text:
function BayesianGradebook

In the US and many other educational systems, students are typically given many assessments (homework problems, projects, exams, etc) that are marked with individual scores. These scores are usually averaged (often with weights) to arrive a course score.

It's not a very scientific process. One way of injecting more clarity (if not "fairness") is to appeal to Bayesian statistics. How might that look?

### Bayes' theorem

Suppose a student's "true ability" is represented as a number $\theta$ in the interval $[0,1]$. We start with a prior belief about how $\theta$ is distributed for an individual, represented as the probability density $P(\theta)$. Now let $x$ be an observation--that is, a score on an assessment. Bayes' theorem tells us how to update our beliefs about $\theta$ in light of the new evidence. Mathematically, the new probability distribution is conditioned on the observation, which we write as $P(\theta | x)$ and call the posterior distribution.

Elementary probability gives us $P(\theta|x) = P(\theta,x)/P(x)$, where $P(\theta,x)$ is the joint probability of $\theta$ and $x$ simultaneously. Because a similar expression holds for the opposite conditional $P(x|\theta)$, we rearrange to get Bayes' theorem: $$P(\theta|x) = \frac{ P(x|\theta) P(\theta)} { P(x) }$$ The quantity $P(x|\theta)$ is called the likelihood: for each possible $\theta$, what is the probability that we would observe $x$? Where this likelihood is small, the posterior will be greatly reduced.

It's worth noting that we treat the posterior $P(\theta|x)$ as a distribution in $\theta$, and the quantity $P(x)$ in the denominator of Bayes' theorem can be seen (and computed!) simply as a normalization needed to make the posterior a probability distribution.

### Prior distribution

Here is a chebfun for $\theta$.

theta = chebfun('x',[0 1]);

We define expected value and variance functionals:

E = @(f,prob) sum(f.*prob);
Var = @(f,prob) E( (f-E(f,prob)).^2, prob );

We now choose a prior distribution to initialize our belief about ability $\theta$. Good practice would be to use past empirical data, but we will idealize it as a truncated normal distribution with a fairly large variance.

phi = @(mu,sigma) exp( -((theta-mu)./sigma).^2/2 );
prior = phi(.7,.3);  prior = prior/sum(prior);
plot(prior,'linewidth',2)

### Likelihood function

Given a "true ability" value $\theta$, we can model the probability of observing a score $x$ also with a truncated normal: $$P(x|\theta) = \frac{ \phi_{\theta,\sigma}(x) }{ q(\theta,\sigma)},$$ where $\phi_{\theta,\sigma}$ is the Gaussian function as above with mean $\theta$ and variance $\sigma^2$, and $q$ provides normalization. Because $\phi_{\theta,\sigma}(x) = \phi_{x,\sigma}(\theta)$ for the Gaussian, we can assemble the likelihood function rather easily.

We're going to treat $\sigma$ as a fixed parameter, though of course one could create a Bayesian estimator for it as well.

sigma = 0.06;
q = chebfun( @(theta) sum( phi(theta,sigma) ), [0 1], 'vectorize' );
likelihood = @(x) phi(x,sigma)./q;

### Update process

We will encapsulate the Bayesian update process in a function for repeated use below.

    function belief = bayes(scores)
belief = prior;
m = length(scores);
for k = 1:m
b = belief(:,k).*likelihood(scores(k));
b = b / sum(b);

belief(:,k+1) = b;
Mu(k) = E(theta,b);
Sig2(k) = Var(theta,b);
[~,Mode(k)] = max(b);
end

fprintf('Method       %6s %6s %6s %6s\n','m-3','m-2','m-1','m')
fprintf('------------------------------------------------\n')
fprintf('Bayes Mode    %6.3f %6.3f %6.3f %6.3f\n',Mode(m-3:m))
fprintf('Bayes Mean    %6.3f %6.3f %6.3f %6.3f\n',Mu(m-3:m))
fprintf('Std dev       %6.3f %6.3f %6.3f %6.3f\n',sqrt(Sig2(m-3:m)))

end

### Poor student

We start with a student who has a cluster of fairly low scores.

scores = [.55 .67 .62 .66];

Now we apply Bayes estimation. The output is explained below.

belief = bayes(scores);
Method          m-3    m-2    m-1      m
------------------------------------------------
Bayes Mode     0.556  0.612  0.614  0.626
Bayes Mean     0.556  0.612  0.614  0.626
Std dev        0.059  0.042  0.034  0.030

The distributions show how confidence in estimating the student's ability increases with each new score.

clf, plot(belief,'linewidth',2)
xlabel('\theta')
ylabel('P(\theta|x)')

When we want to condense a distribution into a 'best guess' for the ability $\theta$, there are two obvious choices. One is to take the mode of the distribution; that is, the value of $\theta$ that maximizes probability. However, the standard (and probably better) estimator is to use the expected value of $\theta$. Both are shown in the output above and compared to the traditional method of a running average.

In this case, the different grading methods do not differ in any meaningful way. We do get a little extra information, though: the confidence with which we believe our estimates, as reported by the standard deviation of the variable $\theta$.

### Good student

It might seem like the Bayes method could hurt a very good student, because of the initial prior that assumes a modest ability. In fact, these students turn out to be the chief beneficiaries of the Bayes approach.

If we shift all the scores in the previous example by 30 percentage points, the running averages simply shift too.

scores = 0.3 + scores
belief = bayes(scores);
scores =
Columns 1 through 3
0.850000000000000   0.970000000000000   0.920000000000000
Column 4
0.960000000000000
Method          m-3    m-2    m-1      m
------------------------------------------------
Bayes Mode     0.845  0.915  0.922  0.940
Bayes Mean     0.847  0.915  0.922  0.938
Std dev        0.060  0.043  0.037  0.032

As you can see, the Bayes estimator does start out a bit more skeptical than the running average, but ends up giving the student a higher score.

The reason is the boundary.

cla, plot(belief,'linewidth',2)

Because it's impossible to get a score greater than 1, scores close to the upper limit are shortchanged in the traditional method. Looking back at the definition of the likelihood function, we see that the factor $1/q$ multiplies the normal distribution centered at the score $x$. Because $q$ represents normalization of a truncated Gaussian, it gets smaller near the endpoints, and its reciprocal is larger.

clf, plot(1./q,'linewidth',2)

As a result, when $x$ is large enough to have some "explanatory energy" at the boundary, our belief in the higher values is amplified to compensate for the inability to observe scores greater than 1.

### The comeback kid

It's not uncommon for a student to do much worse on the first assessment than on the rest. Such students sometimes ask the instructor to drop the bad grade as an aberration. Let's do that experiment.

scores(1) = 0.72
belief = bayes(scores);
scores =
Columns 1 through 3
0.720000000000000   0.970000000000000   0.920000000000000
Column 4
0.960000000000000
Method          m-3    m-2    m-1      m
------------------------------------------------
Bayes Mode     0.719  0.843  0.870  0.896
Bayes Mean     0.719  0.844  0.872  0.898
Std dev        0.059  0.043  0.037  0.033

The Bayes method suggests that a small adjustment might be justified. The first score, by itself, isn't all that unlikely from someone of considerably higher ability. Interestingly, the standard deviation suggests we have just as much confidence in this case as with the more reliable good student.

### You gotta have faith

Suppose we are now a lot less confident in how well the outcome represents ability. This could be appropriate for homework problems, for instance, which are typically of low stakes and for which the student could get a lot of extra help. Or, it might represent some empirical observation about the spread of scores on one exam.

sigma = 0.15;
q = chebfun( @(theta) sum( phi(theta,sigma) ), [0 1], 'vectorize' );
likelihood = @(x) phi(x,sigma)./q;

We now have a student with mostly good scores and a few clunkers.

scores = [ 0.88 0.90 0.46 0.86 0.93 0.61 0.95 0.89 0.84 0.76 ];
belief = bayes(scores);
Method          m-3    m-2    m-1      m
------------------------------------------------

end


text:
Example for Logistic Regression

#1

Is it possible reproduce the example:

http://www.portalaction.com.br/analise-de-regressao/412-estimacao-dos-parametros-do-modelo

using the Logistic Regression node in KNIME? Obtain the same results?

0 Likes

#2

Hi @Bryan - I don’t know any Portuguese, but I can take a guess at the overall meaning with some help from Google Translate.

This looks like a maximum-likelihood estimation problem, where coefficients and Fisher information are being calculated manually.

The Logistic Regression learner in KNIME is used to predict a target nominal column. So, if I’m understanding the linked page correctly, the answer to your question is no. If I were trying to recreate the analysis, I’d personally use R - or maybe even do it by hand at first.

1 Like

format: plain


text:
Identifying Acids and Bases Video Lessons

Concept: Identifying Binary Acids.

Problem: Complete the following sentences about strong and weak acids and bases. Strong acids are generally either hydrohalic acids (ex. _____) or oxoacids in which the number of oxygen atoms exceeds the number of ionizable _____ by 2 or more (ex. HNO3). Weak acids are usually one of the following: 1. hydrofluoric acid, _____, 2. acids in which the proton is not bonded to an oxygen atom or a halogen (ex. _____), 3. _____ where the number of oxygen atoms equals or only exceeds by 1 the number of ionizable _____ (ex. _____) or 4. _____ (ex. _____). Strong bases are _____ of group _____ metals (ex. _____) or larger group _____ metals (ex. _____), or oxides of the same metals. Weak bases are usually compounds containing a nitrogen atom with _____ bond(s) and 1 lone pair(s) (ex. _____).

FREE Expert Solution
96% (158 ratings)
Problem Details

Complete the following sentences about strong and weak acids and bases.

Strong acids are generally either hydrohalic acids (ex. _____) or oxoacids in which the number of oxygen atoms exceeds the number of ionizable _____ by 2 or more (ex. HNO3).

Weak acids are usually one of the following: 1. hydrofluoric acid, _____, 2. acids in which the proton is not bonded to an oxygen atom or a halogen (ex. _____), 3. _____ where the number of oxygen atoms equals or only exceeds by 1 the number of ionizable _____ (ex. _____) or 4. _____ (ex. _____).

Strong bases are _____ of group _____ metals (ex. _____) or larger group _____ metals (ex. _____), or oxides of the same metals.

Weak bases are usually compounds containing a nitrogen atom with _____ bond(s) and 1 lone pair(s) (ex. _____).

What scientific concept do you need to know in order to solve this problem?

Our tutors have indicated that to solve this problem you will need to apply the Identifying Acids and Bases concept. You can view video lessons to learn Identifying Acids and Bases. Or if you need more Identifying Acids and Bases practice, you can also practice Identifying Acids and Bases practice problems.

What is the difficulty of this problem?

Our tutors rated the difficulty ofComplete the following sentences about strong and weak acids...as medium difficulty.

How long does this problem take to solve?

Our expert Chemistry tutor, Sabrina took 8 minutes and 29 seconds to solve this problem. You can follow their steps in the video explanation above.

What professor is this problem relevant for?

Based on our data, we think this problem is relevant for Professor Montano's class at UIC.


text:
## anonymous one year ago Find thw anti-derivative of $f(x)=\frac{ -1 }{ x^2 }e ^{1/x}$

1. johnweldon1993

$\large -\int \frac{e^{1/x}}{x^2}$ Make a u-sub of 1/x

2. johnweldon1993

Oh...what have you learned?

3. anonymous

How do you write in this format? f(x)=F(x)+C

4. zepdrix

$\Large\rm \int\limits f(x)dx=F(x)+C$Where F(x) is the anti-derivative of f(x). Did you understand the u-substitution?

5. anonymous

Is there another way to solve it without using the u-substitution?

6. freckles

i can't think of a better way

7. anonymous

The reason that I ask that was because the question was at the beginning of antiderivatives/integration. I learned the u-substitution as a last topic...

8. ganeshie8

you may use advanced guessing what is the derivative of $$\large e^{1/x}$$ ?

9. anonymous

e^1/x

10. ganeshie8

what do you know about chain rule

11. anonymous

Chain rule applies to composite function. It describe of outer evulated at inner, times derivative of inner

12. johnweldon1993

So using that logic...if we consider the "inner" as 1/x The derivative of $$\large e^{1/x}$$ would be $$\large \frac{d(1/x)}{dx} \times e^{1/x}$$ So what is the derivative of $$\large 1/x$$ ?

13. anonymous

lnx?

14. freckles

you found the antiderivative for positive x

15. freckles

that you found the antiderivative of 1/x for positive x the question was to differentiate

16. ganeshie8

derivative if lnx is 1/x derivative of 1/x is not lnx

17. anonymous

ooh... derivative of 1/x is -1/x^2 ...

18. johnweldon1993

Correct...and to finish up from the previous post $\large \frac{d}{dx}e^{1/x} = e^{1/x} \times \frac{d}{dx}\frac{1}{x}$ Since we just found the later $$\large \frac{d}{dx} \frac{1}{x} = -\frac{1}{x^2}$$ we have $\large \frac{d}{dx}e^{1/x} = -\frac{e^{1/x}}{x^2}$ right? Now what do you notice?

19. anonymous

So F(x)= e^{1/x} +C ?

20. johnweldon1993

Perfect! :)

21. anonymous

Thank you! :) I got it now. I was messed up b/w derivatives and anti-derivatives. My bad...

22. johnweldon1993

Yeah sometimes they can be hard to keep straight...but as long as you can derive them you're all set!


text:
# Math Help - Coordinates

1. ## Coordinates

M (3,4) is the midpoint of Line JK, where J has coordinates (1,2) Find the coordinates of K. Please explain. Thank you!


text:
Author
Juan Camilo Henao Londono
AbstractTemplate to be used for the advanced laboratory report of the University of Duisburg-Essen


text:
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# Can you comment on my essay?

Author Message
Intern
Joined: 25 Nov 2013
Posts: 3
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Can you comment on my essay? [#permalink]

### Show Tags

25 Nov 2013, 07:28
Modernization

The issue of globalization has made many social debates arguing for and against it. Some conventional minds and figures see it as an attacking on the grandparents culture. Other, who mostly travelled outside, consider it as something banality. Yet, the debates and arguments will continue, and modernization too whether we accept it or not as if we do not admit it someone else will do.

The first to think about is to define what tradition means. Meaning to keep the habits, behaviors, values and customs in a specific society or group exactly the same as they were in last ages. For instance, the tradition may represented through the closing styles which surely varies from Egypt to Brazil to India. Personally, I do and do not argue for tradition. Supporting the positive effects that it'll have if appreciate and respect our heritage from grandparents, I agree on the traditional perspectives. As it is our national identities which seem non-replaceable many times. We cannot eliminate or hide our culture, beliefs, and values because they are our national identity; because we are nothing with out it. Yet, keeping originality does not necessarily acquire our insulation from the world.

Given the last concept of traditional heritage, we will investigate the seemingly contrasting concept; modernization or globalization. Using simple words, it means to delete and cancel all the old-fashioned cultural boarders among the nations. It is to support the rights of freedom of belief, and getting information, and dependable individual decision. No doubt, this is sometimes risky or ambiguous in results viewpoints. Yet, knowing what the other does and believes is germane in this era. So, to assure having and expecting the positive site results, we should make the necessary compromise weighing the points of weaknesses/strengths if modernization foments, or tradition reserves.

It is hard to give accurate position for or against modern things and manners as there is a wide and spread spectra from positive to negative application. If we believe just in the dogmatic conventional preservance, we would never travel using airplanes, or notebooks because just another nation was the first to discover it. On the opposite side, we can not credit all the modern and strange values that represent other society with other history and another identity.

In conclusion, we should combine the social traditional heritage with the what can be applied from the modernized discoveries and findings. Tradition will stay as a must because it is our nationality and what we can and have. Yet, modernization cannot be avoided or disconsidered in this age.
________________________________________
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Re: Can you comment on my essay? [#permalink]

### Show Tags

25 Nov 2013, 08:24
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Joined: 25 Nov 2013
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Re: Can you comment on my essay? [#permalink]

### Show Tags

01 Dec 2013, 02:43

I did read the page, but I am running for GRE and I came here to share ideas and thoughts about AWA .

In any field of endeavor, it is impossible to make a significant contribution without first being strongly influenced by past achievements within that field.
Internal motives are crucial.

Internal motives are germane.

There is an old quote saying Gardens are not made by sitting in the shade. Honda cars in Japan which carry the name of the founder are evidence that contributing does not depend basically on the past achievements influential effects. They suffered from failure many times but they never quieted. Success depends generally on the basic talents of an individual or teamwork. Yet, the contributions will sometimes make sometimes more work-bearer and resilient.
For one thing I disagree with the claim that a significant contribution should be preceded by strong inspiration within that field, is that the process of contribution depends basically and mostly on the personal potentials of the individuals and team groups. These contributors may and sometimes may not have been influenced by past achievements within their interest area. If they have large potentials and skills doing their work, probably, they will not influencing effects from the past endeavor contributors. For one instance, many of those who may belong to the past achievements have not influenced by the people who preceded themselves. Taha Hussein, the dean of Arab art, and Helen Keler are clear examples for that, though the Ms Keler was superior in suffering. Both of them were blind, but their blindness did prevent them from learning achievement. Both of them did not put their huge and massive endeavors because someone else from the past did that and they just liked that ideal. They have the internal motives for success which are far more important than the external ones. Past achievements belong to those external motives for success.
Following the internal motives, there is no guarantee that the person getting involved in a new field has known and experienced the last achievements influence. Sometimes, we spend large interval of time with a specific field of activity and then we change our minds to do totally different work kind from that we know. In that case, we are already unaware of the past contributions effect in that field, however that is not necessarily will be a barrier for our endeavors success. If we have a huge passion for the work we want and like to do, then whatever the challenges against us, we are going to make something brilliant. One example was my undergraduate supervisor which had his PhD from Hungary. After his graduation from Minia University in Egypt, he was employed to teach a little class aiding the faculty professors. He spent half of his salary on transport from Cairo to Minia and back. The other half he was spending on buying periodicals and journals. He had done that for 8 years. After that, he obtained his MS degree and was able to travel to Hungary. He had no experience about what he is going to do, but he has great passion for it.
Adding to the required passion, and internal motives, the work success requires variable sets of skills. Among them, we can say emotional labor (getting in love with the work), resilience (the ability to recover from probable failure), persistence (keep going and not quieting), ambition, politeness, cooperation, teamwork, negtiotation, argumentation, creative and critical analysis, and analytical writing. However I have never read that last contributor effect should be present among those skills to shine. In his book linchpin Seth Godin argues doing art and giving gifts is the most part of achieving successful achievement witing the work environment.
On the opposite position of giving gifts, doing art, passion, and internal motives, it holds true sometimes that we need external inspiration to achieve good success in our duties and tasks. The last contributions effect may sometimes recover and force us to think positively about our abilities and the possibilities we have. Thus we can figure out the exact positions we are putting our feet assessing where we should forward our next step. Many times when I feel that I will never be able to do that, I just to get some inspiration fromm external sources including experiences and achievements from the past times.
In conclusion, I suppose the past experiences and contributions dont necessarily have the superior effect on our prospective contributions within any area of interest. Because it depends on us who we work currently, our passion, our gifts, and our hard work. However, I also support that we will need them and other external sources of inspiration.

Thank you for passing on.
Re: Can you comment on my essay?   [#permalink] 01 Dec 2013, 02:43
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Topic

# Curie–Weiss law

About: Curie–Weiss law is a(n) research topic. Over the lifetime, 3501 publication(s) have been published within this topic receiving 60002 citation(s).

##### Papers
More filters
Book
04 Sep 1985
Abstract: A general theory of spin fluctuations and thermodynamical properties of itinerant electron magnets is developed, interpolating between the weakly and strongly ferromagnetic limits. A unified expression is given for the Curie temperature and the physical meaning of the Curie-Weiss magnetic susceptibility is discussed. As new phenomena derived from this theory the temperature-induced local magnetic moments as observed in CoS2, CoSe2, etc. and peculiar magnetic and thermal properties of nearly ferromagnetic semiconductors such as FeSi are discussed.

1,267 citations

Journal ArticleDOI
Abstract: A criterion is proposed for determining the onset of ferromagnetism in a material as its temperature is lowered from a region in which the linearity of its magnetic moment versus field isotherm gives an indication of paramagnetism. Within the limits of validity of a molecular field treatment, the Curie temperature is shown to be in general indicated by the third power of the magnetization being proportional to the internal magnetic field. The method has been employed to redetermine the Curie point of nickel from the data of Weiss and Forrer, of ${\mathrm{Fe}}_{3}$${\mathrm{O}}_{4}$ from the data of Smith and of some alloys from the data of Kaufmann and his collaborators and the author.

969 citations

Journal ArticleDOI
TL;DR: Analytical expressions for the Curie temperature and the magnetization in the limit of low carrier density are derived, obtaining excellent quantitative agreement with Monte Carlo simulation results and good qualitative agreement with experimental results.
Abstract: We theoretically study the development of spontaneous magnetization in diluted magnetic semiconductors as arising from a percolation of bound magnetic polarons. Within the framework of a generalized percolation theory we derive analytic expressions for the Curie temperature and the magnetization in the limit of low carrier density, obtaining excellent quantitative agreement with Monte Carlo simulation results and good qualitative agreement with experimental results.

603 citations

Journal ArticleDOI
Abstract: The change of the saturation magnetization of Fe films with the pressure of nitrogen during deposition ranging from 2 × 10−5 to 7 × 10−3 Torr has been investigated systematically. We found a new magnetic material which has the highest saturation magnetization at room temperature, 2050 G, among those of all the magnetic materials. This was attributed to Fe16N2, which has a bct structure, and the magnetic moment associated with Fe atoms of Fe16N2 was deduced to be 3.0 μB.

552 citations

Journal ArticleDOI
Abstract: The first six coefficients in the expansion of the susceptibility χ, and its inverse, χ −1, in ascending powers of the reciprocal temperature, have been determined for the Heisenberg model of a ferromagnetic, for any spin value, S, and any lattice. The first five coefficients appropriate to the magnetic specific heat, C, have also been found. For the body-centred and face-centred cubic lattices, the χ and C coefficients are tabulated for half-integral S from 1/2 to 3. From these coefficients estimates have been made of the reduced Curie temperatures, θs c= k T c/J. It is found that for the simple, body-centred and face-centred cubic lattices the formula reproduces the estimated Curie temperatures fairly accurately. Here X=S(S+1) and z is the lattice coordination-number. It is found that, suitably scaled, the theoretical curves for inverse susceptibility against temperature above the Curie point are rather insensitive to the spin value and to the precise lattice structure. The ratio of their initial to the...

482 citations

##### Network Information
###### Related Topics (5)
Magnetization

107.8K papers, 1.9M citations

94% related
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86% related
##### Performance
###### Metrics
No. of papers in the topic in previous years
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202112
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201916
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201736
201637


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# Root finding using bisection method in Python

This code below is supposed to calculate the root of a function using bisection method. I am a newbie to Python, so I do not know how to raise errors. Also, I would like to have a general review on this algorithm design, where I could have optimised, utilised some tricks or any sort of improvements. I feel like I am really not using the full funcionality of Python somehow.

Any comments on how to write a succinct code is appreciated. Thanks.

"""
Program to find root of a function using bisection method
"""
import sys
import math

def is_equal(a,b):
return abs(b-a) < sys.float_info.epsilon

def bisection(f, a, b, TOL=0.01, MAX_ITER=100):
"""
f is the cost function, [a,b] is the initial bracket,
TOL is tolerance, MAX_ITER is maximum iteration.
"""
f_a = f(a)
f_b = f(b)

iter = 0
while iter < MAX_ITER:
c = (a+b)/2
f_c = f(c)

if math.isclose(f_c,0.0,abs_tol=1.0E-6) or abs(b-a)/2<TOL:
return (c, iter)
else:
if f_a * f_c < 0:
b = c
f_b = f_c
else:
a = c
f_a = f_c
iter = iter + 1
return None

def func(x):
"""
The cost function: f(x) = cos(x) - x
"""
return math.cos(x) - x

root, iter = bisection(func, -1, 1, TOL=1.0E-6)
print("Iteration = ", iter)
print("root = ", root, ", func(x) = ", func(root))

## Type hints

They can help; an example:

def is_equal(a: float, b: float) -> bool:

The return type of bisection should probably be Optional[float].

## Argument format

MAX_ITER and TOL should be lower-case because they are the arguments to a function, not a global constant.

## Early-return

        return (c, iter)
else:

does not need the else, so you can drop it.

        iter = iter + 1

can be

        iter += 1

## Return parens

This does not need parentheses:

        return (c, iter)

The tuple is implied.

• I don't see the point of passing MAX_ITER. Bisection is guaranteed to terminate in $$\\log \dfrac{b - a}{TOL}\$$ iterations.

• I strongly advise against breaking the loop early at math.isclose(f_c,0.0,abs_tol=1.0E-6). It only tells you that the value at c is close to 0, but doesn't tell you where the root is (consider the case when the derivative at root is very small). After all, tolerance is passed for a reason!

If you insist on early termination, at least return the (a, b) interval as well. The caller deserves to know the precision.

• You may want to do this test right before returning (like, is there a root at all):

if math.isclose(f_c,0.0,abs_tol=1.0E-6):
return None
return c

but I'd rather let the caller worry about that.

• abs in abs(b - a) seems excessive.

if a > b:
a, b = b, a

at the very beginning of bisection takes care of it once and forever.

• (a + b) may overflow. c = a + (b - a)/2 is more prudent.

• I don't see where is_equal(a,b) is ever used.


text:
# Taylor expansion of translated fields

First of all, I would like to say that I am somewhat new to four-vector notation.

I have a function of a four-vector that I want to expand.

$$A_\mu (\mathbf{x} + \mathbf{x}_0) = A_\mu (\mathbf{x}) + \mathbf{x}_0 \frac{\partial A_\mu (\mathbf{x})}{\partial \mathbf{x}} + \dots$$

where both $\mathbf{x}$ and $\mathbf{x}_0$ are four-vectors so that e.g. $\mathbf{x} = (ct,x,y,z)$

My question is then what this actually looks like? The second term, should I have an index on the derivative? Is the derivative term a sum of several terms?

Could anyone help me write this out more explicitly, so that I am sure that I don't leave any terms behind?

Consider a scalar field, $\phi$. If we make the active transformation, $x^\mu \to x^\mu-\epsilon^\mu$ the change in the scalar field, infinitesimally, is given by,

$$\phi \to \phi + \epsilon^\mu\partial_\mu \phi + \mathcal{O}(\epsilon^2)$$

In more explicit notation, we may write,

$$\phi\to \phi + \epsilon^0 \frac{\partial \phi}{\partial t}+\epsilon^1 \frac{\partial \phi}{\partial x}+\epsilon^2 \frac{\partial \phi}{\partial y}+\epsilon^3 \frac{\partial \phi}{\partial z}$$

Note $\epsilon^\mu$ is simply the magnitude of the translation in the $\mu$th direction. For a $1$-form such as $A_\nu$, we simply pick up an extra index on the field in our expansion. The above is simply a higher dimensional analogue of the usual Taylor expansion. To see this clearly, note,

$$f(x+h)= f(x)+hf'(x)+\frac{1}{2}h^2f''(x)+\mathcal{O}(h^3)$$

In quantum field theory, a field takes the place of $f$, and $h$ becomes a vector, in which case we pick up derivatives, as shown, with respect to each coordinate.

• It is long ago for me, but shouldn't the time derivative and space derivatives have different signs? – M.Herzkamp Jun 19 '14 at 11:06
• @M.Herzkamp: No, I don't think so, because as I stated the translation is ACTIVE as opposed to PASSIVE. – JamalS Jun 19 '14 at 11:31
• @M.Herzkamp you can see this from the fact that $\partial_\mu=\frac{\partial}{\partial x^\mu}$, which clearly doesn't involve any opposite signs ;) – Danu Jun 19 '14 at 12:46


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# Math Help - Finding the Equation of Tangent Line

1. ## Finding the Equation of Tangent Line

Find an equation of the tangent line to the graph of f at the indicated point
f(x)=x2+1
(2, 5)

This is what I did:
First step - lim (x+△x)2+1-(x+△x)^2)
△x->0 .........................△x
Plugged in 0 to every △x, which turns out to be 1/0, which is undefined

Second step - (x+△x)(x+△x)+1-(x+△x)(x+△x)
....................................△x
Turns into x2+2△x+△x2+1-x2-2△x-△x2
....................................△x
I got lost here, I think I messed up on the 2nd parenthesis, because I thought I should have turned everything negative since there was a minus sign.
But then it cancels everything else except the +1

Can someone help me from here?
Thanks

2. ## Re: Finding the Equation of Tangent Line

$f(x) = x^2+1$

$f(x + \Delta x) = (x+\Delta x)^2 + 1$ (you forgot the 1)

try again ...

3. ## Re: Finding the Equation of Tangent Line

Originally Posted by Chaim
Find an equation of the tangent line to the graph of f at the indicated point
f(x)=x2+1 at (2, 5)
Here is the setup:
$\lim _{\Delta x \to 0} \frac{{f(2 + \Delta x) - f(2)}}{{\Delta x}} = \lim _{\Delta x \to 0} \frac{{4 + 4\Delta x + \left( {\Delta x} \right)^2+1 - 5}}{{\Delta x}}$

4. ## Re: Finding the Equation of Tangent Line

Originally Posted by Plato
Here is the setup:
$\lim _{\Delta x \to 0} \frac{{f(2 + \Delta x) - f(2)}}{{\Delta x}} = \lim _{\Delta x \to 0} \frac{{4 + 4\Delta x + \left( {\Delta x} \right)^2+1 - 5}}{{\Delta x}}$
Hmm, so I see you used 2 as the x.
This would simplify it down to (4△x+△x2)/△x
Which then turns into 4x+△x?

I'm not sure if I did it right, the answer at the back of the book is 4x-3.

5. ## Re: Finding the Equation of Tangent Line

Originally Posted by skeeter
$f(x) = x^2+1$

$f(x + \Delta x) = (x+\Delta x)^2 + 1$ (you forgot the 1)

try again ...
Um, I thought I added the +1 o.o
I used the lim △x->0 (f(x+△x)-f(x))/△x, where this is the definition of a derivative.
Could you explain more on where the +1 goes?

Thanks

6. ## Re: Finding the Equation of Tangent Line

Originally Posted by Plato
Here is the setup:
$\lim _{\Delta x \to 0} \frac{{f(2 + \Delta x) - f(2)}}{{\Delta x}} = \lim _{\Delta x \to 0} \frac{{4 + 4\Delta x + \left( {\Delta x} \right)^2+1 - 5}}{{\Delta x}}$
$\lim _{\Delta x \to 0} \frac{{4 + 4\Delta x + \left( {\Delta x} \right)^2+1 - 5}}{{\Delta x}}=\lim _{\Delta x \to 0} \frac{4\Delta x + \left( {\Delta x} \right)^2}{{\Delta x}}=\lim _{\Delta x \to 0} {4 + \left( {\Delta x} \right)}$

7. ## Re: Finding the Equation of Tangent Line

Originally Posted by Chaim
Hmm, so I see you used 2 as the x.
This would simplify it down to (4△x+△x2)/△x
Which then turns into 4x+△x?
No, it doesn't: 4△x/△x= 4, not "4x". Now take the limit as △x goes to 0. That gives the derivative at x= 2 and so the slope of the tangent line at x= 2.

I'm not sure if I did it right, the answer at the back of the book is 4x-3.

8. ## Re: Finding the Equation of Tangent Line

Originally Posted by Chaim
Um, I thought I added the +1 o.o
I used the lim △x->0 (f(x+△x)-f(x))/△x, where this is the definition of a derivative.
Could you explain more on where the +1 goes?

Thanks
Because $f(x)= x^2+ 1$, it goes in both f(x+△x) and f(x). You added it in one but not the other:
$f(x)= x^2+ 1$ so that $f(x+\Delta x)= (x+\Deltas x)^2+ 1= x^2+ 2x\Delta x+ (\Delta x)^2+ 1$ Then when you subtract the two, the two "1"s will cancel.


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# Alfred George Greenhill

Sir (Alfred) George Greenhill, F.R.S. (29 November 1847 in London – 10 February 1927 in London), was a British mathematician.

George Greenhill was educated at Christ's Hospital School and from there he went up to St John's College, Cambridge in 1866.[1] In 1876, Greenhill was appointed professor of mathematics at the Royal Military Academy (RMA) at Woolwich, London, UK.[2] He held this chair until his retirement in 1908. His 1892 textbook on applications of elliptic functions is of acknowledged excellence. He was one of the world's leading experts on applications of elliptic integrals in electromagnetic theory.[3] He was a Plenary Speaker of the ICM in 1904 at Heidelberg[4] and an Invited Speaker of the ICM in 1908 at Rome, in 1920 at Strasbourg,[5] and in 1924 at Toronto.

In 1879, Greenhill developed a rule of thumb for calculating the optimal twist rate for lead-core bullets. This shortcut uses the bullet's length, needing no allowances for weight or nose shape.[6] Greenhill applied this theory to account for the steadiness of flight conferred upon an elongated projectile by rifling. The eponymous Greenhill Formula, still used today, is:

Cast bullets as cast (left), with gas check (center) and lubricated (right).

${\displaystyle Twist={\frac {CD^{2}}{L}}\times {\sqrt {\frac {SG}{10.9}}}}$

where:

• C = 150 (use 180 for muzzle velocities higher than 2,800 f/s)
• D = bullet's diameter in inches
• L = bullet's length in inches
• SG = bullet's specific gravity (10.9 for lead-core bullets, which cancels out the second half of the equation)

The original value of C was 150, which yields a twist rate in inches per turn, when given the diameter D and the length L of the bullet in inches. This works to velocities of about 840 m/s (2800 ft/s); above those velocities, a C of 180 should be used. For instance, with a velocity of 600 m/s (2000 ft/s), a diameter of 0.5 inches (13 mm) and a length of 1.5 inches (38 mm), the Greenhill formula would give a value of 25, which means 1 turn in 25 inches (640 mm).

## References

1. ^ "Greenhill, George Alfred (GRNL866GA)". A Cambridge Alumni Database. University of Cambridge.
2. ^
3. ^ Greenhill, Alfred George (1907). "The elliptic integral in electromagnetic theory". Bull. Amer. Math. Soc. 8: 447–534. doi:10.1090/s0002-9947-1907-1500798-2. MR 1500798.
4. ^ "The Mathematical Theory of the Top considered historically by A. G. Greenhill". Verhandlungen des dritten Mathematiker-Kongresses in Heidelberg von 8. bis 13. August 1904. Leipzig: B. G. Teubner. 1905. pp. 100–108.
5. ^ "The Fourier and Bessel Functions contrasted by G. Greenhill" (PDF). Compte rendu du Congrès international des mathématiciens tenu à Strasbourg du 22 au 30 Septembre 1920. 1921. pp. 636–655.
6. ^ Mosdell, Matthew. The Greenhill Formula. "Archived copy". Archived from the original on 2011-07-18. Retrieved 2009-08-19. (Accessed 2009 AUG 19)
7. ^ Harkness, J. (1893). "Review: The Applications of Elliptic Functions by Alfred George Greenhill" (PDF). Bull. Amer. Math. Soc. 2 (7): 151–157. doi:10.1090/s0002-9904-1893-00129-8.
8. ^ Wilson, Edwin Bidwell (1917). "Review: Report on Gyroscopic Theory by Sir G. Greenhill" (PDF). Bull. Amer. Math. Soc. 23 (5): 241–244. doi:10.1090/s0002-9904-1917-02930-8.


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Limits of Perturbation theory for hydrogenic atoms

1. May 4, 2009

Cheetox

1. The problem statement, all variables and given/known data
Why can't we use perturbation theory to calculate the effect of the spin orbit interaction in hydrogen like uranium?

2. Relevant equations

3. The attempt at a solution
Is it something to do with the fact that the perturbation must be small compared to the rest of the hamiltonian? But for hydrogen like uranium the ionization energy is still 28000 times that of the spin orbit shift? Or could it be to do with the fact that the bohr radius for uranium approaches the radius of the nucleus?
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. May 4, 2009

millitiz

Well, I am a bit confused by your question, too.
So normally, we can obtain the EXACT result by using J operator and what not. And since we can get the exact wave function, why do we want to do it "approximately?"
The reason that Perturbation probably does not work is that, in order to get your energy, you need to know the wave function (or approximately the wave function) through the equation
E_n = <$$\Phi$$_n l H l $$\Phi$$_n>
We know H, it has many extra terms (especially the L dot S term)
This L dot S term made your previous wave function for Hydrogen invalid.
So without solving J and stuff like that, you actually don't really know what your phi look like (notice that this phi would not look like the original ones, ie, without the spin, because of the L dot S term). So essentially, you still need to go through solving J, and what not. And since you go through the whole thing, you can in fact get the exactly E. So I guess yes you could, it is probably just a bit redundant.


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# Orbital time

The orbital period or revolution period in astronomy is the time in which a celestial body completes its orbit around a reference point (has passed through its orbit once), i.e. the duration of a revolution.

## Basics

It should be noted that there can be different reference points to which the complete 360 ° circuit is measured. B. the period of revolution of the moon with or without taking into account the simultaneous movement of the earth around the sun .

The astronomical coordinate systems are generally not fixed to each other in space. Therefore, the period of rotation is given against a reference system that is as static as possible :

The reference can also be the (apparent) position of the sun ( synodic period ), the knot of individual planetary orbits ( draconic period ), the center of gravity of the entire solar system , its total center of mass ( barycentric period ) or the "rest of the universe " (see inertial system ).

## Table: Orbital times in the solar system

In the special case of the earth's orbit around the sun, the length of the revolution period is one year ; this expression is generalized, for example to a "Mars year", a "Venus year" etc.

The orbital times follow Newton's law of gravitation :

${\ displaystyle U = {\ sqrt {\ frac {4 \ pi ^ {2} a ^ {3}} {G \ left (M_ {1} + M_ {2} \ right)}}}}$

With

The orbital times of the planets are related to one another according to Kepler's third law :

The squares of the periods of revolution are in the same ratio as the cubes of the major semi-axes.

The following table contains the times for the synodic, sidereal or anomalistic periods of rotation of the planets of the solar system , a body in the asteroid belt and of trans-Neptunes , as well as the earth's moon, satellites and the sun (given in days and calendar years ):

• Except for the Earth's moon, the difference between the anomalistic orbital period and the sidereal orbital period is negligible in this accuracy, because the pericentres of the planets and asteroids shift only minimally compared to the period of orbit ( pericenter rotation ).
• In contrast to the moon, the synodic orbital times for Mercury and Venus are significantly longer , but for Mars and the outer planets (the term “inside / outside” refers to the asteroid belt, not the earth), it is increasingly shorter . The exact explanation for this see in the section Planets of the article Synodic Orbital Period .
object Sidereal anomalous orbital period
"in relation to the fixed stars / the orbit geometry"
Synodic period
"in relation to earth and sun"
ISS 00001.51 hours I1 0001.53 hours I2
Geosynchronous 00023.93 hours G1 0024.00 hours
Moon M1 000027,322 days /
000027,554 days M2
0029.53 days
Mercury 000087,969 days 0115.88 days
Venus 000224,701 days 0583.92 days
Earth E1 000365.256 days 000-
Mars 000686,980 days 0779.94 days
Ceres 000004.605 years 0466.72 days
Jupiter 000≈11,862 years 0398.88 days
Saturn 000≈29,458 years 0378.09 days
Uranus 000≈84.014 years 0369.66 days
Neptune 00≈164.793 years 0367.49 days
Pluto 00≈247.94 0years NP 0366.73 days
Orcus 00≈247.97 0years NP 000-
Varuna 00≈283.56 0years NP 000-
Haumea 00≈284.61 0years NP 000-
Quaoar 00≈285.09 0years NP 0366.54 days
Makemake 00≈309.41 0years NP 000-
Eris 00≈557.4 00years NP 000-
Sedna ≈10704 , 000years NP 0365.29 days
Sun S 00≈230 million years 000-
I1 Anomalistic Orbital Time: 91.4887 minutes
I2That is the time between two sunrises for an ISS astronaut. The ISS runs prograde around the earth, so the sun comes “towards” it. It takes 1.61 hours until it arrives again over a parallel
M1For the orbit period of the moon see in detail: Lunar orbit
M2The Draconite Period is the time between two passes through the same lunar node. It plays a role for the eclipses, for the planets and minor planets it is of no particular significance
E1For the orbit period of the earth see in detail: Earth orbit
NPThe orbital periods of objects beyond Neptune are so long that modern astronomy has not yet fully grasped them. The values given are based on planetary theories (such as the VSOP 87 ), which then provide useful results in model calculations. The confirmation by measurement is still pending. On April 11, 2009, Neptune completed its first fully observed period, and has been fairly accurately specified ever since.
S.About the center of the Milky Way, see The Sun in the Milky Way System

## Conversion synodic - sidereal

Sidereal period (1 to 2) and synodic period (1 to 3).
${\ displaystyle T _ {\ mathrm {E}}}$ = sidereal period of revolution of the earth

Outer planets:

${\ displaystyle T _ {\ mathrm {sid}} = {\ frac {T _ {\ mathrm {E}}} {T _ {\ mathrm {syn}} -T _ {\ mathrm {E}}}} \ cdot T _ {\ mathrm {syn}}}$

Inner planets:

${\ displaystyle T _ {\ mathrm {sid}} = {\ frac {T _ {\ mathrm {E}}} {T _ {\ mathrm {syn}} + T _ {\ mathrm {E}}}} \ cdot T _ {\ mathrm {syn}}}$

## Table: Orbit times of the sun, moon, earth and derived time quantities

A table about the mean dates, standard epoch J2000.0 , and the derived quantities of the calendar calculation .

It should be noted that the "revolution time of the sun" is the apparent solar path observed from the earth . It is not created by an orbit, but by the rotation of the earth .

Day month year
Sidereal day Sidereal Month (1) Sidereal year
86164,099s 27.32166 d 365.256366 d
23h 56m 4.099s 27d 7h 43m 11.5s 365d 6h 9m 9s
Sidereal day (2) Tropical month Tropical year
86164.091 s 27.32158 d 365.242199 d
23h 56m 4.091s 27d 7h 43m 4.7s 365d 5h 48m 46s
Sunny day (3) Synodic month (5) Solar year (3)
86400s (4) 29.53059 d 365.242199 d (6)
24h (4) 29d 12h 44m 2.9s 365d 5h 48m 46s (6)
Calendar day Calendar month Calendar year (8)
1 d = 86400 s (7) 30 d / 31 d 365.2425 d
24h (7)   365d 5h 49m 12s
(1) Cycle of the highest and lowest levels of the moon
(2)The term tropical day is not in use.
(3)The terms synodic day and synodic year are not in use.
(4)mean day length, cf. Mean local time
(5)The moon phase cycle , the single period fluctuating around the mean value, is called the lunation
(6) The solar year corresponds to the tropical year.
(7) The calendar day is - in general - defined via the sunny day.
(8th)The middle year of the Gregorian calendar .

## Individual evidence

1. Gerhard Dangl: ISS - Visibility table July 22, 2009 to July 25, 2009. Retrieved on August 5, 2009 .


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# differentiable function

I'm trying to solve this problem

Let $f:[a,b] \to \mathbb R$ a differentiable function with continuous derivative.

Suppose further that $f$ is twice differentiable on $(a,b)$.

Prove that if $f(a)=f(b)$ and $f'(a)=f'(b)=0$, then exist $x_1, x_2 \in (a,b)$ with $x_1 \neq x_2$ such that $f''(x_1) = f''(x_2)$.

I'm trying to solve this problem graphically intuitively ; I tried, and likely the problem is real: if $f(a) = f(b)$ and $f'(a) = f'(b) = 0$, the points $a$ and $b$ are the points of maximum and minimum for the function. but how can you prove that there are $x_1, x_2 \in (a,b)$ with $x_1 \neq x_2$ such that $f''(x_1) = f''(x_2)$?

-
The last paragraph of your question doesn't make much sense. You could consider fixing it. –  user21436 Dec 28 '11 at 17:29
There's this weird spacing when you write $[a,b]$ and $(a,b)$ before the second bracket/parenthesis that I can't get rid of by editing. Weird. –  Patrick Da Silva Dec 28 '11 at 17:30
@Patrick: Done. –  Did Dec 28 '11 at 17:46
@Didier : What was the problem exactly? I looked at the edit but saw nothing, just you replacing some characters by the exact same characters. –  Patrick Da Silva Dec 28 '11 at 18:27
@Patrick: What we saw as b plus a space was in fact a compound of the letter b and of two invisible pseudo-characters attached to it and translated by the system as a space. That is, an encoding gone awry (but please do not ask me for more precise details...). –  Did Dec 29 '11 at 9:16

If the function is constant, everything is easy.

If the function is not constant, it attains a maximum and a minimum value in our interval, and these values are distinct.

The max and min cannot both occur at endpoints, since $f(a)=f(b)$. So there is a local extremum in $(a,b)$, and therefore a point $c\in(a,b)$ such that $f'(c)=0$. Now you should be able to use Rolle's Theorem to show that there exist points $x_1\in(a,c)$ and $x_2\in(c,b)$ such that $f''(x_1)=f''(x_2)=0$.

-
A nitpick: Why do you essentially prove Rolle's Theorem and then appeal to it? –  David Mitra Dec 28 '11 at 18:02
@David Mitra: Of course you are right. I wanted to connect the finding of $c$ with something familiar, to make the geometry more concrete-seeming. –  André Nicolas Dec 28 '11 at 18:07

Hint: Apply the Mean Value Theorem three times (or Rolle's Theorem if you prefer).

Applying it once to $f$ over $[a,b]$ gives you a point $c$ with $a<c<b$ where $f'(c)=0$.

Now apply the Mean Value Theorem to $f'$ on each of the intervals $[a,c]$ and $[c,b]$.

(Note that the Mean Value Theorem gives you a point $c$ strictly between the endpoints of the interval that you're working over.)

It's not true that $a$ and $b$ necessarily give the minimum and maximum of the function. They could in fact give neither. For instance, the graph could resemble a $\sin$ wave over $[0,2\pi]$ that's been "smoothed out" at the endpoints (so that the derivative is zero at the endpoints).

-


text:
Home / Help

Author Lang, Serge. author SL 2(R) [electronic resource] / by Serge Lang New York, NY : Springer New York : Imprint: Springer, 1985 http://dx.doi.org/10.1007/978-1-4612-5142-2 XIV, 431 p. online resource

SUMMARY

SL2(R) gives the student an introduction to the infinite dimensional representation theory of semisimple Lie groups by concentrating on one example - SL2(R). This field is of interest not only for its own sake, but for its connections with other areas such as number theory, as brought out, for example, in the work of Langlands. The rapid development of representation theory over the past 40 years has made it increasingly difficult for a student to enter the field. This book makes the theory accessible to a wide audience, its only prerequisites being a knowledge of real analysis, and some differential equations

CONTENT

I General Results -- 1 The representation on Cc(G) -- 2 A criterion for complete reducibility -- 3 L2 kernels and operators -- 4 Plancherel measures -- II Compact Groups -- 1 Decomposition over K for SL2(R) -- 2 Compact groups in general -- III Induced Representations -- 1 Integration on coset spaces -- 2 Induced representations -- 3 Associated spherical functions -- 4 The kernel defining the induced representation -- IV Spherical Functions -- 1 Bi-invariance -- 2 Irreducibility -- 3 The spherical property -- 4 Connection with unitary representations -- 5 Positive definite functions -- V The Spherical Transform -- 1 Integral formulas -- 2 The Harish transform -- 3 The Mellin transfor -- 4 The spherical transform -- 5 Explicit formulas and asymptotic expansions -- VI The Derived Representation on the Lie Algebra -- 1 The derived representation -- 2 The derived representation decomposed over K -- 3 Unitarization of a representation -- 4 The Lie derivatives on G -- 5 Irreducible components of the induced representations -- 6 Classification of all unitary irreducible representations -- 7 Separation by the trace -- VII Traces -- 1 Operators of trace class -- 2 Integral formulas -- 3 The trace in the induced representation -- 4 The trace in the discrete series -- 5 Relation between the Harish transforms on A and K -- Appendix. General facts about traces -- VIII The PlanchereS Formula -- 1 Calculus lemma -- 2 The Harish transforms discontinuities -- 3 Some lemmas -- 4 The Plancherel formula -- IX Discrete Series -- 1 Discrete series in L2(G) -- 2 Representation in the upper half plane -- 3 Representation on the disc -- 4 The lifting of weight m -- 5 The holomorphic property -- X Partial Differential Operators -- 1 The universal enveloping algebra -- 2 Analytic vectors -- 3 Eigenfunctions of ?f -- XI The Well Representation -- 1 3/2 -- 8 The equation $$- \psi ''(y) = {\text{ }}\frac{{s(1 - s)}}{{{ŷ2}}}\psi (y)\;on\;\left[ {a,\infty } \right)$$ -- 9 Eigenfunctions of the Laplacian in L2?\? = H -- 10 The resolvant equations for 0 < ? < 2 -- 11 The kernel of the resolvant for 0 < ? < 2 -- 12 The Eisenstein operator and Eisenstein functions -- 13 The continuous part of the spectrum -- 14 Several cusps -- Appendix 1 Bounded Hermitian Operators and Schurโ{128}{153}s Lemma -- 1 Continuous functions of operators -- 2 Projection functions of operators -- Appendix 2 Unbounded Operators -- 1 Self-adjoint operators -- 2 The spectral measure -- 3 The resolvant formula -- Appendix 3 Meromorphic Families of Operators -- 1 Compact operators -- 2 Bounded operators -- Appendix 4 Elliptic PDF -- 1 Sobolev spaces -- 2 Ordinary estimates -- 3 Elliptic estimates -- 4 Compactness and regularity on the torus -- 5 Regularity in Euclidean space -- Appendix 5 Weak and Strong Analyticity -- 1 Complex theorem -- 2 Real theorem -- Symbols Frequently Used

Mathematics Algebra Mathematics Algebra

Location

Office of Academic Resources, Chulalongkorn University, Phayathai Rd. Pathumwan Bangkok 10330 Thailand


text:
Quiz 10: Geometry Stuff

## Question 1: Element Ratio

A very useful relationship to be aware of is that in any triangle mesh, the ratio of vertices to edges to faces is roughly

V : E : F = 1 : 3 : 2

(In fact, they'll approach this ratio exactly as the number of elements goes to infinity, independent of the shape of the surface or choice of triangulation.) A formal proof of this fact takes a bit of work (but is not beyond your ability!).

For this quiz question, you should just give a very rough argument for why mesh elements might appear in this ratio. (Hint: think about the manifold property, the notion of regular valence, and the fact that we are only considering triangle meshes.)

## Question 2: Bezier/Hermite Conversion

The second part of your quiz is written up here: BezierHermiteConversion.pdf


text:
Beginner's Guide to mev-boost

Understanding mev-boost as an iteration on the original Flashbots auction, and how it paves the way for full in-protocol PBS

What is mev-boost?#

mev-boost is the version of Flashbots that is adapted for and compatible with Proof of Stake Ethereum. It is related to the design for in-protocol proposer/builder separation, and can be conceptualised as an intermediate step to full in-protocol PBS. By understanding the current Flashbots auction, and by comparing this current state to the plans for PBS, we see how mev-boost provides block-building functionality that would otherwise be unavailable until PBS is completed.

Understanding the Flashbots Auction#

Currently, Flashbots provides a private transaction pool (mev-relay) and a sealed bid blockspace auction mechanism (mev-geth). This allows miners to outsource the work of finding the optimal block construction.

mev-geth introduced a new RPC endpoint, eth_sendBundle - the message sent to this endpoint is called a bundle. Bundles consist of one or many transactions to be executed in an atomic batch. They can be sent to the relay by a searcher, and to the miners by the relay.

Searchers are Ethereum users who prefer to use the FB private transaction pool over the regular p2p pool. They monitor the state of the chain and send bundles to the relayers. There is an option for searchers to express their bids for inclusion via the ether gas price, or by direct eth transfer to the miner's coinbase address.

The relay is a bundle propagation service that receives bundles from searchers and forwards them to miners. The relay is in charge of validating and routing FB bundles. Since searchers do not have to pay for failed bids (they can pay directly to the coinbase address, and make the payment conditional on their bundle succeeding), there is the potential that they could spam the network with invalid bundles. The relay therefore serves as a mitigation to this DOS threat. It will simulate each transaction and only forward valid bundles to the miner.

The miner (or more generally, block producer) is the party who ultimately collects all the bundles and produces a block. Miners traditionally run go-ethereum, and greedily order transactions by gas price. However, miners participating in Flashbots run a version of mev-geth.

These miners that are running mev-geth evaluate incoming bundles using the first-price sealed-bid auction and pick the most profitable bundles to place at the top of the block. The node then compares the Flashbots block to a normal block and begins mining on the most profitable.

The Flashbots bundle information allows the searcher to express blockspace preferences relative to the state of the chain as well as the state of the transaction pool. Before Flashbots, the only way for searchers to express their preference for transaction placement was via the gas price. For transactions that needed to be first in the block, this naturally led to competition in the gas price. However, for transactions that could be anywhere in the block, bidding on gas price might be ineffective. Flashbots Auction decoupled placement from gas price, thus enabling price discovery for discrete MEV opportunities instead of competition for priority. The miner can evaluate all bundles received and combine those which do not conflict in order to produce the most profitable block.

Miners have full access to bundle content and could arbitrarily reorder/steal/censor bundles sent to them by searchers and relayers. However, they are incentivised to comply, as Flashbots monitors for misbehaviour and removes miners who steal.

Proposer/Builder Separation#

Proposer/block-builder separation (PBS) was proposed by Ethereum researchers as a response to the risk that MEV poses to decentralisation of consensus networks. They have suggested that uncontrolled MEV extraction promotes economies of scale that have a centralising effect, as well as causing complications to decentralised pooling.

PBS is a change to the core protocol that aims to mitigate this. Instead of the block proposer (currently the miner, after PoS the validator) also trying to produce a maximally profitable block by itself, it can outsource this to a block building market. Here, block builders would produce bundles consisting of a complete block and a fee for the proposer. The proposer just has to pick the block with the highest fee.

There are number of properties that are desired in the PBS design:

• untrusted proposers and builders should be able to participate successfully (there is little to no risk that a proposer would steal blocks from a builder, and vice versa)
• proposers are not advantaged by having high resources or technical competence
• proposers cannot extract transactions from bundles without paying the fee
• that the new design works with the existing consensus layer.

PBS is still an open and active research area that is developing rapidly. The most recent design/specification for the in-protocol version is two-slot PBS

mev-boost#

Mev-boost is an updated Flashbots architecture that is compatible with PoS Ethereum. It also implements the design goals behind PBS, although in a semi-trusted way. The design is as follows:

We start with the existing Ethereum network and mempool in the above diagram. As discussed previously, in the current Flashbots model searchers take transactions from the public mempool, potentially adding their own, and arrange them into bundles.

The possibility of private orderflow is also anticipated at this point. Private orderflow (also known as exclusive orderflow) refers to transactions that can be included in blocks but are not visible in the public mempool. This could be because the transactions are sent to a certain entity and they may route these transactions to their own nodes, or keep them secret in order to build more profitable blocks for themselves.

Block builders are also introduced - these are services/providers that will aggregate various bundles and transactions into block templates. The builder orders the transactions in a block according to what is most profitable for them. The block templates are then forwarded to a relay.

The relay receives these block templates (also referred to as execution payloads), and will verify their validity. The MEV-boost component is a middleware which handles communication with the relays, the profit-switching logic, and a fallback mechanism in the case of some system failure.

What is the link between PBS and MEV-boost?#

If Flashbots Auction was the first step of "block building as a service", then PBS could be thought of as a response to the success and mainstream adoption of Flashbots. Instead of the entire network utilising one block builder (and thereby creating a centralising effect at the block-building level), the protocol would split block proposing and block building into two separate components. Block proposing would be handled in-protocol by the validators, and it is then much easier for block building to be delegated to a market of competing parties.

However, the current designs of PBS require a change to beacon chain consensus (specifically the fork choice rule). For this reason (as well the designs not being completely finalised yet), it will need to wait until after the merge is successfully completed.

Flashbots MEV-boost acts as a prototype for what a block-building market would look like, and provides the necessary proposer logic and middleware that is missing from PoS Ethereum.

Longer-term, PBS is important to the Ethereum protocol for a number of reasons. Splitting block proposal and block building into separate components actually has desired effects that align with the protocol's goals, such as removing the requirement for validators to hold the full state (stateless Ethereum initiative). PBS is also a necessary step for the current favoured sharding proposal (Danksharding - https://notes.ethereum.org/@vbuterin/proto_danksharding_faq).


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# Tag Info

### Can we create a decision tree for any comparison sorting algorithm even if it is very complicated?

The procedure below can be used to explicitly build the decision tree of any comparison-based algorithm: generate all permutations of the array $[1, 2, \cdots n]$. for every permutation, run the ...
• 5,215
1 vote

### Can we create a decision tree for any comparison sorting algorithm even if it is very complicated?

In theory, yes you should be able to create a decision tree for every comparison based sorting algorithm. You can write a program that will do so (for a fixed comparison-based sorting algorithm of ...
• 293
1 vote
Accepted

### Can we create a decision tree for any comparison sorting algorithm even if it is very complicated?

In general, a decision tree would enumerate all possible permutations $(N!)$ for given input of $N$ elements, out of which only $1$ would be desirable. A sorting algorithm doesn't generate all ...
• 512

Top 50 recent answers are included


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# big o notation of fraction $\frac{\log(N+1)}{\log(N)}$

I want to estimate big-O notation for $\frac{\log(8n^2)}{\log(n)}$.

I think the big-O notation is constant. am I right?

in case of $\frac{\log(N+1)}{\log(N)}$, is it the same and O(1)?

and what about $\frac{\log(N^3+7n+1)}{\log(N^4+N^2)}$, is it O($\log$ $n$)?

It would be appreciated if somebody could generalize the method of estimating big-O notation of fractions.

Yes, you're right. The simplest way to see all this is to use equivalents:

• $\log(8n^3)=\log 8+3\log n\sim_\infty 3\log n$, so $$\frac{\log(8n^3)}{\log n}\sim_\infty\frac{3\log n}{\log n}=3,$$ hence it is $O(1)$.
• $n^3+7n+1\sim_\infty n^3$, hence $\log(n^3+7n+1)\sim_\infty \log(n^3)=3\log n$,
• similarly $\log(n^4+n^2)\sim_\infty 4\log n$,

so $\;\dfrac{\log(n^3+7n+1)}{\log(n^4+n^2)}\sim_\infty \dfrac{3}{4}=O(1)$.

• Thanks for paying attention to the details. Now it is clear. – David Oct 5 '17 at 18:26

In general, for any two polynomials with complex coefficients, $\dfrac{\log|P(n)|}{\log|Q(n)|} =O(1)$ as $n \to \infty$.

You can go further and show that $\dfrac{\log|P(n)|}{\log|Q(n)|} =\dfrac{u}{v}+o(1)$ where $u$ and $v$ are, respectively, the degrees of $P$ and $Q$.


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pediatrics
June 2017, VOLUME139 /ISSUE 6

# Receptivity to Tobacco Advertising and Susceptibility to Tobacco Products

1. John P. Pierce, PhDa,b,
2. James D. Sargent, MDc,
3. Martha M. White, MSa,
4. Nicolette Borek, PhDd,
5. David B. Portnoy, PhD, MPHd,
6. Victoria R. Green, BAe,f,
7. Annette R. Kaufman, PhD, MPHg,
8. Cassandra A. Stanton, PhDh,i,
9. Maansi Bansal-Travers, PhDj,
10. David R. Strong, PhDa,b,
11. Jennifer L. Pearson, PhD, MPHk,l,
12. Blair N. Coleman, PhD, MPHd,
13. Eric Leas, MPHa,b,
16. Meghan B. Moran, PhDk,
17. Charles Carusi, PhDh,
18. Andrew Hyland, PhDj, and
19. Karen Messer, PhDa,b
1. aCancer Prevention Program, Moores Cancer Center, and
2. bDepartment of Family Medicine and Public Health, University of California, San Diego, La Jolla, California;
3. cC. Everett Koop Institute, Norris Cotton Cancer Center, Geisel School of Medicine at Dartmouth, Hanover, New Hampshire;
4. dCenter for Tobacco Products, US Food and Drug Administration, Silver Spring, Maryland;
5. eNational Institute on Drug Abuse, National Institutes of Health, Bethesda, Maryland;
6. fKelly Government Solutions, Rockville, Maryland;
7. gTobacco Control Research Branch, Behavioral Research Program, Division of Cancer Control and Population Sciences, National Cancer Institute, National Institutes of Health, Rockville, Maryland;
8. hWestat, Rockville, Maryland;
9. iDepartment of Oncology, Cancer Prevention and Control Program, Lombardi Comprehensive Cancer Center, Georgetown University Medical Center, Washington, District of Columbia;
10. jDepartment of Health Behavior, Roswell Park Cancer Institute, Buffalo, New York;
11. kDepartment of Health, Behavior, and Society, Bloomberg School of Public Health, Johns Hopkins University, Baltimore, Maryland; and
12. lSchroeder Institute for Tobacco Research and Policy Studies at Truth Initiative, Washington, District of Columbia
1. Dr Pierce conceptualized and designed the study and drafted the initial manuscript; Dr Sargent conceptualized and designed the study including the study instruments and critically reviewed the manuscript; Ms White carried out the initial analyses and reviewed and revised the manuscript; Dr Messer designed the analyses and reviewed and revised the manuscript; Drs Carusi and Hyland lead the PATH project, including the design of survey instruments, and reviewed and revised the manuscript; and Drs Borek, Portnoy, Kaufman, Stanton, Bansal-Travers, Strong, Pearson, Coleman, Trinidad, and Moran and Ms Green, Ms Noble, and Mr Leas reviewed and approved the initial proposed analyses and reviewed and revised the manuscript; and all authors approved the final manuscript as submitted.

## Abstract

BACKGROUND AND OBJECTIVES: Non–cigarette tobacco marketing is less regulated and may promote cigarette smoking among adolescents. We quantified receptivity to advertising for multiple tobacco products and hypothesized associations with susceptibility to cigarette smoking.

METHODS: Wave 1 of the nationally representative PATH (Population Assessment of Tobacco and Health) study interviewed 10 751 adolescents who had never used tobacco. A stratified random selection of 5 advertisements for each of cigarettes, e-cigarettes, smokeless products, and cigars were shown from 959 recent tobacco advertisements. Aided recall was classified as low receptivity, and image-liking or favorite ad as higher receptivity. The main dependent variable was susceptibility to cigarette smoking.

CONCLUSIONS: A large proportion of US adolescent never tobacco users are receptive to tobacco advertising, with television advertising for e-cigarettes having the highest recall. Receptivity to advertising for each non–cigarette tobacco product was associated with susceptibility to smoke cigarettes.

• Abbreviations:
CI
confidence interval
aOR
P1
period 1
P2
period 2
PATH
Population Assessment of Tobacco and Health
• #### What’s Known on This Subject:

The consensus that cigarette marketing is 1 cause of adolescent smoking is the basis for marketing constraints imposed on these products in the United States and elsewhere. Little is known about the influence of marketing for non–cigarette tobacco products.

Advertising for non–cigarette tobacco products is reaching non–tobacco-using US adolescents, especially through television. High percentages of non–tobacco-using adolescents recognize tobacco ad images, including e-cigarette ads. Receptivity to these ads is associated with susceptibility to future cigarette smoking.

Susceptibility to smoking is a validated measure that predicts the risk of smoking initiation as many as 3 to 4 years before first experimentation.2325 A series of questions identifies “committed never users” as those who have never been curious about use, have strong intentions not to use, and who would resist an offer to use from a best friend. All other never users are considered susceptible. Susceptibility and subsequent experimentation vary across sociodemographic variables, receptivity to tobacco marketing,16,17 exposure to other tobacco users,26 use of another psychoactive substance,27 use of another tobacco product,28 and psychosocial variables such as externalizing problem behaviors (eg, rule-breaking, aggression),29 internalizing problem behaviors (eg, depression, anxiety),30 and sensation-seeking.31

This study explores levels of receptivity to the marketing of e-cigarettes, cigarettes, cigars, and smokeless tobacco products among never-using 12 to 17 year olds, with the use of data from the national Population Assessment of Tobacco and Health (PATH) Study. The study design allowed us to assess adolescent receptivity to advertising for multiple tobacco products and to summarize this in a measure of receptivity to any tobacco advertising. We investigated whether this general receptivity to tobacco advertising was associated with susceptibility to use any tobacco product. Finally, we investigated the associations of advertising receptivity to each product with susceptibility to cigarette smoking.

## Methods

Data are from wave 1 of the PATH study, which is a nationally representative sample of the civilian, noninstitutionalized US population, aged ≥12 years between September 2013 and December 2014.32,33 With oversight from the National Institute of Drug Abuse and the Food and Drug Administration, Westat collected data with the use of audio-computer–assisted self-interviews. Westat’s Institutional Review Board approved the study design/protocol, and the Office of Management and Budget approved the data collection. Households were identified by using an address-based, area-probability sampling method, and a screener survey enumerated household members (response rate = 54%). Generally, all youth aged 12 to 17 years (maximum 2 per household) were selected for interview, and parental consent and youth assent were obtained. Interviews were completed for 78.4% of selected youth. The data were weighted to adjust for the complex sample design and nonresponse to allow population estimates. If the respondent did not answer regarding their age, sex, race, or Hispanic ethnicity, these were obtained from the household screener (n = 704 or 6.6%) or, if not available, by using statistical imputation methods (n = 143 or 1.3%).34 In this article, our analyses were restricted to the 10 751 of the youth sample (N = 13 651) who reported that they had never used any tobacco product.

### Dependent Variable: Susceptibility To Use Tobacco Products and Cigarettes

Following previous research,23,24 never users who had heard of the product were classified as either susceptible to use or committed never users on the basis of their responses to 3 questions assessing their curiosity about the product, intention to try it in the near future, and likely response if a best friend were to offer them the product (see Supplemental Information). Only those with the strongest rejection to all 3 questions were categorized as committed never users to each of the following 8 products (cigarettes, e-cigarettes, pipes, cigars, hookahs, smokeless tobacco, dissolvable tobacco, and bidis/kreteks). The strongest rejection to all questions for all products classified an individual as a committed never tobacco user; all others were considered to be susceptible to use ≥1 products. For this analysis, those who had never heard of a product were considered to be committed never users.

### Key Independent Variable: Receptivity to Tobacco Product Advertising

Respondents were asked “What is the brand of your favorite tobacco advertisement?” A list of brands was provided, with an option to nominate another brand. We assumed that identifying Marlboro or Camel as favorite referred to cigarettes, unless snus was specifically mentioned.

#### Aided Recall and Liking of Contemporary Ads

The above sampling scheme allowed some respondents to see >1 ad for a brand within a category, particularly for the little cigars and cigarillos set in P2. When >1 ad was shown for a brand, aided recall was assessed only on the first ad that was displayed. We computed the estimated recall rate as the weighted proportion of respondents who recalled seeing an ad for the brand out of the total number of respondents who were shown an ad for the brand. Before ranking brands on the frequency of recall, we removed brands with advertisements that were shown to <500 participants. Thus, for the rank order of aided recall, we considered a total of 55 tobacco brands (9 cigarette brands, 9 e-cigarette brands with television ads, 14 e-cigarette brands with print ads, 9 smokeless brands, and 14 cigar brands).

### Other Covariates of Tobacco Use in Adolescence

#### Exposure to Tobacco Use

All respondents were asked: “Does anyone who lives with you now use any of the following: [list of tobacco products]” and “During the past 7 days, about how many hours were you around others who were smoking? Include time in your home, in a car, at school or outdoors.” We report 2 binary variables: 1 for household exposure (no users of tobacco in the household versus any) and 1 for general exposure (no exposure to smoking by others in the past 7 days versus any).

#### Use of Other Drugs or Alcohol

Ever use was ascertained for alcohol, marijuana, as well as for misuse of prescription drugs (ie, Ritalin/Adderall, painkillers, sedatives, tranquilizers), cocaine or crack, methamphetamine or speed, heroin, inhalants, solvents, and hallucinogens by a series of questions: “Have you ever used [substance]?” (See Supplemental Information for the full list of questions). Adolescents who reported ever use of any of these were classified as “any use”; all others were classified as “no use.”

#### Psychosocial Predictors of Tobacco Use

We adjusted for mental health and substance use problems by using scales from the Global Appraisal of Individual Needs–Short Screener.35 The internalizing subscale (α = .81) included 4 items of depressive and anxiety symptoms. The externalizing subscale (α = .70) included experience with 5 conduct and behavioral items. The substance use problems subscale (α = .67) asked 7 questions about problems associated with alcohol or drug use. Adolescents were scored on how many items they had experienced in the past month or past 2 to 12 months (see Supplemental Information for questions). In addition, sensation seeking (α = .74) was assessed by 3 items modified from the Brief Sensation Seeking Scale.36 On all scales, respondents were scored as having no symptoms (0), low to moderate symptom levels (1–2), or high symptom levels (≥3).

### Statistical Analyses

Analyses were performed by using SAS Software, version 9.3 (SAS Institute, Cary, NC).37 We used the survey weights supplied with the data, and computed variances and P values by using the recommended Balanced Repeated Replication method with Fay = 0.3.27,33,38 Weighted percentages were calculated with SAS version 9.3 PROC SURVEYFREQ. Modified Wilson confidence limits for proportions were calculated on the weighted estimates by using PROC SURVEYFREQ. Separate logistic regression models tested the association of receptivity to any tobacco advertising with susceptibility to any tobacco product, as well as with susceptibility to cigarette smoking. These models controlled for the above-mentioned covariates. χ2 Tests were also conducted to test for significant differences between percentage recall of pairs of selected ad brands or modes. Odds ratios, confidence limits, and P values are reported from the weighted, adjusted model (calculated via PROC SURVEYLOGISTIC).

## Results

### Demographic Characteristics

This sample of adolescent never tobacco users was balanced on sex and had a high proportion of younger adolescents (12–13 years: 39.5%; 14–15 years: 34.1%; 16–17 years: 26.4%) (Table 1). Just over half were non-Hispanic white (53.6%), 22.4% were Hispanic, 14.4% were non-Hispanic African American, 5.2% were Asian, and 4.5% were other races/ethnicities. Most adolescents (82.0%) had parents who completed high school, and 60.7% had parents with at least some college education. Never tobacco users who were not receptive to any tobacco advertising (54%) had the lowest susceptibility to any tobacco use (34.3%; 95% confidence interval [CI]: 32.9%–35.6%). Among never users with low receptivity (34%), 50.4% (95% CI: 48.5%–52.2%) were susceptible. Among the 10% of never users who had a moderate receptivity, 65.4% (95% CI: 62.2%–68.4%) were susceptible. Among the <2% of never users with high receptivity to tobacco advertising, 87.7% (95% CI: 81.2%–92.2%) were susceptible to any tobacco product.

TABLE 1

Logistic Regression Predicting Susceptibility to Any Tobacco Product Among Never-Using Adolescents

Overall, 41.0% of 12 to 13 year olds and approximately half of both 14 to 15 year olds and 16 to 17 year olds were classified as being receptive to any tobacco advertising (Table 2). Approximately one-third of each age group had a low level of receptivity. There were significantly fewer 12 to 13 year olds with moderate or high receptivity (8.8%) compared with 16- to 17-year-old adolescents (15.0%) (P < .001).

TABLE 2

Receptivity to Tobacco Advertising by Product Type Among Never Tobacco Users

In multivariable logistic regression (Table 1) controlling for potential confounding variables, even low receptivity to any tobacco ads was significantly associated with increased concurrent susceptibility to use any tobacco product (adjusted odds ratio [aOR]: 1.38; 95% CI: 1.25–1.53). Moderate and high receptivity levels had higher and significant odds ratios (moderate: aOR = 2.39, P < .001; high: aOR = 6.73, P < .001).

### Aided Recall for Tobacco Advertising by Brand

TABLE 3

Ads for Tobacco Brands With the Highest Aided Recall by Age Group Among US Never Tobacco Users

### Association of Advertising Receptivity With Susceptibility to Cigarette Smoking

In our multivariable logistic regression controlling for potential covariates, moderate to high receptivity to cigarettes (aOR: 1.57; 95% CI: 1.25–1.98), e-cigarettes (aOR: 1.53; 95% CI: 1.21–1.94), and smokeless tobacco (aOR: 1.58; CI: 1.23–2.03) was significantly associated with concurrent susceptibility to smoke cigarettes (Table 4), which was not the case for moderate to high receptivity to cigar advertising. A low level of receptivity to any of the 4 forms of tobacco advertising was not associated with concurrent susceptibility to smoke cigarettes.

TABLE 4

Variables Associated With Susceptibility to Cigarette Smoking Among Adolescent Never Tobacco Users

## Discussion

Despite marketing restrictions on cigarettes and other products, tobacco advertising continues to reach one-third to one-half of adolescents, depending on age. E-cigarette advertising generated the highest reach. This dominance of e-cigarettes is in stark contrast to the advertising expenditures for each product over the study period: >$9 billion/year for cigarettes,39$415 million/year for smokeless products,40 and only \$60 million/year for e-cigarettes4 (although this latter amount appears to be growing substantially each year). One explanation for receptivity being highest for the product with the lowest marketing budget4 is that the product with minimal marketing restrictions may have a significant advantage. Products with less-restricted marketing can use marketing synergies across multiple media channels, which are likely more cost-effective than campaigns for products with significant marketing restrictions.41 It has been noted that television is still among the most effective advertising platforms in the United States.42 In this study, 4 of the top 10 ads recalled were e-cigarette television ads, which made up only a portion of the e-cigarette marketing expenditures, a budget that was 2 orders of magnitude below the cigarette marketing budget.4,39 Our results suggest that, even though cigarettes are not allowed to be advertised on television, interest in them may be increased through observing ads for other tobacco products. This topic, along with how receptivity may affect harm perceptions, will be explored in separate longitudinal analyses of PATH study data.

Two major study strengths are its large nationally representative sample of adolescents and the use of a stratified random sample from a near census of recent tobacco advertising images to gauge reach and receptivity. A limitation is that our findings are cross-sectional and the directionality of associations cannot be determined. However, susceptibility has been a consistent precursor of risk of smoking initiation2 and, in the PATH study wave 1, susceptibility to use tobacco among 12 to 17 year olds appears to be equivalent to the prevalence of experimentation for those aged a few years older.43 Moreover, this sample will be followed longitudinally, allowing us to confirm whether both low and moderate/high receptivity to tobacco advertising predicts later tobacco use behavior among these never smokers.

## Conclusions

In this survey, 41% of US 12 to 13 year olds, as well as half of older adolescents who had never used tobacco, were receptive to tobacco adverting and this receptivity was associated with increased susceptibility to cigarette smoking, regardless of the type of tobacco product advertised. Indeed, there was no difference in the association between receptivity for advertising for the different tobacco products and susceptibility to cigarette smoking. E-cigarette advertising was the most recalled by US adolescents, particularly ads that were shown on television. This high level of recall was achieved despite e-cigarette advertising expenditures being a fraction of those for cigarette marketing.

## Footnotes

• Accepted March 15, 2017.
• Address correspondence to John P. Pierce, PhD, Department of Family Medicine and Public Health, Moores Cancer Center, 3855 Health Sciences Dr, MC 0901, University of California, San Diego, La Jolla, CA 92093-0901. E-mail: jppierce{at}ucsd.edu
• The views and opinions expressed in this presentation are those of the authors only and do not necessarily represent the views, official policy, or position of the US Department of Health and Human Services or any of its affiliated institutions or agencies.

• FINANCIAL DISCLOSURE: The authors have indicated they have no financial relationships relevant to this article to disclose.

• FUNDING: This study is supported with federal funds from the National Institute on Drug Abuse, National Institutes of Health, and the Food and Drug Administration, Department of Health and Human Services, under a contract to Westat (contract HHSN271201100027C). Funded by the National Institutes of Health (NIH).

• POTENTIAL CONFLICT OF INTEREST: The authors have indicated they have no potential conflicts of interest to disclose.


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# Solve for: (a-b)/(a)-(b-a)/(b)

## Expression: $\frac{ a-b }{ a }-\frac{ b-a }{ b }$

Write all numerators above the least common denominator $ab$

$\frac{ b \times \left( a-b \right)-a \times \left( b-a \right) }{ ab }$

Distribute $b$ through the parentheses

$\frac{ ab-{b}^{2}-a \times \left( b-a \right) }{ ab }$

Distribute $-a$ through the parentheses

$\frac{ ab-{b}^{2}-ab+{a}^{2} }{ ab }$

Since two opposites add up to $0$, remove them from the expression

$\frac{ -{b}^{2}+{a}^{2} }{ ab }$

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2 added 420 characters in body

For a field $k$, by a "Nullstellensatz" over $k$, I mean an explicit description of the Galois connection between subsets of $k^n$ and ideals in the polynomial ring $k[x_1,\ldots,x_n]$. See this MO question of mine for more on this perspective.

Whenever one has a ring of functions on a space $X$, there is an induced Galois connection, hence one can ask for a Nullstellensatz: see page 15 of these notes for a(n unfortunately not yet very good) description of this perspective.

In particular there are analytic Nullstellensatze, for instance. I recently found this interesting note concerning the possibility of a Nullstellensatz for the ring of continuous functions on a compact space. (It shows that things work out nicely for maximal ideals and argues fairly convincingly that there is nothing good for all prime ideals. Somehow it did not completely extinguish my hopes, though.)

Certainly the term "Nullstellensatz" has been used for things which do not come under the aegis of the above. For instance, there is Alon's Combinatorial Nullstellensatz, which I unfortunately don't understand very well at all other than as some distant relative of the Chevalley-Warning theorem.

By the way, you say that most of the 200 papers are not referring (albeit obliquely) to Hilbert's Nullstellensatz? I find that somewhat hard to believe. Can you give some examples of far off uses of the term?

(Added: searching "anywhere" for Nullstellensatz on MathSciNet gives 632 matches. I should really turn in soon, so I can't look at them much now. I was surprised by how many of the most recent ones are referring to the Combinatorial Nullstellensatz, but apparently this is only 29 papers in all. By my very cursory glance, most of them do seem to be roughly in line with what I was suggesting above.)

1

For a field $k$, by a "Nullstellensatz" over $k$, I mean an explicit description of the Galois connection between subsets of $k^n$ and ideals in the polynomial ring $k[x_1,\ldots,x_n]$. See this MO question of mine for more on this perspective.

Whenever one has a ring of functions on a space $X$, there is an induced Galois connection, hence one can ask for a Nullstellensatz: see page 15 of these notes for a(n unfortunately not yet very good) description of this perspective.

In particular there are analytic Nullstellensatze, for instance. I recently found this interesting note concerning the possibility of a Nullstellensatz for the ring of continuous functions on a compact space. (It shows that things work out nicely for maximal ideals and argues fairly convincingly that there is nothing good for all prime ideals. Somehow it did not completely extinguish my hopes, though.)

Certainly the term "Nullstellensatz" has been used for things which do not come under the aegis of the above. For instance, there is Alon's Combinatorial Nullstellensatz, which I unfortunately don't understand very well at all other than as some distant relative of the Chevalley-Warning theorem.

By the way, you say that most of the 200 papers are not referring (albeit obliquely) to Hilbert's Nullstellensatz? I find that somewhat hard to believe. Can you give some examples of far off uses of the term?