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# What is the problem with gravity?

1. Mar 20, 2012

### eiyaz

For the longest time educators had repeatedly taught me of the problem with gravity. We have four fundamental forces electromagnetism, strong interaction, weak interaction, and gravitation.

We are able to combine three of these forces but not gravity hence the ongoing development of quantum gravity.

Yet according to general relativity, gravity is NOT a force, but a fictitious force no different from centrifugal force according to wikipedia.

http://en.wikipedia.org/wiki/Fictitious_force

If gravity is not real and is not mediated by a particle what is their to combine? Do we have any evidence that gravity is "real" and is mediated by a particle? If not what is the problem?

What am I missing here?

2. Mar 20, 2012

### mathman

3. Mar 20, 2012

### eiyaz

Not only does that not address the question (I've read that article already) it is also extremely condescending to link wikipedia without quoting the paragraph which answers my question. If you don't understand the question you do not have to be a troll about it.

My questions is if gravity is a fictitious force what issues arise at the quantum level? I may need a more basic explanation. Another issue is the fact that we are able to combine 3 fundamental interactions up to the moments after the big bang, but not gravity. Why do we need to combine gravity with the other three if gravity is not a real force?

In fact if someone can answer just this question:

Why do we need to combine gravity with the other three if gravity is not a real force?

That would be great. I am not able to find the answer on wikipedia so please quote the paragraph if you are linking wikipedia.

Last edited: Mar 20, 2012
4. Mar 20, 2012

### pervect

Staff Emeritus
We have indirect evidence of the existence of gravitational waves - though no direct evidence, yet.

Graitational waves are expected to carry energy, and our detection of them is via the energy loss they cause. (We don't see them direclty yet, but we see the drain in energy as the orbits of the binary pulsars decay, and the amount of decay is what we predict with GR).

So gravity may not be as "fictitious" as you seem to think it is from wikipedia, unless you think "fictitious" things can carry energy. In which case your defintion of "fictitous" is a bit different than mine :-).

5. Mar 20, 2012

### Mark M

Hi eiyaz,

During the 20th century, physicists had combined classical quantum with Special Relativity into quantum electrodynamics. In QED, particle interactions occur in a Minkowski Space, a flat spacetime with no gravitational effects.

This worked fine until it was time to combine general relativity with quantum field theory. In quantum field theory, particles interactions are analysed with a Feynman Diagram:

So, to analyze gravity, it had to be quantized - reduced to particles, gravitons. So the problem comes when you realize gravitons also have a gravitational field (due to their momentum/energy) , so essentially, "gravity gravitates".

So, you get a graviton curving spacetime, further producing gravitons to transmit gravity, then producing more gravitons, and so on.

This leads to a Feynman diagram with many loops, which results in an ultraviolet divergence, a mathematical error that is beyond repair.

6. Mar 20, 2012

### nitsuj

If I see what you're getting at eiyaz, I am with you.

Here is my confusion.

I misunderstand what a "force carrier" is, because I find it odd gravity could fall (pun) under the same term.

I prefer to imagine gravity as a property of space & not a "member" of the force mediating particles.

That's just the way measurements work I guess.

7. Mar 20, 2012

### marcus

Perhaps it is best to think of gravitation as the dynamic geometry of the world. The ability of geometry to interact with matter. "Matter tells geometry how to curve, geometry tells matter how to flow."

So UNIFICATION (of forces) is the wrong idea--the wrong goal for a research program--in that case.

But unification of GR and QM does not need to mean a "unification of forces" as force. It can simply mean to formulate geometry as a quantum field theory---to attain a quantum geometry that is not based on any prior fixed classical geometry. A fully dynamical quantum theory of geometry (interacting with matter) ON WHICH matter fields can be defined.

This is pretty much what the quantum relativists who are working out various approaches to QG are doing.

Yes, but the "ongoing development" you mention is not a unification-of-forces program in any naive sense. One can artificially require LQG to cough up gravitons by pinning it down to approximately flat geometry, but the graviton is not fundamental to the theory. It is formulated at the fundamental level without such mathematical devices.

There is no contradiction. Quantum relativists work on LQG, which does not assume gravity is fundamentally mediated by a particle defined on a fixed prior geometry. They are working out the quantum geometry of the space upon which the matter fields are defined.

"If gravity is not real...?" Well what is real is geometry. And since gravity=geometry, therefore gravity is real. We have to understand space and time. The final question quoted seems vacuous to me. No need to ask it. We don't have to worry about the naive conception of gravity, being mediated, being combined... etc.

Last edited: Mar 20, 2012
8. Mar 20, 2012

### Mark M

Marcus,

If you believe that gravity can be described purely geometrically, then how do you believe Quantum Field Theory works in high energies in small enough distances? For example, a field with sufficiently high energy will contribute to $$T\mu \nu$$ and curve spacetime by
$$G\mu \nu =8\pi GT\mu \nu$$
Since elementary particle interactions are shown in Feynman diagrams, does this not require a quantized gravity particle to properly calculate? (Sorry if LQG provides a better way of doing this, I'm not to familiar with it beyond the basics.)

Also, the semi-classical Einstein equations
$$G=\frac{8\pi G}{c^{4}}\left \langle T\mu \nu \right \rangle\psi$$
imply egienstates of the matter fields in question. This would require an egienstate of the gravitational field, which I'm not exactly sure sure can be done without quantization.

Also, there is the issue that everything else in the universe in quantized. Why not gravity?

9. Mar 21, 2012

### eiyaz

Thanks for great response it cleared somethings up for me!!

My definition of fictitious is any force not directly mediated by a particle. Does the energy have to come from a particle? Could mass causing curvatures in space time result in the wave?

Is this definition a result of a mathematical contingency that forms when attempting to quantized gravity? Or does it suggest that the graviton is in fact "real"?

Isn't this what string theory has been trying to do for the last 30 years?

Also I follow Peter Woit's blog Not Even Wrong, who hates string theory. He states that LQG is not a ToE and in fact string theory is the only attempt at ToE (possibly failed attempt), but if gravity is not a fundamental force wouldn't LQG + SM be fairly complete? If three of the four fundamental forces have particles that mediate it (gluon, W' and Z' bosons, and photon), would it be wrong to say that gravitation is not a fundamental force and merely a property of space-time?

What is the general consensus among the physics community regarding gravity? Do most physicists believe that gravity is mediated by the massless graviton? Or do most believe in quantized GR with no particle?

Last edited: Mar 21, 2012
10. Mar 21, 2012

### Mark M

Yes, that's the idea. Like a boat making ripples in water, a large enough body (such as a supernova explosion) should produce ripples in spacetime.

It suggests the graviton would absolutely be a real particle - the messenger particle of gravity. It's essentially what "tells" matter to follow curvatures in spacetime, just as the photon communicates the electromagnetic force. It has not yet been detected, but is theorized to have no mass, no charge, and a spin of 2. In string theory, it also has the distinction of being represented by a closed string, rather than the open strings of everything else.

Yes, but String theory's unification comes as a bonus, it's original attraction was to explain the graviton.

11. Mar 21, 2012

### marcus

Thanks for responding, Mark. I rarely check stuff here because don't want to get spread thin. Enough to do with Beyond forum and Cosmo forum.

I think you followed what I was saying quite understandingly so you can anticipate how I would reply:

"Everything else in the universe in quantized. Why not geometry?"

Since gravity is geometry, your argument helps to confirm my suspicion that geometry has to be quantized in order to make further fundamental progress in understanding the interaction between geometry and matter.

About your equations, the whole QFT business is build on the assumption of flat space. Fortunately the gravity coupling is small! Gravity is weak. So this works very well. Flat space approximation only works when effect of particles on geometry can be ignored. Whole perturbative approach based on this.

As long there are only a few high energy particles the geometric backreaction can be ignored. Approximate flatness. But what about conditions slightly before and after the start of expansion? With extremely high energy density. That is where quantum geometry comes into its own. Removing the initial singularity and understanding what was going on then. Just my two cents. I just watch the current research on this and try to follow the most significant lines of investigation. Maybe we should take this over to Beyond forum or cosmology if we feel like pursuing it further.

Last edited: Mar 21, 2012
12. Mar 22, 2012

### Mark M

Thanks for responding!

Yes, I agree it will eventually turn out that spacetime is discrete. I guess this comes from my prejudice against infetisimal quantities, as they make no sense in reality. Considering no energy or matter can exist at a sub-planckian distance or quantity, it's rather pointless to suggest an infinite expanse as you get smaller and smaller. Further, the Uncertainty Principle makes it so that at such short distances, spacetime would become so dramatically curved it loses meaning, which essentially leaves you with discrete units. It seems far more logical to suppose spacetime is in fact quantized. As Mark Kac said:

"Be wise, discretize!"

Though, I still do not feel this can be used against the existence of the graviton. I feel strongly that it exists. For example, Maxwell formulated the equations of electromagnetism under the assumption that EM waves were continuous. Of course, this didn't have any effect on his equations, and they were extraordinarily accurate. But, after QFT, we know that a messenger particle, the photon, is needed to transmit the force, to "tell" matter when to respond to other EM fields.

I feel the same situation applies to spacetime - something must "tell" (for lack of a better word) matter how to follow curves in spacetime.

Sure, this discussion probably fits better in Beyond than in an SR/GR sub-thread.


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Method of Difference – Sequences and Series

Here you will learn method of difference in sequences and series with examples.

Let’s begin –

Method of Difference

Some times the $$n^{th}$$ term of a sequence or a series can not be determined by the method, we have discussed earlier. So we compute the difference between the successive terms of given sequence for obtained the $$n^{th}$$ terms.

If $$T_1$$, $$T_2$$, $$T_3$$,…….,$$T_n$$ are the terms of a sequence then some times the terms $$T_2$$ – $$T_1$$,
$$T_3$$ – $$T_1$$…… constitute an AP/GP. $$n^{th}$$ term of the series is determined & the sum to n terms of the sequence can easily be obtained.

Case 1 :

(a) If difference series are in A.P., then

Let $$T_n$$ = $$an^2$$ + bn + c, where a, b, c are constant

(b) If difference of difference series are in A.P.

Let $$T_n$$ = $$an^3$$ + $$bn^2$$ + cn + d, where a, b, c, d are constant

Case 2 :

(a) If difference are in G.P., then

Let $$T_n$$ = $$ar^n$$ + b, where r is common ratio & a, b are constant

(b) If difference of difference are in G.P., then

Let $$T_n$$ = $$ar^n$$ + bn + c, where r is common ratio & a, b, c are constant

Determine constant by putting n = 1, 2, 3 ……. n and putting the value of $$T_1$$, $$T_2$$, $$T_3$$……. and sum of series $$S_n$$ = $${\sum}T_n$$

Example : Find the sum to n terms of the series : 3 + 15 + 35 + 63 + …..

Solution : The difference between the successive terms are 15 – 3 = 12, 35 – 15 = 20, 63 – 35 = 28, …. Clearly,these differences are in AP.

Let $$T_n$$ be the nth term and $$S_n$$ denote the sum to n terms of the given series

Then, $$S_n$$ = 3 + 15 + 35 + 63 + ….. + $$T_{n-1}$$ + $$T_n$$ ……(i)

Also, $$S_n$$ =     3 + 15 + 35 + ….. + $$T_{n-1}$$ + $$T_n$$ ……(ii)

Subtracting (ii) from (i), we get

0 = 3 + [12 + 20 + 28 + …… + $$T_{n-1}$$ + $$T_n$$] – $$T_n$$

$$\implies$$ $$T_n$$ = 3 + $$(n-1)\over 2$${2*12+(n-1-1)*8} = 3 + (n-1){12+4n-8}

$$\implies$$ $$T_n$$ = 3 + (n-1)(4n+4) = $$4n^2 – 1$$

$$\therefore$$ $$S_n$$ = $$\sum_{k=1}^{n}$$ $$T_k$$ = $$\sum_{k=1}^{n}$$ ($$4k^2 – 1$$) = 4$$\sum_{k=1}^{n}$$$$k^2$$ – $$\sum_{k=1}^{n}$$ 1

$$\implies$$ $$S_n$$ = 4{$$n(n+1)(2n+1)\over 6$$} – n = $$n\over 3$$ ($$4n^2 + 6n – 1$$)

Related Questions

Find the sum of n terms of the series 3 + 7 + 14 + 24 + 37 + …….

Find the sum of n terms of the series 1.3.5 + 3.5.7 + 5.7.9 + ……

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# Thread: Using elementary taylor series for taylor polynomial

1. ## Using elementary taylor series for taylor polynomial

Ok, so the problem is to use a sixth-degree Taylor polynomial centered at c for the function f to obtain the required approximation. Here's the info:

Function: $f(x)=x^2e^{-x}$
Center: $c=0$
Approximation: $f(1/4)$

Now, my main question is since there is an elementary Taylor series for $e^x$, which with a center at 0 is $\frac{e^x x^n}{n!}$, then the Taylor series for $e^{-x}$ must be $\frac{e^{-x} -x^n}{n!}$. So for $x^2e^{-x}$, could it be as simple as placing $x^2$ in front of the Taylor series for $e^{-x}$ since n is the value that is changing, so any term with only x can be treated as a constant?

2. Hello,

Originally Posted by Mathstud28
$\sum_{n=0}^{\infty} (-1)^{n} \frac{x^{n}}{n!}$* $x^2$
Some extra \ and a missing one

Originally Posted by Mathstud28
$\sum_{n=0}^{\infty} (-1)^{n} \frac{x^{n}}{n!}$* $x^2$ and then plug in $\frac{1}{4}$
No need to put \ at the end of each portion. It's just here to start frac, sum, etc...

3. Ok, I'm lost, lol. So how did you arrive at that answer, so I know how to do it, and not just what the answer is?

4. ## Because

since $e^{x}\$ $\sum_{n=0}^{/infty}\frac{x^{n}}{n!}\$ then $e^{-x}/$≡]≡ $\sum_{n=0}^{/infty}\frac{(-x)^{n}}{n!}\$=]≡ $\sum_{n=0}^{/infty}(-1)^{n}\frac{x^{n}}{n!}\$ by splitting -x into -1*x and seperating the exponent...so then you just multiply that by x² to get $x^2e^{-x}\$= $x^2\$ $\sum_{n=0}^{/infty}(-1)^{n}\frac{x^{n}}{n!}\$ and to approximate f(1/4) you just imput 1/4 into ]= $x^2\$ $\sum_{n=0}^{/infty}(-1)^{n}\frac{x^{n}}{n!}\$

5. Originally Posted by Mathstud28
$x^2e^{-x}\$= $\sum_{n=0}^{\infty}(-1)^{n}\frac{x^{n}}{n!}\$ $x^2\$

and if you put in $\sum_{n=0}^{10}(-1)^{n}\frac{x^{n}}{n!}\$ $x^2\$ you get.048675 and if you input $\frac{1}{4}\$ into $x^2e^{-x}\$ you get the same answer

and if you really wanted to be good you could distribute and get $x^2e^{-x}\$= $\sum_{n=0}^{\infty}(-1)^{n}\frac{x^{n+2}}{n!}\$
Ok, so would it be mathematically correct if I distributed the $x^2$ into the $x^n$ like shown below to clean it up?

$x^2e^{-x}\$= $\sum_{n=0}^{\infty}(-1)^{n}\frac{x^{2n}}{n!}\$

6. ## No

Its a common mistake but $x^2x^{n}\$ does not equal $x^{2n}\$ it equals $x^{n+2}\$... $x^{2n}\$= $(x^{n})^2\$

7. Originally Posted by Mathstud28
Its a common mistake but $x^2x^{n}\$ does not equal $x^{2n}\$ it equals $x^{n+2}\$... $x^{2n}\$= $(x^{n})^2\$
Ah that's right, sheesh I'm forgetting my simple exponent rules.

Ok, here we go.

$x^2e^{-x}\$= $\sum_{n=0}^{\infty}(-1)^{n}\frac{x^{n+2}}{n!}\$

$S1 (x) = x^2 - x^3$

$S2 (x) = x^2 - x^3 + \frac{x^4}{2!}$

$S3 (x) = x^2 - x^3 + \frac{x^4}{2!} - \frac{x^5}{3!}$

$S4 (x) = x^2 - x^3 + \frac{x^4}{2!} - \frac{x^5}{3!} + \frac{x^6}{4!}$

$S5 (x) = x^2 - x^3 + \frac{x^4}{2!} - \frac{x^5}{3!} + \frac{x^6}{4!} - \frac{x^7}{5!}$

$S6 (x) = x^2 - x^3 + \frac{x^4}{2!} - \frac{x^5}{3!} + \frac{x^6}{4!} - \frac{x^7}{5!} + \frac{x^8}{6!}$

Now then, $S6 (1/4) = 0.0487$ approximately....look about right?

8. ## If you look at my post

I actually not only showed that it is .047 but I also showed the $x^2\$ distributed through the Summation is $x^{n+2}\$ haha maybe you should read posts more carefully haha...but yes you are completely right good job!

9. Originally Posted by Mathstud28
I actually not only showed that it is .047 but I also showed the $x^2\$ distributed through the Summation is $x^{n+2}\$ haha maybe you should read posts more carefully haha...but yes you are completely right good job!
I could have sworn you did, actually, but I scrolled up through the thread before I did submit that post and all I saw was a slew of syntax error posts and one that said "disregard this" which could have had that post before you edited it out. Easy tiger. :P thanks though....my teacher didn't teach us Taylor polynomials, just decided to hold us responsible for them, hence the question in the first place.

10. ## O no its fine

My teacher never taught me either...then again I am sixteen lol....hooray for independent study!


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# Hypothesis testing with binary pattern

Let's say you have a sequence $$\{x_i\}_{i = 1}^N$$ of ones and zeros. You know that $$P(x_i = 1) = \frac{1}{3}$$.

You want to test two hypotheses

$$H_0$$: The sequence is $$iid$$ with $$P(x_i = 1) \ \forall \ i$$

$$H_1$$: The sequence is full of $$001$$ sub-sequences, but has noise. For example, the sequence might start off as $$001001$$, then have a period of random noise (where $$P(x_i = 1) = \frac{1}{3}$$).

After the period of noise, the sequence matches the $$001001$$ pattern again. Importantly, the $$1$$ values might land on different values than when it started.

That is, we are not testing whether $$mod(i, 3) = 0 \implies x_i = 1$$. We are merely testing whether the "normal" state of this sequence is batches of $$001001001$$ but those batches could start at any value.

Another way of putting this is that we allow for "extra" bits to be inserted.

Ideally, I would like to find a test statistic that enables me to reject the null in this scenario. One possible solution, suggested in the comments below, is to count the number of $$001$$ sequences, and compare to the number of $$001$$ sequences you would find under the null.

This is an interesting solution. I will have to think more about how one might construct a confidence interval or assymptotics for this statistic.

Update: After thinking about the scenario that we are working with more, I think I can safely assume that only $$0$$ values are inserted. This is a much simpler problem.

$$H_0$$: The sequence is $$iid$$ with $$P(x_i = 1) = \sum_{i = 1}^N x_i$$. We guess it's Bernoulli distributed according to the number of $$1$$s we observe in the data.

$$H_1$$: The sequence has lots of $$001001001$$ segments, but extra $$0$$ values get inserted at random into the sequence (which creates offsets as above).

I don't think this materially changes the approach too much, but means we no longer need to worry about the distribution of the "Noise" sequences.

• Think carefully about what you want to conclude. "On average, one-third of the bits are 1, rest are 0" is not the same as "Every third bit is a 1, rest are 0". The latter is true if and only if the sequence is $001001001001\cdots$ which is easy to test for; the former is true not only for $001001001001\cdots$ but also for $0010000000111\cdots$ as well as for numerous other sequences. What do you wish to test for? – Dilip Sarwate Mar 26 at 20:09
• Thanks for asking about the clarification. I definitely want to reject the sequence $0010000000111$. Recall that after errors, the sequence "re-starts" doing $001001001$, but the placement of $1$s might not be the same. I.e. before, you might have $1$ at $1, 4, 7$. Then $8$ is an errant $0$. After this, you would expect $1$ at $9, 12, 15...$. – genauguy Mar 26 at 20:29
• Hi: you can use the test of proportions where, in this $p = 1$. under the null. But be careful here because $\hat{p}$ will be the number of 001 sequences out of the total number of three digit sequences. So, if you had say, 000001001000, then this is four three digit sequences and two of the four are 001 so your $\hat{p} = 2/4 = 0.5$. From there, you can carry out the proportions test at whatever significance level you choose. If you are not familiar with the test, any intro stat book will explain the steps. The only catch for your is what $\hat{p}$ is which I explained above. – mark leeds Mar 26 at 21:29
• Are you assuming that you know the alignment of where the $1$s should be? – IanJ Mar 26 at 21:57
• Also would you ever allow a slip? I.E. An extra or missing bit. – IanJ Mar 26 at 22:23

Given what you explained, you can count the number of times the sequence is 001, 100 or 010. Then, $$\hat{p}$$ is defined as that count divided by the number of sequences, $$N$$.

If we define the null as the hypothesis that the proportion of good sequences is 1.0, then the test is

$$H_0: p = 1.0$$ versus $$H_1: p \neq 1.0$$

As far as testing the null, ( I'm assuming N is large here so that the normal approximation to the binomial holds ) you can calculate the test statistic

as $$Z = \frac{ \hat{p} - p_{0}}{\sqrt{\frac{\bar{p}(1-\bar{p})}{N}}}$$

and use $$p_{0} = 1$$ and $$\bar{p} = 0.5$$. Then, reject the null if $$|Z| > Z_{\frac{\alpha}{2}}$$ where $$\alpha$$ is the critical value of the test.

The CI would be $$\hat{p} ~ \pm Z_{\frac{\alpha}{2}} \times {\sqrt{\frac{\bar{p}(1-\bar{p})}{N}}}$$.

If a digit in one of the three digit sequences is is missing or there's an extra, and you know this, then you can just count that sequence as a bad sequence.

• Thank you so much! One question. Our sequence is not IID. Given that you observe 001, the probability that the next sequence is 010 is quite high. If I assume that this sequence is ergodic (which I think just requires something mild about our the "noise" periods, then maybe I can just take a random sample of spaced out trios, right? Those are as good as independent. – genauguy Mar 27 at 4:31
• Hi: If the three digit sequences are not independent, then the proportions test can't be used. But I think what you're referring to as a fix to this is to randomly sample the sequences in order to sort of make them indepedent ? This sounds reasonable but you'll need a lot of sample data. I'm not sure why you refer to simulations ? Do you not have actual data ? – mark leeds Mar 27 at 18:40
• No, we do have actual data. Sorry by "random sample" I mean randomly selected subsets of the series. – genauguy Mar 27 at 18:42
• yes, you can sample with replacement to hopefully deal with the correlation. Note that this is called bootstrapping so you may want to google for an intro to that if you're not familar with the approach. You will sample to get what is a referred to ad a "boostrapped statistic" and then use that to test your hypothesis, construct a CI etc. The underlying distribution is no longer binomial ( because of the correlation ) but the boostrapped distribution can be used to test the hypothesis. There are some good intros out there. If you can't find one, let me know and I can try to find one. – mark leeds Mar 27 at 19:20
• Yes I am familiar with bootstrapping. I should have connected the dots there.I will attempt this! – genauguy Mar 27 at 19:26

You seem to describe a discrete hidden Markov chain X with hidden states (let's name them 1,2,3) and transition matrix

λ*[0 1 0; 0 0 1; 1 0 0] + (1 - λ)*ones(3,3)/3

and observation operator

H(x) = x == 3

If λ == 1, you have a deterministic 1 2 3 1 2 3 1 2 3 latent sequence and will observe 0 0 1 0 0 1 0 0 1.... If λ == 1, the latent states are i.i.d 1 2 1 2 3 3 2 1 3 and you'll observe an i.d.d sequence of 0 0 0 0 1 1 0 0 1...

If you put a prior on λ, you can use Bayesian inference to estimate it, or otherwise maximum likelihood if you want. Related: Baum-Welch algorithm.

You can play with

using Random, LinearAlgebra, Distributions, StatsBase

P =  λ*[0 1 0; 0 0 1; 1 0 0] + (1 - λ)*ones(3,3)/3
"""
samplefrom(p)

Produces a sample from the probability distribution
with prob .mass vector p.

Helpful: to sample from the conditional distribution given state i

j = samplefrom(P[i, :]) # : for all j

"""
samplefrom(p) = sample(1:3, weights(p))

"""
samplefromchain(p, P, n)

Samples an n-step Markov chain with starting distribution p
with transition matrix P.
"""
function samplefromchain(p, P, n) # n is number of trans
s = samplefrom(p)
statevector = [s]
for i in 1:n
s = samplefrom(P[s, :])
push!(statevector, s)
end
return statevector
end
$$$$
`

I would take a different approach (less formal?) than @mschuaer.

1. Generate the autocorrelation of your sequence.
2. Shuffle your sequence by sampling from it without replacement (x1000)
3. Compare the actual autocorrelation with the shuffle.


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# Using the system function $H(z)$ to the solve the output of a DT LTI system

I have a question which asks me to find the magnitude and phase response of

$$H(z) = \frac{1}{1-0.5z^{-1}}$$ for which I solved by doing $$H(e^{j\Omega}) = \frac{1}{1-0.5e^{-j\Omega}} = \frac{1}{1-0.5\cos(\Omega)+j0.5\sin(\Omega)}$$ $$|{H(e^{j\Omega})}| = \frac{1}{\sqrt{(1-0.5\cos(\Omega))^{2}+(0.5\sin(\Omega))^{2}}} = \frac{1}{\sqrt{\frac{5}{4} - \cos(\Omega)}}$$ $$\angle H(e^{j\Omega}) = -\tan^{-1} \left(\frac{\frac{1}{2}\sin(\Omega)}{1-\frac{1}{2}\cos(\Omega)}\right)$$ But, I am unsure how to solve the second part of the problem, which asks me to compute the system response to $$f[k] = A\cos\left(\frac{\pi}{2}k +\frac{\pi}{10}\right)$$

Let's first briefly recall the meaning of $H(z)$. Let $y[n]$ be the output of the LTI system. Given an input $x[n]$, the output can be written $$y[n] = h[n] *x[n] = \sum_{k=-\infty}^\infty h[k]x[n-k]$$ where $h[n]$ is the impulse response of the LTI system en $*$ represents the convolution. Let's now assume that $x[n]$, the input of the LTI system, is of the form $z^n$. In this case, $$y[n] = \sum_{k=-\infty}^\infty h[k]z^{n-k} = z^n\left(\sum_{k=-\infty}^\infty h[k]z^{-k}\right) = z^nH(z)$$ where $H(z)$ denotes the transfer function of the LTI system (the z-transform of the impulse response). It can be seen that that $z^n = e^{j\Omega}$ is an eigenfunction of the system, that is, it is only affected by a complex factor (i.e. $H(z)$, the so-called eigenvalue) at the output of your system.
Now, your input $f[n]$ can be written as $$f[n] = \mathcal{R}\{Ae^{j(\frac{\pi}{2}n + \frac{2\pi}{10})}\} = \mathcal{R}\{Ae^{j\frac{2\pi}{10}}e^{j\frac{\pi}{2}n}\}$$ and $Ae^{j\frac{2\pi}{10}}e^{j\frac{\pi}{2}n}$ matches the form of an eigenfunction (with a complex factor $Ae^{j\frac{2\pi}{10}}$). The output of your system to $Ae^{j\frac{2\pi}{10}}e^{j\frac{\pi}{2}n}$ is then given, using linearity of your system and the polar notation of $H$, by $$y[n] = Ae^{j\frac{2\pi}{10}}H(e^{j\frac{\pi}{2}})e^{j\frac{\pi}{2}n} = Ae^{j\frac{2\pi}{10}}|H(e^{j\frac{\pi}{2}})|e^{\arg\{H(e^{j\frac{\pi}{2}})\}}e^{j\frac{\pi}{2}n}.$$ Finally, again by linearity, the output of your system to $f[n]$ is given by the real part of that.


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# Complete Heyting algebra

(Redirected from Locale (mathematics))

In mathematics, especially in order theory, a complete Heyting algebra is a Heyting algebra which is complete as a lattice. Complete Heyting algebras are the objects of three different categories; the category CHey, the category Loc of locales, and its opposite, the category Frm of frames. Although these three categories contain the same objects, they differ in their morphisms, and thus get distinct names. Only the morphisms of CHey are homomorphisms of complete Heyting algebras.

Locales and frames form the foundation of pointless topology, which, instead of building on point-set topology, recasts the ideas of general topology in categorical terms, as statements on frames and locales.

## Definition

Consider a partially ordered set (P, ≤) that is a complete lattice. Then P is a complete Heyting algebra if any of the following equivalent conditions hold:

• P is a Heyting algebra, i.e. the operation ( x ∧ − ) has a right adjoint (also called the lower adjoint of a (monotone) Galois connection), for each element x of P.
• For all elements x of P and all subsets S of P, the following infinite distributivity law holds:
$x \wedge \bigvee_{s \in S} s = \bigvee_{s \in S} (x \wedge s).$
• P is a distributive lattice, i.e., for all x, y and z in P, we have
$x \wedge ( y \vee z ) = ( x \wedge y ) \vee ( x \wedge z )$
and the meet operations ( x ∧ − ) are Scott continuous for all x in P (i.e., preserve the suprema of directed sets) .

## Examples

The system of all open sets of a given topological space ordered by inclusion is a complete Heyting algebra.

## Frames and locales

The objects of the category CHey, the category Frm of frames and the category Loc of locales are the complete lattices satisfying the infinite distributive law. These categories differ in what constitutes a morphism.

The morphisms of Frm are (necessarily monotone) functions that preserve finite meets and arbitrary joins. Such functions are not homomorphisms of complete Heyting algebras. The definition of Heyting algebras crucially involves the existence of right adjoints to the binary meet operation, which together define an additional implication operation ⇒. Thus, a homomorphism of complete Heyting algebras is a morphism of frames that in addition preserves implication. The morphisms of Loc are opposite to those of Frm, and they are usually called maps (of locales).

The relation of locales and their maps to topological spaces and continuous functions may be seen as follows. Let

$f\colon X\to Y$

be any map. The power sets P(X) and P(Y) are complete Boolean algebras, and the map

$f^{-1}\colon P(Y)\to P(X)$

is a homomorphism of complete Boolean algebras. Suppose the spaces X and Y are topological spaces, endowed with the topology O(X) and O(Y) of open sets on X and Y. Note that O(X) and O(Y) are subframes of P(X) and P(Y). If ƒ is a continuous function, then

$f^{-1}\colon O(Y)\to O(X)$

preserves finite meets and arbitrary joins of these subframes. This shows that O is a functor from the category Top of topological spaces to the category Loc of locales, taking any continuous map

$f\colon X\to Y$

to the map

$O(f)\colon O(X)\to O(Y)$

in Loc that is defined in Frm to be the inverse image frame homomorphism

$f^{-1}\colon O(Y)\to O(X).$

It is common, given a map of locales

$f\colon A\to B$

in Loc, to write

$f^{*}\colon B\to A$

for the frame homomorphism that defines it in Frm. Hence, using this notation, O(ƒ) is defined by the equation O(ƒ)* = ƒ−1.

Conversely, any locale A has a topological space S(A) that best approximates the locale, called its spectrum. In addition, any map of locales

$f\colon A\to B$

determines a continuous map

$S(A)\to S(B),$

and this assignment is functorial: letting P(1) denote the locale that is obtained as the powerset of the terminal set 1 = { * }, the points of S(A) are the maps

$p\colon P(1)\to A$

in Loc, i.e., the frame homomorphisms

$p^*\colon A\to P(1).$

For each aA we define the set UaS(A) that consists of the points pS(A) such that p*(a) = { * }. It is easy to verify that this defines a frame homomorphism AP(S(A)), whose image is therefore a topology on S(A). Then, if

$f\colon A\to B$ is a map of locales,

to each point pS(A) we assign the point S(ƒ)(q) defined by letting S(ƒ)(p)* be the composition of p* with ƒ*, hence obtaining a continuous map

$S(f)\colon S(A)\to S(B).$

This defines a functor $S$ from Loc to Top, which is right adjoint to O.

Any locale that is isomorphic to the topology of its spectrum is called spatial, and any topological space that is homeomorphic to the spectrum of its locale of open sets is called sober. The adjunction between topological spaces and locales restricts to an equivalence of categories between sober spaces and spatial locales.

Any function that preserves all joins (and hence any frame homomorphism) has a right adjoint, and, conversely, any function that preserves all meets has a left adjoint. Hence, the category Loc is isomorphic to the category whose objects are the frames and whose morphisms are the meet preserving functions whose left adjoints preserve finite meets. This is often regarded as a representation of Loc, but it should not be confused with Loc itself, whose morphisms are formally the same as frame homomorphisms in the opposite direction.

## Literature

Still a great resource on locales and complete Heyting algebras.
• G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove, and D. S. Scott, Continuous Lattices and Domains, In Encyclopedia of Mathematics and its Applications, Vol. 93, Cambridge University Press, 2003. ISBN 0-521-80338-1
Includes the characterization in terms of meet continuity.
• Francis Borceux: Handbook of Categorical Algebra III, volume 52 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, 1994.
Surprisingly extensive resource on locales and Heyting algebras. Takes a more categorical viewpoint.


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# How do you differentiate (y^2+1)^2-x=0?

Apr 10, 2018

The implicit derivative $\frac{\mathrm{dy}}{\mathrm{dx}}$ is $\frac{1}{4 y \left({y}^{2} + 1\right)}$.

#### Explanation:

First, rearrange a little:

${\left({y}^{2} + 1\right)}^{2} - x = 0$

${\left({y}^{2} + 1\right)}^{2} = x$

Now, take the implicit derivative (which basically means treat $y$ as a function of $x$):

$\frac{d}{\mathrm{dx}} \left[{\left({y}^{2} + 1\right)}^{2}\right] = \frac{d}{\mathrm{dx}} \left[x\right]$

$\frac{d}{\mathrm{dx}} \left[{\left({y}^{2} + 1\right)}^{2}\right] = 1$

$2 {\left({y}^{2} + 1\right)}^{2 - 1} \cdot \frac{d}{\mathrm{dx}} \left[{y}^{2} - 1\right] = 1$

$2 {\left({y}^{2} + 1\right)}^{1} \cdot \frac{d}{\mathrm{dx}} \left[{y}^{2} - 1\right] = 1$

$2 \left({y}^{2} + 1\right) \cdot \frac{d}{\mathrm{dx}} \left[{y}^{2} - 1\right] = 1$

$\frac{d}{\mathrm{dx}} \left[{y}^{2} - 1\right] = \frac{1}{2 \left({y}^{2} + 1\right)}$

$\frac{\mathrm{dy}}{\mathrm{dx}} \cdot 2 y = \frac{1}{2 \left({y}^{2} + 1\right)}$

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\frac{1}{2 \left({y}^{2} + 1\right)}}{2 y}$

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{1}{4 y \left({y}^{2} + 1\right)}$

That's the derivative. Hope this helped!


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# Why is it a problem if the outputs of an activation function are not zero-centered?

In this lecture, the professor says that one problem with the sigmoid function is that its outputs aren't zero-centered. Are the explanation provided by the professor regarding why this is bad is that the gradient of our loss w.r.t. the weights $$\frac{\partial L}{\partial w}$$ which is equal to $$\frac{\partial L}{\partial \sigma}\frac{\partial \sigma}{\partial w}$$ will always be either negative or positive and we'll have a problem updating our weights as she shows in this slide, we won't be able to move in the direction of the vector $$(1,-1)$$. I don't understand why since she only talks about one component of our gradient and not the whole vector. if the components of the gradient of our loss will have different signs which will allow us to adjust to different directions I'm I wrong? But the thing that I don't understand is how this property generalizes to non zero-centered functions and non-zero centered data?

Yes, if the activation function of the network is not zero centered, $$y = f(x^{T}w)$$ is always positive or always negative. Thus, the output of a layer is always being moved to either the positive values or the negative values. As a result, the weight vector needs more update to be trained properly and the number of epochs needed for the network to get trained also increases. This is why the zero centered property is important, though it is NOT necessary.

Zero-centered activation functions ensure that the mean activation value is around zero. This property is important in deep learning because it has been empirically shown that models operating on normalized data––whether it be inputs or latent activations––enjoy faster convergence.

Unfortunately, zero-centered activation functions like tanh saturate at their asymptotes –– the gradients within this region get vanishingly smaller over time, leading to a weak training signal.

ReLU avoids this problem but it is not zero-centered. Therefore all-positive or all-negative activation functions whether sigmoid or ReLU can be difficult for gradient-based optimization. So, To solve this problem deep learning practitioners have invented a myriad of Normalization layers (batch norm, layer norm, weight norm, etc.). we can normalize the data in advance to be zero-centered as in batch/layer normalization.

Reference:

A Survey on Activation Functions and their relation with Xavier and He Normal Initialization

• Thank you for your answer. Can you provide more details please ? I don't understand how if the $y=f(x^{T}w)$ function isn't zero-centered then the outputs will always be either positive or negative. If we can imagine that $f$ has a symetrical graph (like a gaussian function) shifted towards the positives for example, then it'll all depends on the "width" am I wrong ? A more concrete example: $y=tanh(z-2)$. Then if $x^{T}w=[-4,2,4]$ we'll have $tanh(x^{T}w)=[-1,0,1]$. Do we always expect that $x^{T}w$ will be very close to zero so that we aren't in the later case? – Daviiid Mar 28 at 15:37
• like For the sigmoid function, the gradient saturates as $σ(x)→1$ or $σ(x)→0$ . We can simply tell this by the slope of the function as these points are approached, and also compute the derivative. We know that $σ′(x)=σ(x)(1−σ(x))$ , so $σ′(x)→0$ at these extreme values. – Faizy Mar 28 at 16:50
• Therefore, if inputs are not zero-centered and with sigmoid , saturation will be more likely than if our inputs were zero-centered. but early on in learning, we generally want larger gradients, since our error is high. This can drastically slow down $\eta$. Since zero-centering the data will bring the inputs of the sigmoid away from where they can saturate, zero-centering could increase the gradient signal during the early stages of learning, providing for faster learning. – Faizy Mar 28 at 16:58
• you know that for sigmoid the bound in the range of (0,1). Hence it always produces a non-negative value as output. – Faizy Mar 28 at 17:21
• @Faizy thank you for your answer. I think the term zero-centered may be a little confusing because we may have zero-centered data but that have big magnitude, imagine the set of points $\{ -10, -5, -3, 3, 5, 10\}$. Then we may say that this set is zero-centered, and it'll still give us saturation. But I understand what you mean. Thank you for all the explanation. – Daviiid Mar 28 at 18:42


text:
The Annals of Statistics

Selective inference with a randomized response

Abstract

Inspired by sample splitting and the reusable holdout introduced in the field of differential privacy, we consider selective inference with a randomized response. We discuss two major advantages of using a randomized response for model selection. First, the selectively valid tests are more powerful after randomized selection. Second, it allows consistent estimation and weak convergence of selective inference procedures. Under independent sampling, we prove a selective (or privatized) central limit theorem that transfers procedures valid under asymptotic normality without selection to their corresponding selective counterparts. This allows selective inference in nonparametric settings. Finally, we propose a framework of inference after combining multiple randomized selection procedures. We focus on the classical asymptotic setting, leaving the interesting high-dimensional asymptotic questions for future work.

Article information

Source
Ann. Statist., Volume 46, Number 2 (2018), 679-710.

Dates
Revised: February 2017
First available in Project Euclid: 3 April 2018

https://projecteuclid.org/euclid.aos/1522742433

Digital Object Identifier
doi:10.1214/17-AOS1564

Mathematical Reviews number (MathSciNet)
MR3782381

Zentralblatt MATH identifier
06870276

Subjects
Primary: 62M40: Random fields; image analysis
Secondary: 62J05: Linear regression

Citation

Tian, Xiaoying; Taylor, Jonathan. Selective inference with a randomized response. Ann. Statist. 46 (2018), no. 2, 679--710. doi:10.1214/17-AOS1564. https://projecteuclid.org/euclid.aos/1522742433

References

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• Belloni, A., Chernozhukov, V. and Wang, L. (2011). Square-root lasso: Pivotal recovery of sparse signals via conic programming. Biometrika 98 791–806.
• Benjamini, Y. and Hochberg, Y. (1995). Controlling the false discovery rate: A practical and powerful approach to multiple testing. J. Roy. Statist. Soc. Ser. B 57 289–300.
• Benjamini, Y. and Stark, P. B. (1996). Nonequivariant simultaneous confidence intervals less likely to contain zero. J. Amer. Statist. Assoc. 91 329–337.
• Bühlmann, P. (2013). Statistical significance in high-dimensional linear models. Bernoulli 19 1212–1242.
• Chatterjee, S. (2005). A simple invariance theorem. Preprint. Available at arXiv:math/0508213.
• Chung, E. and Romano, J. P. (2013). Exact and asymptotically robust permutation tests. Ann. Statist. 41 484–507.
• Cox, D. R. (1975). A note on data-splitting for the evaluation of significance levels. Biometrika 62 441–444.
• Dwork, C., Feldman, V., Hardt, M., Pitassi, T., Reingold, O. and Roth, A. (2015). Preserving statistical validity in adaptive data analysis [extended abstract]. In STOC’15—Proceedings of the 2015 ACM Symposium on Theory of Computing 117–126. ACM, New York.
• Fithian, W., Sun, D. and Taylor, J. (2014). Optimal inference after model selection. Available at arXiv:1410.2597.
• Fithian, W., Taylor, J., Tibshirani, R. and Tibshirani, R., (2015). Selective sequential model selection. Available at arXiv:1512.02565.
• Götze, F. (1991). On the rate of convergence in the multivariate CLT. Ann. Probab. 19 724–739.
• Harris, X. T. (2016). Prediction error after model search. Preprint. Available at arXiv:1610.06107.
• Harris, X. T., Panigrahi, S., Markovic, J., Bi, N. and Taylor, J. (2016). Selective sampling after solving a convex problem. Preprint. Available at arXiv:1609.05609.
• Javanmard, A. and Montanari, A. (2014). Confidence intervals and hypothesis testing for high-dimensional regression. J. Mach. Learn. Res. 15 2869–2909.
• Lee, J. D., Sun, D. L., Sun, Y. and Taylor, J. E. (2016). Exact post-selection inference, with application to the lasso. Ann. Statist. 44 907–927.
• Lehmann, E. L. (1986). Testing Statistical Hypotheses, 2nd ed. Wiley, New York.
• Lockhart, R., Taylor, J., Tibshirani, R. J. and Tibshirani, R. (2014). A significance test for the lasso. Ann. Statist. 42 413–468.
• Meinshausen, N. and Bühlmann, P. (2010). Stability selection. J. R. Stat. Soc. Ser. B Stat. Methodol. 72 417–473.
• Meinshausen, N., Meier, L. and Bühlmann, P. (2009). $p$-values for high-dimensional regression. J. Amer. Statist. Assoc. 104 1671–1681.
• Pakman, A. and Paninski, L. (2014). Exact Hamiltonian Monte Carlo for truncated multivariate Gaussians. J. Comput. Graph. Statist. 23 518–542.
• Rosenthal, R. (1979). The file drawer problem and tolerance for null results. Psychol. Bull. 86 638.
• Sun, T. and Zhang, C.-H. (2012). Scaled sparse linear regression. Biometrika 99 879–898.
• Tian, X., Bi, N. and Taylor, J. (2016). Magic: A general, powerful and tractable method for selective inference. Preprint. Available at arXiv:1607.02630.
• Tian, X., Loftus, J. R. and Taylor, J. E. (2015). Selective inference with unknown variance via the square-root lasso. Preprint. Available at arXiv:1504.08031.
• Tian, X. and Taylor, J. (2015). Asymptotics of selective inference. Available at arXiv:1501.03588.
• Tian, X. and Taylor, J. (2018). Supplement to “Selective inference with a randomized response.” DOI:10.1214/17-AOS1564SUPP.
• Tibshirani, R. (1996). Regression shrinkage and selection via the lasso. J. Roy. Statist. Soc. Ser. B 58 267–288.
• Tibshirani, R. J., Rinaldo, A., Tibshirani, R. and Wasserman, L. (2015). Uniform asymptotic inference and the bootstrap after model selection. Preprint. Available at arXiv:1506.06266.
• Tibshirani, R. J., Taylor, J., Lockhart, R. and Tibshirani, R. (2016). Exact post-selection inference for sequential regression procedures. J. Amer. Statist. Assoc. 111 600–620.
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Supplemental materials

• Supplement to “Selective inference with a randomized response”. We provide additional sampling schemes, technical details for plugin variance estimators and proofs for all the theorems and lemmas in the supplementary materials.


text:
You are reading the documentation for version 1.7 of OpenStructure. You may also want to read the documentation for: 1.1 1.2 1.3 1.4 1.5 1.6 1.7.1 1.8 1.9 1.10 1.11 2.0 2.1 2.2 devel

# Editors¶

The structure, topology and connectivity of entities is edited via editors. This includes operations such as changing atom positions, connecting atoms with bonds as well as adding and removing chains, residues and atoms. There are two flavors of editors, one for the internal coordinate system (ICSEditor) and one for the external coordinate system (XCSEditor).

## Edit Modes¶

Editors support two modes: An unbuffered edit mode and a buffered edit mode. In the unbuffered edit mode, dependent information such as the spatial organizer and the internal coordinate system (in case of the XCSEditor) are updated after every change. In buffered edit mode, the updates are delayed until one of the following happens:

The editors follow the RIAA (resource allocation is initialization) principle: Whenever an editor is requested an internal reference counter is incremented. In the destructor, this reference count is decremented. When the count drops to zero, the dependent information is updated.

In Python, one can not rely on the destructors being called. It is advised to always put a call to XCSEditor.UpdateICS() or ICSEditor.UpdateXCS() when the editing is finished. Alternatively, starting from Python version 2.6, one can use the with statement to make sure the destructor are called and the dependent information is updated.

## Basic Editing Operations¶

The basic functionality of editors is implemented in the EditorBase class.

class EditorBase

Inherited by XCSEditor, ICSEditor.

InsertChain(chain_name)

Add new chain to the entity

Parameters: chain_name (string) – The chain’s name. In the scope of an entity, chain names are unique. If a chain of the same already exists an IntegrityError is raised. ChainHandle
AppendResidue(chain, residue_name[, res_num])

Append residue to the end of the chain. If res_num is not given, the residue number will be set to the residue number of the last added residue plus one. The insertion code is the same.

Parameters: chain (ChainHandle) – Must be a valid chain residue_name (string) – 3-letter-code of the residue, e.g. GLY. ResidueHandle
RenameResidue(residue, new_name)

Change the name of residue to new_name. Just overwrites the 3-letter-code without changing anything else. Be aware that the sequence/ 1-letter-code will not change automatically.

Parameters: residue (ResidueHandle) – Must be a valid residue new_name (string) – is the new name. Free to choose and not verified to be a valid identifier.
RenameChain(chain, new_name)

Change the name of a chain to new_name while avoiding duplicated identifiers. If new_name is already in use by any chain, an exception will be generated.

Parameters: chain (ChainHandle) – Must be a valid chain new_name (string) – is the new name
SetChainType(chain, type)
Parameters: chain (ChainHandle) – Must be a valid chain type (ChainType) – Chain type to set
SetChainDescription(chain, description)
Parameters: chain – Must be a valid chain description – Description to be added
InsertAtom(residue, atom_name, pos, element="", occupancy=1.0, b_factor=0.0, is_hetatm=False)

Insert new atom and add it to residue. For atoms with alternative atom locations use InsertAltAtom(). If the element parameter is a valid element, the atom properties mass, charge, and radius are set to default values for that element. If element is an empty string (or an invalid element), the properties are set to rather meaningless default values. You may later set any of the properties manually.

Parameters: residue (ResidueHandle) – is the parent residue and must be valid atom_name (string) – is the atom name. While free to choose a name, it is advised to properly name the atoms according to IUPAC rules as several algorithms as well as most builders in the conop module rely on proper naming. pos (Vec3) – is the position of the atom in global coordinates element (class:string) – is the atom’s element. If set to a a valid element, atom properties such as mass, charge, radius are set based on default values for that element. If the element string is empty, or unknown, the properties are filled with rather meaningless default values. occupancy (float) – The occupancy of the atom. between 0 and 1 b_factor (float) – temperature factor. is_hetatm (bool) – whether the atom is an atom coming from a HETATM record. AtomHandle
InsertAltAtom(residue, atom_name, alt_group, pos, element="", occupancy=1.0, b_factor=0.0)

Insert new atom with alternative position indicator

Parameters: residue (ResidueHandle) – is the parent residue and must be valid atom_name (string) – is the atom name. While free to choose a name, it is advised to properly name the atoms according to IUPAC rules as several algorithms as well as most builders in the conop module rely on proper naming. alt_group (string) – group is the name of the alternative atom position group. If no group of that name exists, it will be created. pos (Vec3) – is the position of the atom in global coordinates element (class:string) – is the atom’s element. If set to a a valid element, atom properties such as mass, charge, radius are set based on default values for that element. If the element string is empty, or unknown, the properties are filled with rather meaningless default values. occupancy (float) – The occupancy of the atom. between 0 and 1 b_factor (float) – temperature factor. AtomHandle
InsertAltAtom(residue, atom, alt_group)

Insert new atom with alternative position indicator

Parameters: residue (ResidueHandle) – is the parent residue and must be valid atom (AtomHandle) – Must be a valid Atom alt_group (string) – group is the name of the alternative atom position group. If no group of that name exists, it will be created. AtomHandle
AddAltAtomPos(alt_group, atom, pos, occupancy=1.0, b_factor=0.0)

Parameters: alt_group (string) – group is the name of the alternative atom position group. If no group of that name exists, it will be created. atom (AtomHandle) – It is required that the atom has been inserted via InsertAltAtom, If the atom is a conventional atom without alternative location, a Error will be thrown. pos (Vec3) – is the position of the atom in global coordinates occupancy (float) – The occupancy of the atom. between 0 and 1 b_factor (float) – temperature factor.
AddTorsion(name, atom1, atom2, atom3, atom4)

Add a named torsion to the entity. The atoms must have bonds between atom1-atom2-atom3-atom4.

Parameters: name (str) – The torsion name, e.g. PHI or PSI atom1 (AtomHandle) – First atom. must be valid atom2 (AtomHandle) – Second atom. must be valid atom3 (AtomHandle) – Third atom. must be valid atom4 (AtomHandle) – Fourth atom. must be valid
DeleteAtom(atom)

Deletes the atom from the entity and removes all bonds and torsions this atom is involved.

Parameters: atom (EntityHandle) – A valid atom
DeleteAllAtoms(residue)

Deletes all atoms of this residue from the entity and remove all bonds and torsions for where an atom of the residue is involved.

Parameters: residue (ResidueHandle) – A valid residue
DeleteAtoms(atoms)

Deletes a set specified atoms. All associated torsions and bonds will be removed as well

Parameters: atoms (AtomHandleList) – A valid set of atoms
DeleteResidue(residue)

Deletes the residue, it’s atoms and removes all bonds and torsion where one atom of the residue is involved

Parameters: residue (ResidueHandle) – A valid residue
DeleteResidues(chain)

Deletes all the residues of given chain. All associated atoms, residues, torsions and bonds will also be deleted

Parameters: chain (ChainHandle) – A valid chain
DeleteChain(chain)

Delete the given chain, and all its residues

Parameters: chain (ChainHandle) – A valid chain
ReorderResidues(chain)
ReorderResidues()

Reorder residues of the chain (the entity) such that their residues numbers are continuously increasing. This function might be useful in cases of PDB files that do not list the residues from N to C terminus but rather use the residue number to describe their position in the chain.

Parameters: chain (ChainHandle) – A valid chain
ReorderAllResidues()

Reorder residues of the all chains such that their residues numbers are continuously increasing. This function might be useful in cases of PDB files that do not list the residues from N to C terminus but rather use the residue number to describe their position in the chain.

RenumberAllResidues(start, keep_spacing)

Renumber residues of all chains

Parameters: start (int) – Residues of every chain will be renumbered, whereas the first residue gets the residue number start. keep_spacing (bool) – If set to false, residues will continously Otherwise the spacings between the residues are kept
RenumberChain(chain, new_numbers)

Renumber residues of the given chain

Parameters: chain (ChainHandle) – A valid chain new_numbers (ResNumList) – A valid ResNumList
RenumberChain(chain, start, keep_spacing)

Renumber residues of the given chain

Parameters: chain (ChainHandle) – A valid chain start (int) – Residues of given chain will be renumbered, whereas the first residue gets the residue number start. keep_spacing (bool) – If set to false, residues will continously Otherwise the spacings between the residues are kept
GetMode()

Get edit mode of editor

Returns: EditMode
RenameAtom(atom, new_name)

Change the name of atom to new_name without changing anything else.

Parameters: atom (AtomHandle) – Must be a valid atom new_name (string) – is the new name. Free to choose and not verified to be a valid atom identifier.
SetResidueNumber(residue, num)

Change the number of residue to num without changing anything else.

Parameters: residue (ResidueHandle) – Must be a valid residue num – Residue number
Connect(atom1, atom2)
Connect(atom1, atom2, bond_order)

Add a bond between two atoms.

Parameters: atom1 (AtomHandle) – Must be a valid atom atom2 (AtomHandle) – Must be a valid atom bond_order (int) – bond order (e.g. 1=single, 2=double, 3=triple)

## Editor for the External Coordinate System¶

The XCSEditor defines the interface for manipulating the external coordinate system. The external coordinate system directly operates on atom positions in Euclidian space.

class XCSEditor

Inherits EditorBase

ApplyTransform(transform)

Apply a transformation to the entity. The transformation is applied to all atoms positions.

Parameters: transform (geom.Mat4) – The transformation to be applied
ApplyTransform(transform)

Apply a transformation to the entity. The transformation is applied to all atoms positions.

Parameters: transform (Transform) – The transformation to be applied
SetTransform(transform)

Set the entity transformation. See also ApplyTransform()

Parameters: transform (geom.Mat4) – The transformation to be applied
SetTransform(transform)

Set the entity transformation. See also ApplyTransform()

Parameters: transform (Transform) – The transformation to be applied
FixTransform()

Set transformed positions to new original positions

UpdateICS()

Immediately update internal coordinate system

ForceUpdate()

Force spatial organizer and ICS update workaround for delayed editor call from Python garbage collection

SetAtomPos(atom, pos)
SetAtomPos(atom_list, pos_list)
SetAtomTransformedPos(atom, pos)
SetAtomTransformedPos(atom_list, pos_list)

Set the transformed position of atoms. This method will also update the original position of atoms by applying the inverse of the entity transform.

Setting all positions at once is by far faster than call the function for each atom, but it is only available if OpenStructure was compiled with an enabled USE_NUMPY flag (see here for details). The fastest option to work with all atom positions externally is to extract the list of atoms with ost.mol.EntityHandle.GetPositions() (with sort_by_index = False). Then extract a buffered editor and use the same list of atoms with a modified numpy array as input to this function. Example:

# get atom list and positions
atom_list = ent.atoms
positions = ent.GetPositions(False)
# modify positions but keep ent and atom_list unchanged
# ...
# apply changes to entity all at once
edi = ent.EditXCS(mol.BUFFERED_EDIT)
edi.SetAtomPos(atom_list, positions)
edi.UpdateICS()

Parameters: atom (ost.mol.AtomHandle) – A valid atom handle atom_list (ost.mol.AtomHandleList or list of int) – A valid atom handle list or a list of atom indices. pos (Vec3) – The new position pos_list (numpy.array) – An array of positions (shape [len(atom_list), 3], preferably contiguous array in memory (C order)).
SetAtomOriginalPos(atom, pos)
SetAtomOriginalPos(atom_list, pos_list)

Set the original (untransformed) position of atoms. This method will also update the transformed position by applying the entity transform to the original pos. See SetAtomPos() for more details.

## Editor for the Internal Coordinate System¶

The ICSEditor is used to manipulate the internal coordinate system that is defined by bond lengths and angles. You can create an editor with the EditICS method of the EntityHandle.

The use XCSEditor and ICSEditor are mutually exclusive. This means that whenever a XCSEditor has pending changes, the results of using an ICSEditor is undefined and vice versa.

Note

For speed reasons, the internal coordinate system is not initialised until the first call to EntityHandle.EditICS(). This involves the build-up of a directed-graph for the bond network as well as calculating the internal coordinate matrices.

class ICSEditor

Inherits EditorBase

SetTorsionAngle(torsion, angle)

Set the angle of the given torsion. If the edit mode of the editor is set to buffered, the external coordinates remain unchanged. If set to unbuffered, the external coordinates are immediately recalculated.

See: UpdateXCS() torsion (TorsionHandle) – A valid torsion angle (float) – The angle in radians RuntimeError when the torsion handle is invalid
SetTorsionAngle(atom1, atom2, atom3, atom4, angle, update_others=True)

Set the angles of the given atoms. All connectors at the third atom (A3) will be adjusted accordingly. If you only want to adjust the bond between A3 and A4, and leave the other bonds untouched, use the SetDihedralAngle() function

See: UpdateXCS() atom1 (AtomHandle) – The first atom. Must be valid atom2 (AtomHandle) – The second atom. Must be valid atom3 (AtomHandle) – The third atom. Must be valid atom4 (AtomHandle) – The third atom. Must be valid angle – The angle in radians update_others (bool) – Update all the the other torsion angles consequently angle: Real
RotateTorsionAngle(torsion, delta, update_others=True)

Rotate torsion angle

See: UpdateXCS() torsion (TorsionHandle) – A valid torsion delta (Real) – delta update_others (bool) – Update all the the other torsion angles consequently
RotateTorsionAngle(atom1, atom2, atom3, atom4, angle, update_others=True)

Rotate torsion angle

See: UpdateXCS() atom1 (AtomHandle) – The first atom. Must be valid atom2 (AtomHandle) – The second atom. Must be valid atom3 (AtomHandle) – The third atom. Must be valid atom4 (AtomHandle) – The third atom. Must be valid angle – The angle in radians update_others (bool) – Update all the the other torsion angles consequently angle: Real
UpdateXCS()

Apply all remaining changes to the internal coordinate system and recalculate external coordinates. In unbuffered edit mode, calling this method has no effect.

SetBondLength(bond, length)

Sets the length of a bond. If the edit mode of the editor is set to buffered, the external coordinates remain unchanged. If set to unbuffered, the external coordinates are immediately recalculated.

See: UpdateICS() bond (BondHandle) – A valid bond handle length (float) – The bond length in Angstrom. RuntimeError when the bond handle is invalid
SetAngle(atom1, atom2, atom3, angle)

Sets the angle between 3 atoms. The first atom must be connected to the second, the second to the third with a bond. If the edit mode of the editor is set to buffered, the external coordinates remain unchanged. If set to unbuffered, the external coordinates are immediately recalculated.

See: UpdateICS() atom1 (AtomHandle) – The first atom. Must be valid atom2 (AtomHandle) – The second atom. Must be valid atom3 (AtomHandle) – The third atom. Must be valid angle – The angle in radians RuntimeError when one of the atoms is invalid or there is no bond between atom1 and atom2 or atom2 and atom3.

## Search

Enter search terms or a module, class or function name.

## Previous topic

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Queries


text:
generate required files for the Visual Studio(R) project of a LabVIEW(R) block - MapleSim Help

Home : Support : Online Help : MapleSim Toolboxes : MapleSim Connector for LabVIEW and NI VeriStand Software : LabVIEWConnector Package : Deprecated Components : LabVIEWConnector/CreateLabVIEWVCproject(deprecated)

LabVIEWConnector[CreateLabVIEWVCproject] - generate required files for the Visual Studio(R) project of a LabVIEW(R) block

 Calling Sequence CreateLabVIEWVCproject(system, in_vars, out_vars, ModelName, output_dir, opts)

Parameters

 system - list; model equations in_vars - list; input variables with respect to time out_vars - list; output variables with respect to time ModelName - string; name of the library output_dir - string; output directory of the Visual Studio project opts - (optional) equation(s) of the form name = value

Description

 • Important: The CreateLabVIEWVCproject command has been deprecated. Use the superseding commands LabVIEWConnector[GenerateCode] and LabVIEWConnector[CompileCode] instead.
 • The CreateLabVIEWVCproject command returns a string that contains a library of the C procedures required to create a LabVIEW EMI or SIT block for a continuous system using a solver that you provide.
 Before the functions are generated, the system tries to perform index 1 reduction to obtain an ordinary differential equation (ODE) system to reduce the amount of time required to compute the system.
 The input and output variables must be functions with respect to time, for example, y(t) and u(t).


text:
A Study on Ship Path Planning Algorithm based on Real-time Ocean Environment

Title & Authors
A Study on Ship Path Planning Algorithm based on Real-time Ocean Environment
Kim, Dongjun; Seol, Hyeonju; Kim, Jinju;

Abstract
Unlike terrestrial transportation, marine transportation should consider environment factors in order to optimize path planning. This is because, ship`s path planning is greatly influenced by real-time ocean environment-sea currents, wave and wind. Therefore, in this study, we suggest a ship path planning algorithm based on real-time ocean environment using not only $\small{A^*}$ algorithm but also path smoothing method. Moreover, in order to improve objective function value, we also consider ship`s moving distance based on ship`s location and real-time ocean environment data on grid map. The efficiency of the suggested algorithm is proved by comparing with $\small{A^*}$ algorithm only. This algorithm can be used as a reasonable automatics control system algorithm for unmaned ship.
Keywords
Path Planning;Real-time Ocean Environment;$\small{A^*}$ Algorithm;Path Smoothing Method;
Language
Korean
Cited by
References
1.
Bui, L. D. and Kim, Y. G., "A New Approach of BK Products of Fuzzy Relations for Obstacle Avoidance of Autonomous Underwater Vehicles," International Journal of Fuzzy Logic and Intelligent Systems, Vol. 4, No. 2, pp. 135-141, 2004.

2.
Botea, A., Muller, M., and Schaeffer, J., "Near Optimal Hierarchical Path-Finding, Journal of Game Development, Vol. 1, No. 1, pp. 7-28, 2004.

3.
Cho, J. H., Ji, M. S., and Kim, Y. G., "An Route Planning for the Navigation System of Autonomous Vessel," Journal of Fuzzy Logic and Intelligent Systems, Vol. 15, No. 4, pp. 418-424, 2005.

4.
Christiansen, M., Fagerholt, K. and Roren, D., "Ship Routing and Scheduling: Statue and Perspectives," Tansportation Science, Vol. 38, No. 1, pp. 1-18, 2004.

5.
Hart, P. E., Nilson, N. J. and Raphael, B., "A Formal Basis for the Heuristic Determination of Minimum Cost Paths," IEEE Transaction of Systems Science and Cybernetics, Vol. 4, No. 2, 1968

6.
Hanssen, G. L. and James R. W., "Optimum Ship Routing," The Institute of Navigation, Vol. 13, No. 3, 1960.

7.
Jang, I. S. and Park, S. H., "A Genetic Algorithm for Searching Shortest Path in Public Transportation Network," Korean Management Science Review, Vol. 18, No. 1, 2001.

8.
Journee, J. M. J., and Meijers, J. H. C., Ship Routing for Optimum Performance, Delft University of Technology, 1980.

9.
Jung, J. S. and Rhyu, K. S., "A Study on the Optimal Navigation Route Decision using A* Algorithm," Journal of the Korea Society of Computer and Information, Vol. 4, No. 1, pp. 38-46, 1999.

10.
Kim, K. T. and Jeon, G. W., "A Path Planning to Maximize Survivability for Unmanned Aerial Vehicle based on 3-dimension Environment," IE Interface, Vol. 24, No. 4, pp. 304-313, 2011.

11.
Kim, S. K., Lee, Y. S., Kong, G. Y., Kim, J. P. and Jung, C. H., "A Study on the Ship's Speed for Reducing the Feul Oil Consumption in Actual Ships," Journal of the Korean Society of Marine Environment & Safety, Vol. 18, No. 1, pp. 41-47, 2012.

12.
Kwon, Y. J. and Kim, D. Y. "A Research on the Approximate Formulae for the Speed Loss at Sea," Journal of Ocean Engineering and Technology, Vol. 19, No. 2, pp. 90-93, 2005.

13.
Lee, H. Y. and Kim, S. H., "On the Optimum Ship Routing by Network Modeling", Journal of the Korean Institute of Navigation, Vol. 25, No. 3, pp. 211-223, 2001.

14.
Maki, A., Akimoto, Y., Nagata, Y., Kobayashi, S., Kobayashi, E., Shiotani, S., Ohsawa, T., Umeda, N., "A New Weather-Routing System That Accounts for Ship Stability based on a Real-Coded Genetic Algorithm," Journal of Marine Science and Technology, Vol. 16, No. 3, pp. 311-322, 2011.

15.
Montes, A. A., "Network Shortest Path Application for Optimum Track Ship Routing," Naval Postgraduate School, 2005.

16.
Padhy, C. P., Sen, D. and Bhaskaran, P. K., "Application of Wave Model for Weather Routing of Ships in the North Indian Ocean," Natural Hazards, Vol. 44, pp. 373-385, 2008.

17.
Park, J. M. and Kim, N. W., "A Comparison and Analysis of Ship Optimal Routing Scenarios Considering Ocean Environment," Journal of the Society of Naval Architects of Korea, Vol. 51, No. 2, pp. 99-106, 2014.

18.
Tolga, Y and Abdullah, S., "An Implementation of Path Planning Algorithms for Mobile Robots on a Grid based Map," Citeseer, 2008.

19.
Townsin, R. L. and Kwon, Y. J., "Estimating the Influence of Weather on Ship Performance," Royal Institution of Naval Architects, Vol. 135, pp. 191-209, 1993.


text:
# 742

Math 742

Commutative Algebra and Galois Theory

MWF 11-11:50, Van Vleck B129

Prof: Andrei Caldararu. Office hours: Wednesday 2:30-4:00pm, room VV 605.

Grader: Eric Ramos. Office hours: Monday 12:00-1:00pm VV 416, or by appointment.

Late homework may be given directly to the grader, along with either (i) the instructor's permission, or (ii) a polite request for mercy.

This course, the second semester of the introductory graduate sequence in algebra, will cover the basic aspects of commutative ring theory and Galois theory. The textbook we'll use for the Commutative Algebra portion will be Atiyah-Macdonald "Commutative Algebra". For Galois Theory I plan on using "Fields and Galois Theory" by J.S. Milne which can be found here.

For Galois theory you may also look at Emil Artin's notes which are available here.

## SYLLABUS

In this space we will record the theorems and definitions we covered each week, which we can use as a list of notions you should be prepared to answer questions about on the Algebra qualifying exam. The material covered on the homework is also an excellent guide to the scope of the course.

WEEKS 0.5-1:

Category theory: the notions of category, functor, natural transformation. Full, faithful, essentially surjective functors. Examples. Groupoid. Categories enhanced in (abelian groups, vector spaces, topological spaces). Monomorphism, epimorphism.

WEEKS 1-2:

Commutative rings and homomorphisms, integral domains, fields. Ideals, prime and maximal. Existence of maximal ideals. Local rings. Some geometric pictures (Spec and Specm). Nilradical. The nilradical is the intersection of all primes. Relatively prime ideals, Chinese remainder theorem. Extension and contraction, pictures of what happens for the inclusion Z -> Z[i]

WEEK 3:

Modules, module homomorphisms. Module Hom, covariance and contravariance. Finitely generated modules, Nakayama's lemma and variants. Short exact sequences. Tensor product and exactness properties.

WEEK 4:

An A-B-bimodule M induces a functor Mod-A -> Mod-B given by -- \otimes_A M. The equivalence of categories Mod-k and Mod-M_n(k). Restriction and extension of scalars. Algebras. Tensor product of algebras, the case of C \otimes_R C. Exactness of tensor products, flatness.

WEEK 5

Simple rings. Structure of finite dimensional simple rings over a field. Brauer group, computation of Br(R) = Z/2Z. Semisimple modules and rings. Wedderburn and Artin-Wedderburn theorems. Maschke's theorem on semisimplicity of k[G] for a finite ring. Tensor product of algebras and interpretation as fiber product of affine schemes. Localization (definition and basic properties).

WEEK 6

Rings and modules of fractions. Definitions and universal properties. Examples A_f, A_p. Modules of fractions. The operation S^{-1} is exact, corresponds to tensoring with S^{-1}A, so S^{-1}A is flat. Local properties, examples. Ideals in rings of fractions, in particular primes in A_p are primes in A which are contained in p.

WEEK 7

Integral dependence and valuations. Various characterizations of integral dependence, integral closure forms a ring, which is integrally closed. Relations to rings of fractions. Going up and going down. Valuation rings definition and basic properties. Existence. The integral closure of A is the intersection of all valuation rings which contain A. Nullstellensatz.

Below you will find a repository of homework problems.

## HOMEWORK 0 (not to be turned in)

1) Show that a fully faithful functor F: C -> D captures the property we called "essentially injective": if F(A) is isomorphic to F(B) for objects A, B of C, then A is isomorphic to B. (If you wanted to think of "functors which are injective on objects", and replaced equality with isomorphism, you'd get this notion of "essentially injective".)

2) Prove that a natural transformation eta: F => G (where F, G are functors C -> D) such that eta_A is an isomorphism for every A in C, is a natural isomorphism. (A natural transformation is called a natural isomorphism if there exists another natural transformation mu: G => F such that eta o mu = id_G, mu o eta = id_F.) In this case we say that F and G are naturally isomorphic.

3) Prove that a functor F that is fully faithful and essentially surjective is an equivalence, in the sense that there exists a functor G: D -> C and natural isomorphisms between F o G and id_D, and between G o F and id_C. (You will need to use the axiom of choice.)

4) Consider the functor ** : Vect -> Vect which takes a vector space V to its double dual V^**. Show that it is isomorphic to the identity when restricted to the subcategory of finite dimensional vector spaces. (We already constructed a map of functors id => ** in class.) If V is not finite dimensional, can you characterize the image of the natural map V -> V^**?

5) Fill in the blanks: "A category C, enhanced in abelian groups, with only one object, is the same things as a ...........". Same question, with "abelian groups" replaced by "k-vector spaces" for a fixed field k.

## HOMEWORK 1 (due Feb 6)

Atiyah-Macdonald, page 10: 1, 2, 6, 10, 12, 15, 16, 17, 18, 21

## HOMEWORK 2 (due Feb 13)

Atiyah-Macdonald, page 10: 19, 22, 26, 27, 28.

## HOMEWORK 3 (due Feb 27)

Atiyah-Macdonald, page 31: 2, 3, 4, 8, 9, 10, 11 (second part is hard!), 12, 13, 14, 20. Bonus -- if you know about Tor -- do #24.

## HOMEWORK 4 (due Mar 14)

Some non-commutative algebra exercises. You may find some references to read here. Or, even better, you can go and read Chapter I of the excellent book "Noncommutative Algebra" by Benson Farb and R. Keith Dennis. (On reserve in the library.) Most of these exercises are from this book.

1) Let A, A' be k-algebras, with subalgebras B, B', respectively. If the centralizers of B, B' are C, C' (in A, A', resp.), then the centralizer of B\otimes_k B' in A\otimes_k A' is C\otimes C'.

2) Find all the two-sided ideals of M_n(R), where R is a ring. Conclude that if D is a division algebra, M_n(D) is simple. (Hint: all two-sided ideals are of the form M_n(I) for some two-sided ideal I of R.)

3) Let N be a submodule of the R-module M. If N and M/N are semisimple, does it follow that M is semisimple?

4) Let M be a module such that every submodule is a direct summand. Show that M is semisimple as follows:

(a) Show that every submodule of M inherits the property that every submodule is a direct summand.

(b) Show that M contains a simple submodule: choose any finitely generated non-zero submodule M' of M. Let M" be a maximal submodule of M' such that M" is not equal to M' (why does it exist?). Then M'/M" is simple.

(c) Let M_1 be the submodule of M generated by all simple submodules. Show that M_1 = M.

5) Let A be a simple k-algebra with center k such that [A:k] = p^2 for a prime number p. Prove that A is either a division algebra or A is M_p(k).

## HOMEWORK 5 (due Mar 28)

Atiyah-Macdonald page 31: 5, 6, 10; page 43: 1, 3, 5, 12, 13, 14

## HOMEWORK 6 (due Apr 4)

Atiyah-Macdonald page 43: 15, 18, 19, 20, 21, 22, 25; page 67: 3, 9

## HOMEWORK 7 (due Apr 11)

Atiyah-Macdonald page 67: 1, 2, 17, 28, 30, 31.

Also prove that the integral closure of R = k[x,y]/(y^2-x^3) is isomorphic to k[t], as follows. Consider the map R -> k[t] given by x|->t^2, y|->t^3. Show that it is injective, so we can consider R as a subring of k[t]. Moreover, they have the same field of fractions. Now on one hand, k[t] is integrally closed, so the closure of R must be included in it. On the other hand, t is integral over R, so k[t] is contained in the integral closure of R.

## HOMEWORK 8 (due Apr 18)

Atiyah-Macdonald page 84: 2, 4, 5, 7, 14, 26, 27

## HOMEWORK 9 (due Apr 25)

Milne page 24: 1.1--1.4; page 31: 2.1--2.6

## HOMEWORK 10 (due May 9)

1. Standard Facts about Finite Fields

Let p be a prime. Observe that if f(x) is an irreducible polynomial of degree d over F_p (the field with p elements), then (F_p)[x]/(f(x)) is a field with p^d elements, which we call F_(p^d).

a. Show that x^(p^d) - x splits into a product of distinct linear factors over F_(p^d) by showing that every element of F_(p^d) is a root of this polynomial.

b. Show that the splitting field of x^(p^d) - x is F_(p^d). Conclude that the field with p^d elements is unique up to isomorphism (i.e., calling it "the field" is justified).

c. Let sigma be the Frobenius automorphism a->a^p. Prove that the Galois group of F_(p^d)/F_p is cyclic of degree d and is generated by sigma. Conclude that the subfields of F_(p^d) are the fields F_(p^k) where k divides d.

d. Prove that x^(p^d) - x factors over F_p as the product of all the monic irreducible polynomials over F_p whose degree divides d. Use this to find the number of irreducible cubic polynomials over F_7.

[Note: as all of these are "standard facts" you can likely look all of them up. Do this only after you have tried to prove them from scratch.]

2. Let k be a field, f be a polynomial of degree n in k[x], and K be the splitting field of f over k.

a. Show that [K:k] divides n!.

b. Show that in order to have [K:k] = n!, it is necessary but not sufficient for f to be irreducible.

3. Find the Galois group of x^7 - 2 explicitly as a permutation group on the roots.

4. a. Show that the splitting field K of x^8-2 is Q(2^(1/8), zeta_8) where zeta_8 is a primitive eighth root of unity.

b. Despite the facts that [Q(2^(1/8)) : Q] = 8 and [Q(zeta_8) : Q] = 4, prove that [K:Q] is actually 16, not 32. (Optional: also explain how you will avoid making similar mistakes in the future, if you have made them in the past.)

c. Find generators for Gal(K/Q) and write explicitly their permutation action on the roots of x^8-2.

5. Let E=k(alpha) where alpha is algebraic over k.

a. If [E:k] is odd, prove that k(alpha^2) = k(alpha).

b. Show more generally that k(alpha^2) = k(alpha) if and only if the minimal polynomial for alpha has an odd-degree term. (In other words, if it has a term b*x^c where b is nonzero and c is odd.)

c. Is it true in general that if m and n are relatively prime and alpha is algebraic of degree m over k, then k(alpha) = k(alpha^n) ?

6. Find the splitting fields and Galois groups of the following polynomials (if they exist):

a. x^3 - 3 over Q.

b. x^3 - x + 1 over Q.

c. x^3 - 3 over Q(sqrt 3).

d. x^3 - 3 over Q(sqrt(-3)).

e. x^4 - 2 over Q.

f. x^4 - 7 over Q.

## HOMEWORK 11 (due May 9)

1. a. Find the splitting field K of x^4 - 4x^2 - 1 over Q.

b. Show that Gal(K/Q) is (isomorphic to) the dihedral group of order 8.

c. Find the 8 nontrivial subfields of K and say which 4 of them are Galois over Q.

2. Prove or disprove: Every field of degree 4 over Q has a subfield of degree 2 over Q.

3. Let K/F be an algebraic extension. We say that an element alpha in K is "abelian" if F[alpha] is a Galois extension of F and the Galois group Gal(F[alpha]/F) is abelian. Prove that the set of abelian elements of K is a field.

4. a. Let F < K < L be a tower of field extensions with [L:F] finite, and let alpha be an element of L. If p(x) is the minimal polynomial of alpha over F, prove that K tensor_F F(alpha) is isomorphic to K[x]/p(x) as a K-algebra.

b. Let K1 and K2 be two algebraic extensions of a field K contained in a field L of characteristic zero. Prove that the K-algebra K1 tensor_K K2 has no nonzero nilpotent elements.