived volatiles has been identified as the lipoxygenase pathway which is positively regulated by phytohormones such as jasmonic acid37. In general, although phloem feeders activate the salicylic acid-dependent shikimic acid pathway, the important role played by jasmonic acid in defence against aphids has also been demonstrated38. Globally, phytohormone crosstalk has been shown to be considerably involved in the biosynthesis of plant volatiles, although the mechanism by which the microbe interacts with the plant, which subsequently affects plant-insect interaction, has not been fully elucidated39. Beneficial micro-organisms such as mycorrizae and rhizobacteria have been demonstrated to play a role in modulating plant-induced systemic resistance, which is mediated by phytohormone signaling40, 41. However, studies focusing on the role of soil microorganisms in the modification of plant volatile emission are still scarce. Previous findings have shown that non-pathogenic root-associated microbes can have a positive or a negative effect on the attraction of third trophic level organisms through changes in the composition of the blend of herbivore-induced plant volatiles14, 42. Despite the necessity for further research in order to elucidate the mechanisms underlying these contrasting effects, the role assigned to microorganisms involves changes in defence-related signaling. Given that stress responses share signalling pathways regulated by defence-related phytohormones43, soil organisms could have the capacity to modulate the synthesis of fatty acid-derived volatiles. From a multitrophic perspective, the differences in the emission of selected aliphatic compounds by plants under aphid attack initiate a volatile blend attractive to a generalist predator. Thus, C. carnea adult females exhibit some degree of preference for certain volatile blends. It is well known that volatile plant secondary metabolites such as terpenes and aromatic compounds can be detected by the olfactory system of C. carnea adults to locate suitable hosts44, 45. Although these adults are not predatory, some research evidences mechanisms to attract females and concentrated them locally in the field to increase egg-laying intensity, to get advantages of the potential of C. carnea larvae as biological control agents46. It is possible to infer from our study that C. carnea adult females respond to a volatile blend, in which the presence of aliphatic compounds is a determining factor.
However, most studies focusing on below- and above-ground interactions involve single microbes, which differ greatly from natural soil conditions, highlight the importance of evaluating the entire soil microbiome when studying microbe-plant-insect interactions. Little research has been devoted to the impact of the soil microbiome on primary metabolite production in plants, which, in turn, determines insect feeding behavior8. It is still important to determine whether the soil microbiome modulates the biosynthesis of secondary metabolites involved in plant-insect interactions. In this study, we aimed to assess whether alterations in the microbial community composition of a sterile soil following vermicompost amendment modify generalist predator responses to shifts in volatile compound emission induced by a phytophagous attack. The effects of the vermicompost-borne microbial community on both the entire and functional rhizospere microbiome structure were evident after three months. Microbiome composition structurally evolved in line with the composition of the inoculums, while the chemical characteristics of the potting soil remained unchanged. Canonical Correspondence Analysis was used to determine whether community composition affects volatile blend composition. The results show two distinct groups corresponding to the two soils, each positively correlated with specific microbial classes at either the DNA or RNA level. It was then necessary to determine whether the plant-mediated effects of microbes on above-ground herbivores are dependent on microbial community composition and the role, if any, played by the microbial physiological state. The soil samples and volatiles matched different microbial classes depending on the total or active nature of the microbial population. As mentioned above, communication is well known to occur between plants and microorganisms, in which their signalling molecules play an important role. Active microorganisms may then be critically involved in this molecular dialogue, whose precise mechanisms in herbivore-induced plant volatile emission still need to be determined. In this study, Vermicompost treatment samples were clustered with the aliphatic volatiles 2-decanone, 2-dodecanone and tretadecane, and positively correlated with active fungi ascribed to classes Chytridiomycetes and Agaricomycetes and active bacteria of classes Saprospirae, Deltaproteobacteria, Actinobacteria, Acidobacteria, Bacilli and Clostridia. Previous studies of the relationship between microbiome members and plant volatile induction by herbivory have described the effects of soil bacteria ascribed to classes Alpha- and Gamma-proteobacteria, as well as endophytic fungi, but not those of the taxa mentioned above42, 47, 48, although it should be noted that these studies focus on the role of single species. However, from a metagenomics perspective, we did not find any remarkable differences in the structure of Alpha- and Gamma-proteobacteria bacterial communities, with virtually all the bacterial species ascribed to these classes sharing the soils of the two treatments (Supplementary Fig. S3).
Under our experimental conditions, we found some evidence to show the contribution of a group of functional rhizosphere bacteria and fungi to the diversity of plant volatile patterns. Given the absence of previous data on the role of free-living classes of fungi in indirect induced plant defence, we focused on a more in-depth analysis of the bacterial population. In this regard, we were unable to identify any differences in bacterial patterns between Vermicompost and Control rhizospheres, with both soils appearing to share virtually all bacterial species ascribed to the discriminant taxa, particularly at the RNA level (Supplementary Figs S2, S3). Although some authors have debated the key role played by rare species within the entire microbiome in plant-herbivore interactions35, we cannot attribute any critical function to species ascribed to those classes identified as discriminant between treatments. On the contrary, using a metagenomic approach, we showed that the principal difference mainly related to the relative abundance of functional microorganisms. In this regard, the estimators of alpha diversity49 indicated higher within-community diversity in the Vermicompost rhizosphere with respect to that found in the Control treatment. The Chao 1 index50 and the Faith’s Phylogenetic Diversity51, based upon the number of rare OTUs and expressing the number of tree units found in a sample, respectively, confirmed that phylogenetic and functional structures of the bacterial communities differed between treatments. In addition, the beta diversity patterns, especially those including the phylogenetic distances between organisms, pointed to a medium-to-high degree of dissimilarity between the bacterial population structure in Vermicompost and Control rhizospheres.
## Concluding Remarks
In this study, the effects of below-ground microbes on indirect plant defences were evaluated. We aimed to determine whether alterations in the microbial community composition of a sterile soil following vermicompost amendment modify plant-insect interactions through shifts in volatile compound emission. We found that differences in the composition of the rhizosphere active microbiome correlate with those in the emission of selected aliphatic compounds by plants under aphid attack, which initiates a volatile blend attractive to a generalist predator. Although further research is required, our results suggest that functional interactions between soil microbes play a significant role in regulating the biosynthesis of volatile plant secondary metabolites. The functional soil microbiome is a factor which needs to be investigated in order to assess whether the plant-mediated effects of soil microorganisms on above-ground herbivores are species- or population structure-dependent.
## Materials and Methods
### Plants
The plant species used in the experiments was 3-month-old oleander (Nerium oleander L. Apocynaceae) grown in a greenhouse nursery (25–30 °C, 60–80% RH, 16:8 h L:D). The roots were washed by being dipped in sterile water to remove soil and were then individually transplanted to a pot filled with the potting soil mixture. At harvest (90 days after planting), the plants were removed and separated into shoots and roots. The shoots were oven-dried at 60 °C for 48 h and their mass determined on an analytical balance.
### Potting soil
A mixture of sand and loamy clay soil (1:1 v:v), previously sterilized by fractional sterilization (tyndallization; 100 °C, 60 min, 3 days), was used as potting medium. The soil was a calcareous loam (Typic Xerorthent)52 collected from an agricultural field (0–20 cm in depth) in Granada, Spain. Tyndallization involves killing vegetative cells and some spores at the initial heating stage; additional heat resistant spores germinate and are killed at a later heating stage. This low temperature sterilization technique preserves soil structure and quality more effectively than autoclaving at 121 °C53. The tyndallized soil characteristics were as follows: 0.9 g kg−1 SOC (soil organic carbon), 1.6 g kg−1 total N, pH (H2O) 7.5.
We used a vermicompost from olive-mill waste produced at the EEZ-CSIC facility (Granada, Spain), as described in Vivas et al.54, as microbial inoculum. In order to attain a soil organic carbon content of 30 g kg−1, 1,000 g of the soil mixture was placed in 2-l black pots and thoroughly mixed with the vermicompost at a rate corresponding to 50 g kg−1 (Vermicompost treatment). Soil amended with the same amount of vermicompost sterilized by tyndallization was used as control (Control treatment). Soil moisture content was adjusted to approximately 60% and maintained at this level by irrigation with sterilized deionized water during the experiment. Three replicates per treatment were arranged in randomized blocks in the greenhouse (25 °C, 60–80% RH, 16:8 h L:D).
### Predators
The Chrysoperla carnea Steph. (Neuroptera: Chrysopidae) larvae were supplied by Koppert Biological Systems (La Mojonera, Almería, Spain). Larvae were individually reared in Petri dishes and fed on Ephestia kuehniella Zell. (Lepidoptera: Pyralidae) eggs. Upon emergence, C. carnea adults were collected daily and kept in boxes (28 cm diameter, 15 cm high) with an ovipositing surface; they were then fed on honey:pollen (1:1, v-v) and mineral water and maintained in a controlled environment cabinet at 25 °C, 50–60% RH and 16:8 h L:D for 2–3 days. Adult C. carnea were sexed by examining the ventral abdominal tip surface. Only females were used for bioassays.
### Phytophagous insects
Aphis nerii Boy. (Homoptera: Aphididae) adults were taken from 20-y-old N. oleander plants located in Gójar, Granada, Spain. 20 individuals were reared on 1-y-old oleander plants maintained in a chamber at 25 °C, 50–60% RH and 16:8 h L:D. The plants were covered with fine mesh netting to prevent A. nerii emigration. When the plants were badly damaged by aphids, they were replaced by fresh plants after aphid migration to the healthy plants. Approximately 7 generations of aphids were produced before being used for the experiments.
### Experimental design
A closed-system Y-tube olfactometer (ID 3 cm; stem 10 cm, arms 8 cm; stem-arm angle 130°) was used to assess choice of predator C. carnea between the two treatments after a 3-month N. oleander growing period (Fig. 1). Two glass chambers (40 × 40 × 140 cm), sufficiently large to accommodate aboveground plant tissues were connected from the top to the Y-shaped glass tubing of the olfactometer by transparent polytetrafluoroethylene. An SPME fiber was inserted into each arm of the olfactometer in order to collect volatiles. Using air pressure, synthetic pure air, at an airflow rate of 1.2 l min−1 per channel55, was drawn in though the bottom of the chambers.
Firstly, uninfested plants were tested in the chambers connected to the olfactometer, and the volatiles were recovered. To infest N. oleander, 20 wingless A. nerii adults were introduced at the top of the plant. After inoculation, the plants were re-introduced into the glass chambers. The aphids were allowed to feed for 48 h, and tests were conducted three days after inoculation.
Behavioural tests of predator C. carnea were carried out under artificial light between 09:00 and 18:00 h at 28 ± 2 °C. A white circular paperboard arena was placed around the olfactometer to prevent visual disturbances. Adult C. carnea females were inserted into the single branch of the olfactometer and were left to choose between the two branches of the device, with a maximum observation period of 5 min. If the insects, which were used only once and then discarded, did not attain a length of at least 4 cm along the arm connected to the test chambers, they were excluded from the data analysis. To rule out directional bias, the olfactometer was washed in hot water, rinsed in sterilized deionized water and dried in an oven at 60 °C before each experiment. The Y-tube was also rotated 180° after each test. The position of the chambers was changed after every two tests. The pairwise experiment on the behaviour of two plants was repeated three times every other day. In each assay, approximately 7 C. carnea females were used. A total of 22 behavioural assays were carried out with C. carnea females from three different rearing groups (blocks).
### Volatiles
The volatiles emitted by N. oleander were sampled using SPME on day 3 for 9 h from 09:00 to 18:00 h. The SPME fibers (50/30 μm DVB/CAR/PDMS Stableflex 23Ga, Autosampler, 3pk, SUPELCO, Bellefonte, PA, USA) were preconditioned prior to analysis at 250 °C for 30 min. After the equilibration period, the fibers were exposed to the headspace of each Y-tube olfactometer arm. After completion of sampling, the fiber was withdrawn into the needle and inserted into the GC–MS system injection port in splitless injection mode at an injector temperature of 250 °C.
Gas chromatography (GC) analyses were conducted on a Varian 450-GC gas chromatograph fitted with a 1079 injector in split/splitless mode, a CTC Analytics CombiPal refrigerated autosampler and a Varian 240 Ion Trap as a mass spectrometer detector. A FactorFour VF-5ms (30 m × 0.25 mm × 0.25 μm) fused silica capillary column was also used.
The initial gas chromatography oven temperature was 50 °C for 5 min which was then increased to 260 °C at 10 °C min−1. It was then raised to a temperature of 300 °C at 30 °C min−1, which was maintained for 1 min with an injection of 1 μl (300 °C) in splitless mode (1 min). The carrier gas was He at 1 ml min−1. Electron impact ionization and detection in full scan (m/z 40 to 450) modes were carried out. The transfer line and trap temperatures were 290 and 210 °C, respectively. Peaks were identified by comparing the volatile sample mass spectra with spectra in the NIST08 Mass Spectral Database (MS Workstation 6.9.1. software). When necessary, the retention index (RI) was calculated for each volatile compound using the retention times of a homologous series of n-alkanes and by comparing the RI with that of compounds analyzed under similar conditions in the literature to confirm the identity of volatile compounds.
### Rhizosphere soil samples
Rhizosphere soil was collected in two steps. First, the root system was separated from the bulk soil by gentle shaking, and the remaining soil was then removed from the roots by more vigorous shaking. Soil still adhering to the roots was removed using a sterile dissecting probe and collected for use as rhizosphere soil. Root-associated soil samples from each pot were placed in separate polyethylene bags and immediately stored at −80 °C for subsequent molecular analyses.
### 16S/18S rRNA gene sequence analysis
For each rhizosphere soil sample replicate, total DNA was separately extracted from four 1 g subsamples using the bead-beating method following the manufacturer’s instructions for the MoBio UltraClean Soil DNA Isolation kit (MoBio Laboratories, Solana Beach, CA, USA). The extracts were pooled and further concentrated at 35 °C to a final volume of 20 μl, with the aid of a Savant Speedvac® concentrator. Total RNA was extracted from four 2 g subsamples of each replicate according to the manufacturer’s instructions for the MoBio RNA PowerSoil Total RNA Isolation kit (MoBio laboratories, Solana Beach, CA, USA). To remove residual DNA, the DNase I enzyme was added using the Roche RNase-Free DNase set (Roche Applied Science, Penzberg, Germany) according to the manufacturer’s instructions. The extracts were pooled and further concentrated at 35 °C to a final volume of 80 μl with the aid of a Savant Speedvac® concentrator. The cDNA was synthesized from 1–2 μg of total RNA-DNase using the Transcriptor High Fidelity cDNA Synthesis Kit according to the manufacturer’s instructions (Roche Applied Science, Penzberg, Germany). The synthesis reaction was carried out at 50 °C for 30 min. The concentration and quality of the final DNA/RNA/cDNA samples were checked by a Nanodrop® ND-100 spectrometer (Nanodrop Technologies, Wilmington, DE, USA).
The metagenome and transcriptome libraries of the bacterial 16S rRNA gene were generated in pooled soil samples using the primers S-D-Bact-0341-b-S-17/S-D-Bact-0785-a-A-21 reported by Klindworth et al.56 (non-underlined sequences) and were fused with underlined Illumina adapter overhang nucleotide sequences. To amplify V3-V4 hypervariable regions of the 16S rRNA gene, the following primer sequences were used: 5′-TCGTCGGCAGCGTCAGATGTGTATAAGAGACAGCCTACGGGNGGCWGCAG-3′ and 5′ GTCTCGTGGGCTCGGAGATGTGTATAAGAGACAGGACTACHVGGGTATCTAATCC-3′. The amplified region was approximately 464 bp. For each library, triplicate soil PCR products with unique indexes were mixed in equal nanogram quantities and sequenced on the Illumina MiSeq platform using a 2 × 250 nucleotide paired-end protocol (Era7 Bioinformatics, Granada, Spain).
Metagenomic and transcriptomic analyses of the fungal 18S rRNA gene were performed on the degenerate primers 563 f (5′-GCCAGCAVCYGCGGTAAY-3′) and 1132r (5′-CCGTCAATTHCTTYAART-3′) designed by Hugerth et al.57. To prepare libraries for Illumina sequencing, primers 563 f and 1132r were fused with the Illumina adapter overhang nucleotide sequences. The primers were used to amplify the V4 region of the 18S rRNA gene, with the amplicon expected to measure approximately 569 bp. For each library, triplicate soil PCR products with unique indexes were mixed in equal nanogram quantities and sequenced on the Illumina MiSeq platform using a 2 × 300 nucleotide paired-end protocol (Era7 Bioinformatics, Granada, Spain).
All original Illumina sequence data were deposited in the Sequence Read Archive (SRA) service of the European Bioinformatics Institute (EBI) database (BioProject ID: PRJNA313153, accession numbers SRX1795432, SRX1795541, SRX1798892, SRX1798901, SRX1795397, SRX1795517, SRX1798890, SRX1798895).
### Chemical analyses
Air-dried rhizosphere soil samples were used to determinate chemical properties. Total N and SOC were determined with the aid of the Leco-TruSpec CN elemental analyzer (LECO Corp., St Joseph, MI, USA). Total mineral content was determined by the digestion method with HNO3 65%:HCl 35% (1:3; v-v) followed by analysis using inductively coupled plasma optical emission spectrometry (ICP-OES) (ICP 720-ES, Agilent, Santa Clara, USA).
### Data analysis
The results of the chemical and volatile analyses were the means of 3 replicates. The data were subjected to factorial analysis of variance (ANOVA) using PAST (Paleontological Statistics) software program v3.1461.
Data on the behaviour of C. carnea were analysed to account for differences between treatments by using generalized linear mixed models (GLMMs). A Poisson error structure and log-link function were used to build these models, with the response variable being the count of C. carnea females located at the end of each Y-tube olfactometer branch62. We generated a set of models composed of different combinations of the fixed “treatment” and “block” factors and the random “assay” factor. By their nature, blocks should be regarded as a random factor; however, because this variable contains only three levels, a fixed factor is recommended63. We also tested a set of models with the block variable treated as a random factor and obtained the same results. The most complex of the eight plausible models we constructed, containing all possible combinations of the variables mentioned above, was the following:
$$C.\,carnea\,individuals=\alpha +{\beta }_{1}\,treatment+{\beta }_{2}\,block+{\varepsilon }_{assay}$$
where $$\alpha$$ represents the intercept of the model; $${\beta }_{1}$$ is the estimated value of the treatment effect; $${\beta }_{2}$$ is the estimated value of the block effect; and $${\varepsilon }_{{assay}}$$ is the estimated error associated with the assays carried out. Alternative models were compared using the Akaike Information Criterion (AICc) corrected for small sample size64. Models showing a difference in AICc > 2 indicate that the worst model has virtually no support and can be ruled out. The selected model was tested to account for unsuited error structure approach with DHARMa package ver. 0.1.3 written for the R environment65, given that the error structure chosen was appropriate for this type of analysis.
An additional analysis of bacterial sequences was carried out using QIIME v1.9.166. The raw files from Illumina paired-end sequencing (R1 and R2) were merged. Quality filtering was then performed using Phred67, 68, with a Phred quality score of Q20. The FASTA files obtained were brought together in a single file. With the aid of UCLUST69, an OTU clustering procedure was performed with a 97% similarity threshold. To facilitate further analysis, a representative set of sequences was selected. An OTU table in biom format70 was obtained to further analyse alfa and beta diversity using different metrics. For the phylogenetic analysis of the representative set of sequences, an alignment using PyNAST71 was carried out, the highly variable regions of sequences were removed and, finally, the phylogenetic tree was obtained with the aid of FastTree72. The phylogenetic tree in newick format is necessary to calculate alfa diversity using the PD_whole_tree metric (Faith’s Phylogenetic Diversity) and beta diversity using the unweighted_unifrac and weighted_unifrac methods.
Analysis of similarity (ANOSIM) and similarity percentage (SIMPER) analyses were performed on total and active microbial communities (OTUs, 16S and 18S ribosomal amplicon pyrosequencing) using PAST software v3.14. Distance indices were calculated with the aid of the Bray-Curtis method. Statistical significance was computed by permutation of group membership with 9,999 replicates. ANOSIM generated an R value, whose magnitude indicates the degree of separation between groups, with a score of 1 indicating total separation and 0 no separation.
The relationship between plant volatile blends and rhizosphere microbiome composition (OTUs, 16S and 18S ribosomal amplicon pyrosequencing) was determined by canonical correspondence analysis in CANOCO 4.573. OTU patterns were fitted to volatile blends by using the Monte Carlo permutation test (499 permutations) at a 95% confidence level. Community similarities were graphed by using an ordination triplot technique with scaling focused on inter-sample differences.
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## Acknowledgements
This work was supported by ERDF-cofinanced grant RECUPERA 2020 from the Spanish Ministry of Economy and Competitiveness-CSIC and Project AGR 1419 from Junta de Andalucía, Spain. E. Rodríguez held a postdoctoral contract (DOC-INIA program) granted by Spanish National Institute for Agricultural and Food Research and Technology (INIA) and the European Social Fund. The volatiles and chemical analyses were made at the Scientific Instrumentation Service, EEZ-CSIC, Granada, Spain. We would also like to thank Dr. Fernando Reyes from Fundación Medina for his help in discussing the results of the analysis of volatiles and Michael O´Shea for assisting in the translation of the original manuscript into English.
## Author information
### Affiliations
1. #### Estación Experimental del Zaidín (EEZ), CSIC, 18008, Granada, Spain
• Emilio Benítez
• , Daniel Paredes
• , Diana Aldana
• , Rogelio Nogales
• , Mercedes Campos
• & Beatriz Moreno
2. #### Estación Experimental Las Palmerillas, Cajamar, Almería, Spain
• Mónica González
3. #### Instituto de Investigación y Formación Agraria y Pesquera, Centro IFAPA La Mojonera, Almería, Spain
• Estefanía Rodríguez
### Contributions
E.B., B.M., E.R. and M.C. conceived the study and planned the experiments. E.B., B.M., D.P. and D.A. performed the experiments. E.B., B.M., D.P., M.C. and E.R. analysed data. R.N., M.R., D.A. and E.R. contributed data. E.B., B.M. and D.P. wrote the manuscript. All authors interpreted the results and reviewed the manuscript.
### Competing Interests
The authors declare that they have no competing interests.
### Corresponding author
Correspondence to Emilio Benítez.
|
# Kerodon
$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
Corollary 5.5.8.8. Let $\operatorname{\mathcal{C}}$ be a simplicial category having the property that, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$ is an $\infty$-category. Let $\operatorname{\mathcal{C}}'$ denote the simplicial subcategory of $\operatorname{\mathcal{C}}$ having the same objects, with morphism simplicial sets given by $\operatorname{Hom}_{\operatorname{\mathcal{C}}'}(X,Y)_{\bullet } = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }^{\simeq }$. Then the inclusion of simplicial categories $\operatorname{\mathcal{C}}' \hookrightarrow \operatorname{\mathcal{C}}$ induces an isomorphism of $\infty$-categories $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}') \simeq \operatorname{Pith}( \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}) )$.
Proof. Let $\sigma$ be an $n$-simplex of the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$, which we identify with a simplicial functor $F: \operatorname{Path}[n]_{\bullet } \rightarrow \operatorname{\mathcal{C}}$ carrying each $i \in [n]$ to an object $C_{i} \in \operatorname{\mathcal{C}}$. If $T \subseteq [n]$ is a nonempty subset having smallest element $i$ and largest element $k$, let us write $F(T)$ for the corresponding vertex of the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C_ i, C_ k)_{\bullet }$. If $S \subseteq T$ is a subset containing $i$ and $k$, let us write $F(S \subseteq T): F(T) \rightarrow F(S)$ for the corresponding edge of the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C_ i, C_ k)_{\bullet }$. Let us abuse notation by identifying $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}')$ with a simplicial subset of $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$. Unwinding the definitions, we see that $\sigma$ is contained in $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}')$ if and only if the following condition is satisfied:
$(1)$
For every inclusion $S \subseteq T$ of nonempty subsets of $[n]$ having the same smallest element $i$ and largest element $k$, the edge $F(S \subseteq T): F(T) \rightarrow F(S)$ is an isomorphism in the $\infty$-category $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( C_ i, C_ k)_{\bullet }$.
Using the thinness criterion of Proposition 5.5.8.7, we see that $\sigma$ belongs to the pith $\operatorname{Pith}( \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}))$ if and only if the following a priori weaker condition is satisfied:
$(2)$
For every triple of elements $0 \leq i \leq j \leq k \leq n$, the edge
$F( \{ i,k \} \subseteq \{ i,j,k\} ): F( \{ i, j, k\} ) \rightarrow F( \{ i, k \} )$
is an isomorphism in the $\infty$-category $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C_ i, C_ k)_{\bullet })$.
To complete the proof, it will suffice to show that $(2) \Rightarrow (1)$. Assume that $(2)$ is satisfied, and suppose that we are given nonempty subsets $S \subseteq T$ of $[n]$ having the same smallest element $i$ and largest element $k$. We wish to show that $F(S \subseteq T)$ is an isomorphism in the $\infty$-category $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( C_ i, C_ k)_{\bullet }$. Since the collection of isomorphisms contains all identity morphisms and is closed under composition (Remark 1.3.6.3), we may assume without loss of generality that the difference $T \setminus S$ contains exactly one element $j$. Set $S_{-} = \{ s \in S: s < j \}$ and $S_{+} = \{ s \in S: s > j \}$. Let $i'$ be the largest element of $S_{-}$, and let $k'$ denote the smallest element of $S_{+}$. Unwinding the definitions, we see that the edge $F(S \subseteq T)$ is the image of $F( \{ i',k' \} \subseteq \{ i',j,k'\} )$ under the functor
$\operatorname{Hom}_{\operatorname{\mathcal{C}}}( C_{i'}, C_{k'})_{\bullet } \xrightarrow { F(S_{+}) \circ \bullet \circ F(S_{-}) } \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C_ i, C_ k)_{\bullet },$
and is therefore an isomorphism by virtue of assumption $(2)$. $\square$
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# Duality between orbifold and quasi-Hopf algebra (twisted quantum doubles)
A quick Question:
Background:
It is known (in theoretical physics) that the algebraic framework underlying discrete H gauge theories with 2+1D Chern-Simons term is the quasi Hopf algebra $D^\omega(H)$, i.e. the Chern-Simons term introduces a 3-cocycle $\omega \in H^4(BH,\mathbb{Z}) \simeq H^4(H,\mathbb{Z}) \simeq H^3(H,U(1))$ in the cohomology group on the Hopf algebra $D(H)$. People in theoretical physics also call the quasi Hopf algebra $D^\omega(H)$ as another name: twisted quantum doubles, such as A Kitaev's (of Caltech) Annals of Physics 303, 2 (2003), Annals of Physics 321, 2 (2006). The background understanding of these topics (to me) would go to Dijkgraaf-Witten theory original paper.
My question here is inspired by the observation in this arXiv paper published in Nucl.Phys. B. It stated that: "From the point of view of conformal field theory it is of interest to mention that the fusion rules of $D^\omega(\mathbb{H}_8)$ for p = 1 coincide with the level 1 SU(2)/($\mathbb{Z}_2 \times \mathbb{Z}_2$)-orbifold (cited a paper by Dijkgraaf, Vafa, Verlinde, Verlinde) after modding out the appropriate $\mathbb{Z}_2$ generated by 1 (see Table 2 here)). Apparently, the algebraic structure of such non-holomorphic orbifolds is still determined by the ‘holomorphic’ Hopf algebra, be it deformed by a non-trivial 3-cocycle. To our knowledge, this has not been noticed before."
A detailed Question:
It seems to me that there may have some duality between: $$\text{quasi Hopf algebra } D^\omega(\mathbb{H}_8) \text{ for p = 1} \leftrightarrow \text{level 1 SU(2)/(\mathbb{Z}_2 \times \mathbb{Z}_2) orbifold}$$ Here $p = 1$ is the 3-cocycles labeled of $H^3(\mathbb{H}_8,U(1))=\mathbb{Z}_8$ for $p$(mod 8) in $\mathbb{Z}_8$. How about other 7 classes other than $p=1$ in $p$(mod 8)?
• Are there other some dualities exist for $$D^\omega(\mathbb{H}_8) \leftrightarrow \text{? orbifold}$$ $$D^\omega(D_8) \leftrightarrow \text{? orbifold}$$ $$D^\omega(\mathbb{Z}_2^3) \leftrightarrow \text{? orbifold}$$ What is the general relation (if any, start with a finite group $H$)? $$D^\omega(H) \leftrightarrow \text{? orbifold}$$
$D_8$ is a dihedral group with 8 group elements. $D^\omega(D_8)$ should have three labels of $p_1$,$p_2$,$p_3$ from $H^3(D_8,U(1))=\mathbb{Z}_4 \times \mathbb{Z}_2 \times \mathbb{Z}_2$. And $D^\omega(\mathbb{Z}_2^3)$ should have 7 labels of $p_j$ from $H^3(\mathbb{Z}_2^3,U(1))=\mathbb{Z}_2^7$.
ps. Excuse me that my mathematical background is not equivalent to a math PhD (but trained in physics), but this should be a research level question in mathematical physics. Please feel free giving comments/answers. Thank you for all who reply and support!
-
The question of finding orbifold constructions of fusion rings is very natural and interesting. On the other hand, I don't see much evidence of a general duality here, since the rings are relatively small. – S. Carnahan Dec 31 '13 at 22:50
But are there some known examples appear between the twos? e.g. In my post I had provided one example. Examples are fine, it needs not to be very general. Physicists appreciate (many) examples more than a theorem. :) – Idear Dec 31 '13 at 22:54
See also this post Phy.SE on orbifolds of SU(2)/G' and SO(3)/G' if there are available data in the literature, please let me know. Many thanks! – Idear Dec 31 '13 at 22:56
What it seems like you mean by duality is that the representation categories of $D^\omega(\mathbb H_8)$ for some $\omega$ and $SU(2)/(\mathbb Z_2 \times \mathbb Z_2)$ are described by the same fusion ring. To clarify then, it seems like what you are asking is this: Given $G,H$ finite groups and $\omega\in H^3(H,U(1))$, is there a G-orbifold theory with fusion ring isomorphic to the fusion ring for $D^\omega (H)$ and if so, what is there relationship? – Matthew Titsworth Jan 2 '14 at 4:22
The reason that I ask is that below, Marcel uses Muger's result to provide a $G$-orbifold CFT whose representation category is the representation category of $D^\omega(G)$. However, the statement from which you draw your inital observation goes no further than saying that the two are Grothendieck equivalent. – Matthew Titsworth Jan 2 '14 at 4:39
Regarding the general question, from the point of view of conformal field theory there is a rather trivial way to obtain (some) $D^\omega(G)$. Namely, the representation category of the $G$-orbifold of a holomorphic (trivial representation category) rational conformal field theory is $\mathrm{Rep}(D^\omega(G))$ for some $[\omega]$, see Corollary 3.6. in http://arxiv.org/abs/0909.2537
But I have now idea if as Scott pointed out as a comment all finite groups $G$ (I suppose yes) can be obtained this way; one has to find a holomorphic theory with an action of $G$, for example the Moonshine CFT for the Monster group etc. Then for a given $G$ I also have no idea which $[\omega]$ arise this way.
I think it also follows conversely, that if a CFT has $\mathrm{Rep}(D^\omega(G))$ as representation category, then it is a $G$-orbifold of a holomorphic theory. But then I don't understand the non-holomorphic examples the op mentioned. In the non-holomorphic examples you have to mod something out to become (the dual of) $D^\omega(G)$
@ Marcel: I thought the $D^\omega(H)$ and (if any) its corresponding $G$-orbifold, (such as the example in my post), the $H$ and $G$ are not necessarily the same groups? Are you identifying $H=G$ for some cases? Thanks. – Idear Jan 1 '14 at 3:21
Any finite group embeds in a sufficiently large symmetric group, and hence in the automorphism group of a sufficiently large tensor product of $E_8$ CFTs. – S. Carnahan Jan 1 '14 at 10:39
@ S. Carnahan, thanks, can you rephrase automorphism group and $E_8$ statement to the context of our posted question? (I could not fully grasp, are you teaching me something?) – Idear Jan 1 '14 at 20:10
@ Marcel, do you know any examples of orbifold describing $D^\omega(\mathbb{Z}_2^3)$, $D^\omega(\mathbb{H}_8)$ or $D^\omega(D_8)$? – Idear Jan 2 '14 at 1:05
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# Strings from domain walls in supersymmetric Yang-Mills theory and adjoint QCD
@article{Anber2015StringsFD,
title={Strings from domain walls in supersymmetric Yang-Mills theory and adjoint QCD},
author={Mohamed M. Anber and Erich R. Poppitz and Tin Sulejmanpasic},
journal={Physical Review D},
year={2015},
volume={92},
pages={021701}
}
• Published 27 January 2015
• Physics
• Physical Review D
We study strings between static quarks in QCD with nf adjoint fermions, including N = 1 supersymmetric Yang-Mills (SYM), in the calculable regime on R-3 x S-L(1), which shares many features with the XY-spin model. We find that they have many qualitatively new features not previously known. The difference from other realizations of Abelian confinement is due to the composite nature of magnetic bions, whose Dirac quantum with fundamental quarks is two, and to the unbroken part of the Weyl group…
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