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# Implement debuginfo path remapping #38348

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### 10 participants

Member
commented Dec 13, 2016
 Fix #38322.
Collaborator
 (rust_highfive has picked a reviewer for you, use r? to override)
removed their assignment Dec 14, 2016
Member
Member
 (note that I'd like to discuss in tools triage tmw, but seems good to me)
Contributor
 Yes, let's discuss it at the meeting.
Contributor
commented Dec 14, 2016
 Seems like functionality we need. Thanks @sanxiyn! Can this be used to let us create seperate rust-dbg packages in rustup? Could we put e.g. std in sysroot/lib/std.rlib and the debuginfo in sysroot/lib/std.debuginfo? Is this unstable? Can it be tested? No test cases here.
Contributor
commented Dec 14, 2016
 Can we get negative test cases too, for when this feature isn't supported? Does it e.g. work on windows?
Contributor
 Can this be used to let us create seperate rust-dbg packages in rustup? Could we put e.g. std in sysroot/lib/std.rlib and the debuginfo in sysroot/lib/std.debuginfo? Separate debuginfo is a different problem. GDB has ways of handling this: https://www.sourceware.org/gdb/onlinedocs/gdb/Separate-Debug-Files.html We should take advantage of that eventually.
Contributor
 @vadimcn suggested : instead of = as separator character, which I personally also like more. That's also how we separate paths for another commandline option (which was it, @vadimcn?)
 sanxiyn Implement debuginfo path remapping d65e2f1
Contributor
 Here is a summary of what we discussed in the tools meeting: We want to support this functionality. We would prefer : over = as the path separator. The feature should be implemented as a -Z flag so we get a testing period before it becomes stable. Once it's implemented as unstable, we'd like to get feedback if it actually solved people's problems. A usability concerns that was raised: The path where Cargo will place the source code of downloaded dependencies is not really predictable. So in and of itself supporting this in the compiler might not be enough to make this feature usable. Also, as @brson said, there should be an auto-test for this. A codegen test that checks the generated LLVM IR would probably work best. @sanxiyn Are you still up to this?
Contributor
commented Dec 15, 2016
 Also, I wanted to note that we should be careful if the prefix includes more than just the root source directory. I am finding stuff like this (below) in debug info of a Rust binary. A naive matching of /Users/rustbuild/src/rust-buildbot/slave/nightly-dist-rustc-mac/build/src/libcore would probably fail to find these? AT_name( "./../src/libstd/lib.rs" ) AT_stmt_list( 0x000000e3 ) AT_comp_dir( "/Users/rustbuild/src/rust-buildbot/slave/nightly-dist-rustc-mac/build/obj" )  and AT_call_file( "/Users/rustbuild/src/rust-buildbot/slave/nightly-dist-rustc-mac/build/obj/../src/libcore/cell.rs" )
Contributor
commented Dec 15, 2016
 @vadimcn suggested : instead of = as separator character, which I personally also like more. That's also how we separate paths for another commandline option (which was it, @vadimcn?) 13477c7#diff-8ad3595966bf31a87e30e1c585628363R1457
Contributor
commented Dec 20, 2016
 13477c7#diff-8ad3595966bf31a87e30e1c585628363R1457 Parse string of the form "[KIND=]lib[:new_name]", Are those actually paths? They look like library names, which are typically more restricted. As I've noted in the issue, to actually be fool-proof here we need to allow each path to be a separate argument.
Contributor
 Does this mean paths that contain : themselves won't be able to be remapped-from or remapped-to? Or is there an escaping mechanism planned? The reason I ask is because we at the reproducible builds project are in the process of specifying the format of a standardised environment variable for this, that different compilers can all support, and we are trying to decide (e.g.) what to use as a separator characters. It would be good to get many different compiler writers' opinions on this. We also have the constraint that multiple mappings would have to be joined up into a single string, for this envvar.
Contributor
commented Dec 29, 2016
 Yeah, looks like we forgot about Windows drive letters :( In today's Tools Team meeting we've discussed several options (none of which is particularly appealing): go back to the '=' separator (cons: '=' may occur in file names), put the "NEW" part into a separate command line argument: -Zdebug-prefix-map=C:\\foo D:\\bar (cons: inconsistent with every other rustc parameter), quoting (but it'd also need escaping and shouldn't use the same characters as the shell), load the map from a .json file, whose name is given on the command line (actually, this isn't bad, aside from implementation complexity). A couple more that occurred to me later: escape the separator character by doubling it up: -Zdebug-prefix-map=C::\\foo:D::\\bar (cons: surprising to users, who must remember to run paths through string replacement when generating command line programmatically (oh, and for extra fun, both ':' and '=' are used in Windows shell string substitution syntax! )) just be careful with drive letters. They follow a pretty specific pattern:  ':' '\'. We can detect and avoid splitting on colons following a letter and followed by a backslash. With that in mind, strings like -Zdebug-prefix-map=C:\\foo:D:\\bar can be split unambiguously, and stuff like -Zdebug-prefix-map=C:\\foo:\\bar, i.e. OLD=C:\foo, NEW=\bar, is unlikely to be used in practice.
Contributor
 We should probably just avoid getting into the whole cross-platform shell/filename escaping business. I'd be for either going the json-file route or just allow an arbitrary number of -Zdebug-prefix-map-from= / -Zdebug-prefix-map-to= pairs which are matched up by their position in the command. (I know that I've argued against position-dependent arguments in the past, but given the alternatives they seem like an OK option)
Contributor
 @vadimcn Also, I wanted to note that we should be careful if the prefix includes more than just the root source directory. That's a very good point.
Contributor
 I still think it's fine just having one argument -Zdebug-prefix-map=A=B with A=B being split at the final = so that A can be arbitrary but B can't contain a =. I think the reduced complexity is worth the loss of function. I haven't seen anyone propose any use-case (even an unreasonable one) where B must be able to contain a =. The consistency with GCC also helps to reinforce this idea globally. If this is still not convincing to the Rust team, then I suggest that instead of doing complex position-based logic, you define -Zdebug-prefix-map-from= and -Zdebug-prefix-map-to= to be lists that must be equal length, then zip them together. That would be easier to parse and reason about. It would allow things like -Zdpm-from=A1 -Zdpm-from=A2 -Zdpm-to=B1 -Zdpm-to=B2 but I don't think that's so bad in practice.
Contributor
 It would allow things like -Zdpm-from=A1 -Zdpm-from=A2 -Zdpm-to=B1 -Zdpm-to=B2 but I don't think that's so bad in practice. That's pretty much what I had in mind.
Contributor
commented Jan 14, 2017
 A couple of points: Are debug maps defined at non-directory boundaries? Ie, would a mapping of foo=bar remap foo-src/lib.rs -> bar-src/lib.rs? It looks like the current implementation would do this, but that seems like it could result in unexpected remappings. I'm a bit concerned about not allowing = (or any other separator character) in a path. I'm also don't like introducing any new layers of quoting. One solution that occurred to me was to default to (say) =, but add -Z remap-separator=x, where x is the separator character. This could be specified multiple times, and takes effect for all following parameters until its replaced. Still ugly, but somewhat like separators in MIME.
Contributor
commented Jan 15, 2017 edited
 It looks like the current implementation would do this [..] Yes, and this is the same behaviour as GCC. If one wanted foo-src to map to something else (including itself, i.e. no change), one can just add another mapping for foo-src that appears later in the list. This should be fine - if both foo and foo-src are going to be compiled in the same compilation unit, there must exist some program that knows about both of them, to be able to set these flags. Otherwise, one of them is not going to need to be remapped anyways. I think it's not worth the complexity to add extra logic in there. At the Reproducible Builds project we have some interest in getting multiple build tools to adopt the same approach (roughly what's been described here, and what GCC does), and adding this complexity would likely hinder adoption.
Contributor
commented Jan 16, 2017 edited
 100% OK with matching gcc behaviour. I guess the user could include the '/' if they wanted to enforce matching directory boundaries.
reviewed
 + for (old, new) in sess.opts.debug_prefix_map.clone() { + if path.starts_with(&old) { + return new + &path[old.len()..]; + }
Contributor

It seems unnecessary to clone the whole map first; it would be better to iterate references, then only clone the "new" once we've found a match.

 @@ -669,7 +678,10 @@ pub fn file_metadata(cx: &CrateContext, path: &str, full_path: &Option) } }); - file_metadata_(cx, path, file_name, &work_dir) + let sess = cx.sess(); + let remap_file_name = remap_path(sess, file_name);
Contributor

This ends up matching against a full path rather than the path passed on the command line. The let file_name expression above ends up using the full_path parameter if its present (which it always appears to be) rather than the relative path (which it goes to some lengths to make relative to work_dir). I'm not sure what the larger intent is here, but the effect is that this doesn't work as expected.

It looks like this was done deliberately in 24e7491 by @luser. This suggests that path remapping should be done in absolute space - though the code is pretty inconsistent. If full_path is None, then I think the resulting path will be relative.

reviewed
 @@ -253,6 +253,7 @@ top_level_options!( // Include the debug_assertions flag into dependency tracking, since it // can influence whether overflow checks are done or not. debug_assertions: bool [TRACKED], + debug_prefix_map: Option<(String, String)> [TRACKED],
Contributor

This only allows a single remapping. It needs to allow an arbitrary number of remappings to be useful.

This was referenced Jan 17, 2017
Open

Open

#### Allow remapping source path prefixes in debug output #38322

Member
 I haven't been following this too closely, but was a conclusion of how to implement this reached? (just triaging some old PRs)
Contributor
 @alexcrichton No, not yet, mostly due to my being busy with other things and not properly following up over at #38322. I have an idea what it could look like, I just need to write it up and confirm that it is what people except.


text:
# How can i find the lenght of the third side of any triangle?

I will know the length of two sides of any triangle that i use, but i will not know any of the angles. I know how to find the length of the third side if I knew the angle where I am sitting, but how can i quickly find the included angle where i am sitting with basic tools or something else?

• Are you asking how to figure out the unknown side without knowing the included angle? – turkeyhundt Nov 30 '14 at 6:24
• It's more like how can I find the included angle in the field. – Parable Nov 30 '14 at 6:26
• Shoover's answer is correct. You cannot figure out the third side without more information. Is it possible that you meant the triangle to be a right triangle, as it appears in your drawing? – Kaj Hansen Nov 30 '14 at 6:30
• No I didn't mean it as a right triangle and I will probably never be able to figure out if there is a right angle in the triangles I use so it's best to assume that it's not a right triangle. – Parable Nov 30 '14 at 6:41
• To figure out the angle "quickly in the field with basic tools", I will suggest either of two instruments. One is a magnetic pocket bearing compass, but a more basic model might also work. You would also need a calculator and/or trig tables. You may be able to measure the angle within 2 degrees of arc. For the compass to be useful there must be visible targets for your sights, and the angle must be in a nearly horizontal plane. A more accurate measurement could be made with a sextant, but that might not be a "basic tool", more in the "something else" category. – LouisB Nov 30 '14 at 7:22

Do you have a compass? You could determine the bearing of each object. Find the difference. And then you have the included angle and can determine the distance.

You can't.

### Here's what you can know

Given only the length of two sides of a triangle, the length of the third side is not fixed. Let a and b represent the lengths of the two known sides such that $a \geq b$. Let c represent the length of the unknown side, the length of c must fall within

$a - b < c < a + b$

Based on the example triangle you give, the third side, c, must be

\begin{align} 475 - 390 <& c < 475 + 390 \\ 85 <& c < 865 \end{align}

# More Detail

With the two given lengths, we can construct a segment and a circle. It is irrelevant where we choose to position the segment. It's endpoints can lie anywhere with the caveat that it has a length exactly equal to the first known side. For your example let the segment have length 475.

With the second side we can represent all possible endpoints as the points of a circle. This makes sense since a circle is the set of all points a given distance away from the center point. So the circle would have radius 390.

Well, you want to know the third side of the triangle, but the third side -- without any other information about the triangle -- could be any segment which starts at the free endpoint of our original segment and has its other endpoint on the circle. You can see why this means that there is more than one possible segment length because not all such segments have equal length. Here are a few examples.

# Closing Remarks

If you want to calculate the third side of the triangle, you need more information than simply two sides. For example, if you know the triangle is a right triangle, or if you know the measure of the included angle between the two known segments, then you can determine the length of the third side.

• Thank you for your response. Again this does not answer my question. If you look at the second sentence in my post you will know what I'm trying to figure out. – Parable Nov 30 '14 at 14:16
• Your question is unclear. That is why people answering are having a difficult time giving you the information you want. This sentence: "I will know the length of two sides of any triangle that I use, but I will not know any of the angles. I know how to find the length of the third side if I know the angle where I am sitting, but how can I figure this out quickly "in the field" with basic tools or something else?" should be revised. – Nick H Nov 30 '14 at 15:41
• @NickH - this isn't a math problem, it's a surveying issue. OP wants to know how to physically measure that angle. The distance is anecdotal. I mean, with no angle, you know, there's no solution. He wants ideas how to measure the angle in the field. I assume without the expensive equipment. +1 by the way. – JoeTaxpayer Nov 30 '14 at 18:58

Knowing only the lengths of two sides of the triangle, and no angles, you cannot calculate the length of the third side; there are an infinite number of answers.

To illustrate, imagine that you have two fixed-length pieces of wood, and you drill a hole near the end of each one and put a nail through the hole. You can then rotate the pieces relative to each other, changing the angle between them, and thus changing the distance between their free ends.

• @Parable \\ Hey, the man DID give you an answer: He told you that, without knowing the angle between the two lines which emanate from your position, you will not have enough information to solve your problem. "... take down your post ...[,]" indeed!? – Senex Ægypti Parvi Nov 30 '14 at 8:27

If the angle between those measurements reach zero (this is, the extreme points line in a line in front of you), such unknown will be 475 - 390 = 85.

The opposite could happen too, 390 measured in one direction and 475 in the other direction. In that case that unknown would be 457 + 390 = 865.

You see, there is no formula to get you that unknown in the present setting. Any value between 85 and 865 would be reasonable depending on the shape of that triangle.

So my kind suggestion is to get hold on another piece of data. As it seems you are able to measure distances, the easiest thing to do is to move yourself to a new position (say 10 yards apart), measure the distances again to your reference points, and measure also the distance you moved. If this is possible for you, then a rephrasing of your problem would help those who want to help you here.

• Thank you for your response. I most likely won't be able to move from my spot to get another reference point, but it's slightly plausible. I know this angle can be figured out with a theodolite. Buying one of these is impractical in the sense of money and weight. Is there another cheaper way to find the included angle where I am sitting? – Parable Nov 30 '14 at 14:28
• Sorry, apart from moving I don't see another option. If you can move to another position, please ensure you get reasonable measurements; this is, they need to be reasonably different from the ones you have (or put other way, you have to move at least 40 or 50 yards from your original point if your measurements are only precise +- 1 yards, I believe). – carlosayam Dec 1 '14 at 3:21

$$x + 475 > 390$$ $$390 + 475 > x$$ $$390 + x > 475$$

so $x$ can be any number bigger than 85 or smaller than 865

$85<x<865$

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## Distributed Proximal Gradient Algorithm for Partially Asynchronous Computer Clusters

Yi Zhou, Yingbin Liang, Yaoliang Yu, Wei Dai, Eric P. Xing; 19(19):1−32, 2018.

### Abstract

With ever growing data volume and model size, an error-tolerant, communication efficient, yet versatile distributed algorithm has become vital for the success of many large-scale machine learning applications. In this work we propose m-PAPG, an implementation of the flexible proximal gradient algorithm in model parallel systems equipped with the partially asynchronous communication protocol. The worker machines communicate asynchronously with a controlled staleness bound $s$ and operate at different frequencies. We characterize various convergence properties of m-PAPG: 1) Under a general non-smooth and non-convex setting, we prove that every limit point of the sequence generated by m-PAPG is a critical point of the objective function; 2) Under an error bound condition of convex objective functions, we prove that the optimality gap decays linearly for every $s$ steps; 3) Under the Kurdyka-Łojasiewicz inequality and a sufficient decrease assumption, we prove that the sequences generated by m-PAPG converge to the same critical point, provided that a proximal Lipschitz condition is satisfied.

[abs][pdf][bib]


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• CommentRowNumber1.
• CommentAuthorAndrew Stacey
• CommentTimeFeb 7th 2012
• (edited Feb 9th 2012)

I hope that the title is legible, if not it is meant to read

$\langle\text{Peer Review},\text{Refereeing}\rangle = 0.$

It’s a common defence of current journals: that journals provide peer review and that’s part of the value that they add to an article.

Total bunkum.

I will admit that my experience of peer review and of refereeing is limited. But it is extremely varied. I’ve participated in refereeing from both sides: as an author and as a referee. I’ve also participated in peer review, mainly as a reviewer. And the two have almost nothing in common.

“Refereeing” is something that is very hard to define. It can vary immensely between actual peer review and simply glancing at the paper in question. Which is the main problem with it: as the questions asked by editors and journals vary so much, and the standard of refereeing is so variable, the variety of things that “This article has been refereed” could mean is so large that it is effectively meaningless.

“Peer review” on the other hand is easier to define. It is when your peers look through your paper and tear it to shreds. It can be a lot of fun - despite my description. So long as everyone sticks with it, it almost always leads to a better paper than was first produced. One of the most extensive peer review sessions that I participated in took place on the n-Category Café. After considerable discussion about the paper in question, John Baez (one of the authors) remarked:

Thanks for the new round of comments, Andrew. I hope that when we submit this paper for publication — where? does anyone have suggestions? — you are the referee. And I hope that when you referee it, you write: “Thanks to the careful vetting this paper has received at the n-Category Café, it needs no further improvements.”

to which I replied:

I suspect that by participating in this discussion then I’ve removed myself from the list of potential referees. But actually you want a completely different referee who, nonetheless, says exactly the same thing.

What I was trying to convey was that the referee should look at the peer review and see that it has already been gone over with a fine-toothed comb and therefore be able to concentrate on the evaluation part of their job: is the subject matter of the paper suitable for this journal? Not: is it written well enough for this journal?

The problem is, of course, that not everyone has access to the resources of the n-Category Café. That’s why I like Tim Gowers’ suggestion in his more modest proposal as it suggests a way of opening this up to all mathematicians. As always, the difficulty is in the details! So could such a system work? What would be the incentives? How could we ensure that peer review was available for all, without the system getting swamped by papers that really need another revision even before they are looked at by another mathematician?

For closing remarks, I would like to note that whilst I think that peer review and refereeing are orthogonal, they can happen simultaneously. I have no idea who the referee was on my first solo paper, but whoever it was - and the editor - worked extremely hard to show me how to make it far, far better than it was originally (and if you think it isn’t a great paper, just be thankful that you never read the version I originally submitted!).

• CommentRowNumber2.
• CommentAuthorTerence Tao
• CommentTimeFeb 9th 2012

As an editor of the Journal of the AMS, I solicited two types of reports for a given submission: quick opinions, which focused on the questions of the significance of result, degree of technical advance, and quality of exposition, and full reports, which focused more on correctness of the arguments, the relationship to existing literature, and suggestions for improving the paper. In practice, of course, the distinction was blurry, as the referees would naturally offer opinions beyond whatever narrow parameters I might specify. The editorial board would then consider all of these factors when voting on a JAMS paper. (This did unfortunately lead to some upset authors wondering why a paper was rejected if it had good full reports, but unenthusiastic quick opinions, but this is par for the course at a top-tier journal.)

But JAMS is one of the “big three” maths journals, with a particularly thorough refereeing process. At the more median level of maths journal, one typically just relies on one or two referee reports, and so the feedback one obtains is indeed much more variable. But it is usually good enough for the basic purpose of deciding whether the paper reaches an acceptable level of quality to be published in a median-level journal, as opposed to the more demanding question of whether it is of such exceptional quality in all aspects that it deserves to go into a top-tier journal.

• CommentRowNumber3.
• CommentAuthorCharles Rezk
• CommentTimeFeb 9th 2012
It should probably be pointed out that the kind of public peer review that you might see on n-Category Cafe is not going to be for everyone. Some folks can thrive on that sort of thing, but others are going to be very reluctant to expose themselves to scrutiny that way. Even if care is taken to make everything constructive and polite. Perhaps especially if they are early in their career, or feel otherwise vulnerable.
• CommentRowNumber4.
• CommentAuthorJohn Baez
• CommentTimeFeb 10th 2012

I agree with Charles Rezk’s remark here. Many of here are somewhat more bold or adventurous or tough-skinned or foolhardy than the average mathematician, so we should always keep that in mind when imagining what systems others will like to use.

1. I very much agree with Andrew: despite the quite different perception still there is between an article made public via the web (say, via arXiv) and so exposed to any possible positive or negative comment by peers (though not formally refereed) and a published article, we should be aware and face the reality that a "published paper" is in most cases just a paper which a single anonymous referee has said "ok, this is good for me".
• CommentRowNumber6.
• CommentAuthorAndrew Stacey
• CommentTimeFeb 11th 2012

What Terry writes about JAMS sounds great - until I start thinking about the details. So a paper in JAMS has undergone a rigorous peer review and I can cite it without fear of embarrassment - or at least, the editors of JAMS will be just as embarrassed as me. But what if a paper in JAMS cites a paper in a “lesser” journal? Can I now assume that the other article has been vetted? And when does this back-date to? Is it true that all articles in JAMS have this level of certification?

It’s too random, and it hints at a two-tier system with an inner circle of “vetted articles”. I wouldn’t mind this too much if this were all explicit and there was a list of which journals were Top Notch and which just published any old rubbish.

One of my biggest gripes with the current system is that it is too random. If you get a good editor and a good referee then, yes, the system can really work well for you. But if you get a busy editor and a rubbish referee, then that’s your bad luck and there’s not a lot you can do. I was really lucky with my first solo article and had a fantastic editor and a fantastic referee and they turned the article into something far better than it originally was, but I’ve had many (more) experiences of the other type, sadly. We’re meant to be professionals, so we should be professional about this. A physicist of my acquaintance is fond of quoting the difference between a professional and an amateur: “An amateur practises until he always gets it right, a professional practices until she never gets it wrong.”.

I also agree with Charles that the kind of peer review that goes on on the n-Café isn’t for everyone. But I would be willing to bet that it is for more people than it currently is, and a system like that which Tim Gowers outlines would make it possible to take part without hosting ones own blog. The great thing about it is that it improves the situation for everyone whether they take part or not, since the papers that have gone through it will be better papers and thus easier to referee, easier to read, and easier to understand. Moreover, the aim of Tim Gowers’ suggestion - as I see it - is as a stage before the paper is submitted so it isn’t about reviewing the paper afterwards but about polishing the paper beforehand.

I vividly remember what it was like to be a new mathematician. I almost felt as though I wanted to publish my papers without telling anyone about them in the fear that someone would come along and say that they were wrong, or I was doing it all wrong, or I was a rubbish mathematician. If I could just sneak a couple of papers into the system without anyone really looking at them, I’d feel a bit more confident! Nonsense, I know, but I do have sympathy with those who don’t want to open their work up to such scrutiny in such a public way.

• CommentRowNumber7.
• CommentAuthorHal Swyers
• CommentTimeFeb 17th 2012
I hope an engineer's opinion is welcome.
Looking at your problem it seems that there are a few overlapping issues:

1) Cost and fairness - Why should a community pay fees for a product the profits off other's labor? Granted there is a need for sustaining an organization that can handle costs of publication, but in this era of online publication, is the current system fair to the community?

2) Credibility - In most cases it is assumed that a paper that has survived some peer review process and has been published in a major journal is credible. In this process, there is a type of economy. The economy centers around the exchange of reputation, which is the reward for the person submitting the paper. The process is important to those who wish to protect the interests of the community.

3) Volume of and timing of content - With the increase of trained professionals, and the increase in the availability of media, the expectation of individuals and the community has changed. Being first is always important, along with publishing frequently. In addtion, the accessibility of material to the community is important in terms of the citation system. Having material behind a cost barrier places individuals behind the wall at a disadvantage. Exchange of information is restricted and the citiation system is probably biased.

I would think that the biggest question would center around how one retains and tasks reviewers. If the review process is entirely volunteer, then the question is how one retains a sufficient critical mass of reviewers to ensure there is adequate throughput of material.

I would offer the solution of building upon a barter system. What is it that people want? I think one element is to have their work reviewed and in order to gain credibility and authority. From a reviewers standpoint I would think it would be to retain credibility and authority and maintain the community standard.

With that in mind, it would seem natural that the buy in to having your paper reviewed is to agree to review papers once you reach a certain level of authority. The review process can still be blind and randomized which would allow for protection of authority and prevent influence and corruption. The first step though would be to identify individuals who would be the first "investors" into the community, who have sufficient credibility and authority to create a pool of "capital" that would attract people into the system.

Those are my thoughts, I hope people don't mind me sharing.


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# Tag Info

8

Generally speaking they refer to the distances from us when the light as emitted. No correction is usually made to say how far away the object is from us now, because this correction would be very small and inconsequential compared to the uncertainty in the original distance measurement. For instance, taking the Andromeda M31 galaxy as an example. Riess et ...

7

In the weak field limit, which applies to all the cases you've described, the difference between the time rates for two observers with a Newtonian gravitational potential energy difference of $\Delta\Phi$ is given by: $$\frac{\Delta t_1}{\Delta t_2} = \sqrt{1 - \frac{2\Delta\Phi}{c^2}} \tag{1}$$ Note that the time dilation is related to the gravitational ...

6

Stellar motion would imply we're in the place where the Big Bang occurred while everything else is speeding away from us. I can see how this would be a worthwile explanation for a creationist, but it doesn't really make much sense building just on physics. Doppler Effect has the same issue - it assumes that every star we can see moves in a speed ...

6

Light always travels at the speed of light when in a vacuum. Space is a pretty good vacuum. So if it's been travelling for 13.7 billion years, then it has travelled 13.7 billion light years. There is no contradiction here. Yes, those galaxies are now 46 billion light years light years away, but this is because the universe has expanded. You can find lots ...

5

Fluctuations of density in the universe naturally become greater with time because matter is attracted to regions that are denser than average, and as a result they get denser still and the other regions less dense. So if there were even tiny density fluctuations in the early universe they would have grown into the density variations we see today - ...

5

Another line of thought is that there are other observations that support the notion of an expanding universe - it does not all hinge on redshift. For instance, I believe numbers 3 onwards would struggle to explain why the light curves of otherwise similar supernovae become stretched (at a variety of wavelengths) at larger distances from us and by an amount ...

5

It doesn't make sense to say "am I experiencing time dialation?" It only makes sense to compare two different observers, and ask whether one of them observes the others' clock to be ticking more slowly, say, when they are looking at minimum distance light rays coming from the other observer. With this in mind, the answers to most of your questions are ...

4

Gravitational redshift See this answer: http://physics.stackexchange.com/a/113941/4552 Photons interacting with something There are various problems with this, including problems explaining surface brightness. For more information, see http://en.wikipedia.org/wiki/Tired_light Transverse Doppler shift There are strict empirical limits on cosmological ...

4

There's (almost) no difference between matter and antimatter, and it would be very difficult to know if a galaxy far away is made of matter or not. The reason we know that the visible is made of matter, and not half-matter and half-antimatter is that it this were the case, then we would see a ton of events where anti-matter galaxies would smash into a ...

4

Neither the space between galaxies nor the space within galaxies is completely empty, but rather is filled with a diffuse gas. If there was an isolated galaxy, or region of galaxies that was made up of antimatter, there would be an interface between the two regions, and this region would be rich in annhilation events, which would be visible in X-Rays. We ...

4

First off, spherical symmetry isn't really the best description. Cosmological models usually assume that the universe is (approximately) homogeneous and isotropic. That's a higher degree of symmetry than spherical symmetry. Spherical symmetry would normally be used to describe something that has a lower degree of symmetry, so that there is a center. The ...

4

A quick note in the light of some of the comments: I'm interpreting the question to be asking about the escape velocity from a black hole containing a naked singularity rather than the escape velocity from the singularity itself. The escape velocity at the singularity is undefined as GR cannot describe the geometry at that point. Anyhow, a convenient way to ...

3

It's tempting to think of spacetime as a thing, and it doesn't help that it's often represented as a rubber sheet in popular science programmes. In relativity (special and general) spacetime is a mathematical concept - it is a manifold equipped with a metric. At the risk of over-simplifying, a manifold is a thing that has dimensionality (four dimensions for ...

3

What are their purpose? The "purposes" of Hyperbolic geometries are many and varied in mathematics, but one stands out far beyond all others, at least historically as the purpose. Hyperbolic geometries were constructed to prove that the Euclid parallel postulate (see "Parallel Postulate" Wiki page) was logically independent of Euclid's other axioms of ...

2

"Only about 10 percent of the total baryonic matter is sufficiently condensed by gravity to form stars and galaxies. More than 90 percent was left between the galaxies." http://hubblesite.org/hubble_discoveries/science_year_in_review/pdf/2008/searching_for_baryonic_matter_in_intergalactic_space.pdf 6% of baryonic matter is within stars according to the ...

2

The purpose of any model of the hyperbolic plane is that some aspect in it will be easy to work with computationally or intuitively, e.g. writing out certain isometries, identifying geodesics, computing volumes, etc. Hyperbolic space is unbounded, a hyperbolic manifold can be bounded. I don't know Yes If you are not using the terms bounded and infinite to ...

2

The singularity comes from the scale factor $a(t)$: $$ds^2 = -dt^2 + [a(t)]^2 ( dr^2 + r^2 d \Omega^2)$$ By solving the Friedmann equations for the scale factor we know that: $$a(t) = a_0 t^{\lambda}$$ where $\lambda$ is some positive number that depends on the matter-radiation ratio of the universe. At $t=0$ the scale factor becomes $a(0)=0$. So at ...

1

You need to include the 'zero' versions of $\rho$ and $a$ in order for the units to works out, or, to put it another way, to account for initial conditions. As regards the 32, you made an error while integrating: $\int a \mathrm{d}a = \frac{a^{2}}{2}$; the factor of 2 taken to the other side under the square root provides a 4 to makes the 8 into a 32.

1

In Zero-Energy Model, negative energy associated with Gravity counterbalances positive energy associated with matter, photons, etc. So, No, Big Bang wasn't cold. You are just looking at partial picture (you just ignored Gravity). This is what Zero-Energy Model says: With traditional Big Bang model (which doesn't contain Inflation), the universe started out ...

1

In the standard homogeneous cosmological models the total energy in an expanding volume is zero. This is true for positive, negative or zero curvature and it must take into account the gravitational energy (which is negative), dark energy, matter and heat. Since the gravitational energy is negative the heat can be positive and increasing as you go back ...

1

$$\dot{\rho}+3(\rho+P)\frac{\dot{a}}{a}=0$$ with $P=0$, leads to $$\dot{\rho}+3\rho\frac{\dot{a}}{a}=0$$ $$\dot{\rho}=-3\rho\frac{\dot{a}}{a}$$ $$\frac{\dot{\rho}}{\rho}=-3\frac{\dot{a}}{a}$$ $$\frac{\frac{d\rho}{dt}}{\rho}=-3\frac{\frac{da}{dt}}{a}$$ $$\frac{1}{\rho}\frac{d\rho}{dt}=-3\frac{1}{a}\frac{da}{dt}$$ ...

1

At this point in time, your questions are subject to research - we just don't know, yet. We see the effects of dark matter (in rotation curves of galaxies, in gravitational lensing and others), so there is "something". But what exactly that "something" constitutes is not clear. There are several theories, and to think of dark matter as some kind of ...

1

I could probably go along with this assertion; except that the observable universe is not infinite. We can only see light reaching us from about 13 billion light years away. The stars/galaxies etc that formed in those distant regions are now even further away from us, but nevertheless, the observable universe is finite. Expansion of the Universe, will light ...

1

The volume of the universe is infinite, therefore there must be an infinite number of worlds. But not all of them are populated; therefore only a finite number are. Any finite number divided by infinity is as close to zero as makes no odds, therefore we can state that the population of the Universe to zero, and anyone you have ever met is merely a figment ...

1

Which scientists have called dark matter and dark energy forces? Dark matter is a form of matter that doesn't feel any force other than gravity (so far as we have observed). It's true that dark matter bends light because it has mass and it makes up about $5/6$ of all the mass in the universe. The gravity from this mass is what bends light through ...

1

Einstein's equation is a mathematical model devised to provide an approximate description of the universe. We know it's approximate because it generically predicts singularities that we believe to be unphysical, and it takes no account of quantum mechanics. So the question is whether the equation:  R_{\mu \nu} -\frac{1}{2}R\,g_{\mu \nu} + \Lambda\,g_{\mu ...

1

Here is the scenario from wikipedia According to inflation theory, the inflaton is a scalar field that is responsible for cosmic inflation in the very early universe. A quantized particle for this field is expected, similar to other quantum fields, called an inflaton. The field provides a mechanism by which a period of rapid expansion from 10^−35 to ...

Only top voted, non community-wiki answers of a minimum length are eligible


text:
# Revision history [back]

### Write custom file to client

I need to configure cents 7 to auth to AD. I have to write custom sssd.conf From init.pp config class is called: class { '::realmd::sssd::config': }~> in config class i need to set contents of sssd.conf :class realmd::sssd::config {

$_sssd_config =$::realmd::sssd_config

file { $::realmd::sssd_config_file: content => file('realmd/files/sssd.conf'), owner => 'root', group => 'root', mode => '0600', notify => Exec['force_config_cache_rebuild'], } exec { 'force_config_cache_rebuild': command => "/usr/bin/rm -f${::realmd::sssd_config_cache_file}",
refreshonly => true,

} }

But on client side I get Error: Could not retrieve catalog from remote server: Error 500 on SERVER: {"message":"Server Error: Evaluation Error: Error while evaluating a Function Call, Could not find any files from realmd/files/sssd.conf at...

It is obvious I have no clue what I am doing here, but be nice to get this working

Thank you


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The four points, two distances problem: theguardian.com/science/2019/o

Can you find all of the ways of arranging four distinct points in the plane so that they form only two distances? The link is not a spoiler but it has a separate link to the solution. "Nearly everyone misses at least one" says Peter Winkler; can you guess the one I missed?

Because I just spent 15 minutes trying to remember how to find out what variables a CMake find_package call sets:

cmake --help-module Find<name>

Goddammit. Why is this information so hard to find?

Robert MacKay's Chaos Machine. Its configuration space is a genus three surface. The dynamics of the machine are equivalent to geodesic flow on the surface, which is Anosov, hence chaotic.

Very happy to notice that Erich's Packing Center (www2.stetson.edu/~efriedma/pac) is still being updated! Two new (and very pretty) packings have been added this year: www2.stetson.edu/~efriedma/tri, www2.stetson.edu/~efriedma/lin

The problem with transiting though Shanghai airport is that all the staff try to talk to me in Chinese

I sure do love when the metro makes an announcement they say only once so by the time you start paying attention all you catch is "The inconvenience is regretted"

Compact packings of the plane with three sizes of discs: arxiv.org/abs/1810.02231, Thomas Fernique, Amir Hashemi, and Olga Sizova

Here, "compact packing" means interior-disjoint disks forming only 3-sided gaps. The circle packing theorem constructs these for any finite maximal planar graph, with little control over disk size. Instead this paper seeks packings of the whole plane by infinitely many disks, with few sizes. 9 pairs of sizes and 164 triples work. Here's one from Fig.3 of the paper.

shitpost about 90s TV

Here's a 1965 RAND report demonstrating a conversational interface with an algebraic processing system. They determine that a conversational user interface vastly improves usability in their tests.

New blog post: Motorcycle graphs and the eventual fate of sparse Life, 11011110.github.io/blog/2018/1

Of course Mac OS X's Calendar interprets "IST" as Irish Standard Time and not Indian Standard Time. The fool is me for expecting different.

I got the urge to start singing "Domo Arigato Mr. Roboto" the moment I landed in Tokyo and I'm pretty sure that's extremely culturally insensitive

A recent Learned League question asked for the name of the infamous Leni Riefenstahl film which documented the second Nazi Party Congress. Some brilliant wrong answers shared on the forums:
- Das Jackboot
- 2 Fascist 2 Fuhrerious
- Dude, Wehrmacht Car?

$r=\sqrt{1+\alpha\cos2\theta}.$ All these curves enclose the same area.

cw: shitpost

cw: shitpost

The quality of a Fourier approximation of a closed curve $$\gamma:\mathbb S^1\to\mathbb R^2$$ depends heavily on its parametrization. Below: a square approximated with 1, 2, and 3 nonzero Fourier modes.

Fun fact: The hairy ball theorem doesn't apply to black holes because black holes have no hair

Spam texts in India are amazing.
"ACCOUNTS IS THE FUTURE OF YOUR CHILD."
wtf

Show more

The social network of the future: No ads, no corporate surveillance, ethical design, and decentralization! Own your data with Mastodon!


text:
How to solve systems of equations by Elimination

Step by step tutorial for systems of linear equations (in 2 variables)

The Elimination Method

You use elimination when you perform an operation on 1 equation then add the equations so that one of the variables cancels.

Elimination Example 1

$$y = x + 1 \\ y = -x$$


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How do you multiply (x^2 + 3x - 2)(2x^2 + 5x + 9)?

Jul 31, 2015

=color(blue)(2x^4+11x^3+20x^2 +17x -18

Explanation:

color(blue)((x^2+3x−2))(2x^2+5x+9)

We need to multiply each term within the first bracket with all three terms of the second bracket.

=color(blue)((x^2))(2x^2+5x+9) +color(blue)(3x)(2x^2+5x+9) −color(blue)((2))(2x^2+5x+9)

$= 2 {x}^{4} + 5 {x}^{3} + 9 {x}^{2} + 6 {x}^{3} + 15 {x}^{2} + 27 x - 4 {x}^{2} - 10 x - 18$

Grouping like terms:
$= 2 {x}^{4} + \left(5 {x}^{3} + 6 {x}^{3}\right) + \left(9 {x}^{2} - 4 {x}^{2} + 15 {x}^{2}\right) + \left(27 x - 10 x\right) - 18$

$= 2 {x}^{4} + 11 {x}^{3} + 20 {x}^{2} + 17 x - 18$


text:
# tilde and hat together over a letter

Hey guys I have a problem. I need to write some letters with both tilde and bar or tilde and hat at the same time but I cannot find any functions that allow so. Do you know any?

• Is it in text or in math? – egreg Apr 15 '18 at 17:00

\documentclass[]{article}
\usepackage[english]{babel}

\begin{document}

$\hat{\tilde{a}}$

$\tilde{\hat{a}}$

\end{document}

• +1. You may want to mention explicitly that it's not actually necessary to load the amsmath package in order to combine \hat and \tilde accents. – Mico Apr 15 '18 at 16:02


text:
You are currently browsing the tag archive for the ‘quasilinear equations’ tag.

When solving the initial value problem to an ordinary differential equation, such as

$\displaystyle \partial_t u = F(u); \quad u(0) = u_0, \ \ \ \ \ (1)$

where ${u: {\bf R} \rightarrow V}$ is the unknown solution (taking values in some finite-dimensional vector space ${V}$), ${u_0 \in V}$ is the initial datum, and ${F: V \rightarrow V}$ is some nonlinear function (which we will take to be smooth for sake of argument), then one can construct a solution locally in time via the Picard iteration method. There are two basic ideas. The first is to use the fundamental theorem of calculus to rewrite the initial value problem (1) as the problem of solving an integral equation,

$\displaystyle u(t) = u_0 + \int_0^t F(u(s))\ ds. \ \ \ \ \ (2)$

The second idea is to solve this integral equation by the contraction mapping theorem, showing that the integral operator ${{\mathcal N}}$ defined by

$\displaystyle {\mathcal N}(u) (t) := u_0 + \int_0^t F(u(s))\ ds$

is a contraction on a suitable complete metric space (e.g. a closed ball in the function space ${C^0([0,T]; V)}$), and thus has a unique fixed point in this space. This method works as long as one only seeks to construct local solutions (for time ${t}$ in ${[0,T]}$ for sufficiently small ${T>0}$), but the solutions constructed have a number of very good properties, including

• Existence: A solution ${u}$ exists in the space ${C^0([0,T];V)}$ (and even in ${C^\infty([0,T];V)}$) for ${T}$ sufficiently small.
• Uniqueness: There is at most one solution ${u}$ to the initial value problem in the space ${C^0([0,T];V)}$ (or in smoother spaces, such as ${C^\infty([0,T];V)}$). (For solutions in the weaker space ${C^0([0,T];V)}$ we use the integral formulation (2) to define the solution concept.)
• Lipschitz continuous dependence on the data: If ${u_0^{(n)}}$ is a sequence of initial data converging to ${u_0}$, then the associated solutions ${u^{(n)}}$ converge uniformly to ${u}$ on ${[0,T]}$ (possibly after shrinking ${T}$ slightly). In fact we have the Lipschitz bound ${\| u^{(n)}(t) - u(t) \|_V \leq C \| u^{(n)}_0 - u_0 \|_V}$ for ${n}$ large enough and ${t \in [0,T]}$, where ${C}$ is an absolute constant.

This package of properties is referred to as (Lipschitz) wellposedness.

This method extends to certain partial differential equations, particularly those of a semilinear nature (linear except for lower order nonlinear terms). For instance, if trying to solve an initial value problem of the form

$\displaystyle \partial_t u + Lu = F(u); \quad u(0,x) = u_0(x),$

where now ${u: {\bf R} \rightarrow V}$ takes values in a function space ${V}$ (e.g. a Sobolev space ${H^k({\bf R}^d)}$), ${u_0 \in V}$ is an initial datum, ${L}$ is some (differential) operator (independent of ${u}$) that is (densely) defined on ${V}$, and ${F}$ is a nonlinearity which is also (densely) defined on ${V}$, then (formally, at least) one can solve this problem by using Duhamel’s formula to convert the problem to that of solving an integral equation

$\displaystyle u(t) = e^{-tL} u_0 + \int_0^t e^{-(t-s)L} F(u(s))\ ds$

and one can then hope to show that the associated nonlinear integral operator

$\displaystyle u \mapsto e^{-tL} u_0 + \int_0^t e^{-(t-s)L} F(u(s))\ ds$

is a contraction in a subset of a suitably chosen function space.

This method turns out to work surprisingly well for many semilinear partial differential equations, and in particular for semilinear parabolic, semilinear dispersive, and semilinear wave equations. As in the ODE case, when the method works, it usually gives the entire package of Lipschitz well-posedness: existence, uniqueness, and Lipschitz continuous dependence on the initial data, for short times at least.

However, when one moves from semilinear initial value problems to quasilinear initial value problems such as

$\displaystyle \partial_t u + L_u u = F(u); \quad u(0,x) = u_0(x)$

in which the top order operator ${L_u}$ now depends on the solution ${u}$ itself, then the nature of well-posedness changes; one can still hope to obtain (local) existence and uniqueness, and even continuous dependence on the data, but one usually is forced to give up Lipschitz continuous dependence at the highest available regularity (though one can often recover it at lower regularities). As a consequence, the Picard iteration method is not directly suitable for constructing solutions to such equations.

One can already see this phenomenon with a very simple equation, namely the one-dimensional constant-velocity transport equation

$\displaystyle \partial_t u + c \partial_x u = 0; \quad u(0,x) = u_0(x) \ \ \ \ \ (3)$

where we consider ${c = c_0}$ as part of the initial data. (If one wishes, one could view this equation as a rather trivial example of a system.

$\displaystyle \partial_t u + c \partial_x u = 0; \quad \partial_t c = 0$

$\displaystyle u(0,x) = u_0(x); \quad c(0) = c_0,$

to emphasis this viewpoint, but this would be somewhat idiosyncratic.) One can solve this equation explicitly of course to get the solution

$\displaystyle u(t,x) = u_0(x-ct).$

In particular, if we look at the solution just at time ${t=1}$ for simplicity, we have

$\displaystyle u(1,x) = u_0(x-c).$

Now let us see how this solution ${u(1,x)}$ depends on the parameter ${c}$. One can ask whether this dependence is Lipschitz in ${c}$, in some function space ${V}$:

$\displaystyle \| u_0(\cdot - c) - u_0(\cdot - c') \|_V \leq A |c-c'|$

for some finite ${A}$. But using the Newton approximation

$\displaystyle u_0(\cdot - c) - u_0(\cdot - c') \approx (c-c') \partial_x u_0(\cdot - c)$

we see that we should only expect such a bound when ${\partial_x u_0}$ (and its translates) lie in ${V}$. Thus, we see a loss of derivatives phenomenon with regard to Lipschitz well-posedness; if the initial data ${u_0}$ is in some regularity space, say ${C^3}$, then one only obtains Lipschitz dependence on ${c}$ in a lower regularity space such as ${C^2}$.

We have just seen that if all one knows about the initial data ${u_0}$ is that it is bounded in a function space ${V}$, then one usually cannot hope to make the dependence of ${u}$ on the velocity parameter ${c}$ Lipschitz continuous. Indeed, one cannot even make it continuous uniformly in ${V}$. Given two values of ${c}$ that are close together, e.g. ${c = 0}$ and ${c=\epsilon}$, and a reasonable function space ${V}$ (e.g. a Sobolev space ${H^k}$, or a classical regularity space ${C^k}$) one can easily cook up a function ${u_0}$ that is bounded in ${V}$ but whose two solutions ${u_0(\cdot)}$ and ${u_0(\cdot-\epsilon)}$ separate in the ${V}$ norm at time ${1}$, simply by choosing ${u_0}$ to be supported on an interval of width ${\epsilon}$.

(Part of the problem here is that using a subtractive method ${\|u-v\|_V}$ to determine the distance between two solutions ${u, v}$ is not a physically natural operation when transport mechanisms are present that could cause the key features of ${u, v}$ (such as singularities) to be situated in slightly different locations. In such cases, the correct notion of distance may need to take transport into account, e.g. by using metrics of Wasserstein type.)

On the other hand, one still has non-uniform continuous dependence on the initial parameters: if ${u_0}$ lies in some reasonable function space ${V}$, then the map ${c \mapsto u_0(\cdot-c)}$ is continuous in the ${V}$ topology, even if it is not uniformly continuous with respect to ${v_0}$. (More succinctly: translation is a continuous but not uniformly continuous operation in most function spaces.) The reason for this is that we already have established this continuity in the case when ${u_0}$ is so smooth that an additional derivative of ${u_0}$ lies in ${V}$; and such smooth functions tend to be dense in the original space ${V}$, so the general case can then be established by a limiting argument, approximating a general function in ${V}$ by a smoother function. We then see that the non-uniformity ultimately comes from the fact that a given function in ${V}$ may be arbitrarily rough (or concentrated at an arbitrarily fine scale), and so the ability to approximate such a function by a smooth one can be arbitrarily poor.

In many quasilinear PDE, one often encounters qualitatively similar phenomena. Namely, one often has local well-posedness in sufficiently smooth function spaces ${V}$ (so that if the initial data lies in ${V}$, then for short times one has existence, uniqueness, and continuous dependence on the data in the ${V}$ topology), but Lipschitz or uniform continuity in the ${V}$ topology is usually false. However, if the data (and solution) is known to be in a high-regularity function space ${V}$, one can often recover Lipschitz or uniform continuity in a lower-regularity topology.

Because the continuous dependence on the data in quasilinear equations is necessarily non-uniform, the arguments needed to establish this dependence can be remarkably delicate. As with the simple example of the transport equation, the key is to approximate a rough solution by a smooth solution first, by smoothing out the data (this is the non-uniform step, as it depends on the physical scale (or wavelength) that the data features are located). But for quasilinear equations, keeping the rough and smooth solution together can require a little juggling of function space norms, in particular playing the low-frequency nature of the smooth solution against the high-frequency nature of the residual between the rough and smooth solutions.

Below the fold I will illustrate this phenomenon with one of the simplest quasilinear equations, namely the initial value problem for the inviscid Burgers’ equation

$\displaystyle \partial_t u + u u_x = 0; \quad u(0,x) = u_0(x) \ \ \ \ \ (4)$

which is a modification of the transport equation (3) in which the velocity ${c}$ is no longer a parameter, but now depends (and is, in this case, actually equal to) the solution. To avoid technicalities we will work only with the classical function spaces ${C^k}$ of ${k}$ times continuously differentiable functions, though one can certainly work with other spaces (such as Sobolev spaces) by exploiting the Sobolev embedding theorem. To avoid having to distinguish continuity from uniform continuity, we shall work in a compact domain by assuming periodicity in space, thus for instance restricting ${x}$ to the unit circle ${{\bf R}/{\bf Z}}$.

This discussion is inspired by this survey article of Nikolay Tzvetkov, which further explores the distinction between well-posedness and ill-posedness in both semilinear and quasilinear settings.


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# Events Archive

## 2013

### Hamilton Colloquium Series - William Young, Scripps Institution of Oceanography, UC, San Diego - "Two Dimensional Turbulence"

Thu, Mar 14, 2013, 4:30 pm to 5:30 pm
In the first part of this talk I will review basic results about two-dimensional turbulence emphasizing the absence of a dissipative anomaly in D=2, and the energy-conserving long-time behavior of solutions of the inviscid equations of motion. Arguments dating back to Onsager predict the formation of an en-semble of vortices separated by potential...

### PCTS/Condensed Matter Seminar - Salvatore R. Manmana, Goettingen University - Far-from-Equilibrium Quantum Magnetism with Ultracold Polar Molecules

Thu, Mar 14, 2013, 1:15 pm to 2:30 pm
Recent theory has indicated how to emulate tunable models of quantum magnetism with ultracold polar molecules.

### Special Condensed Matter Seminar-Roman Lutchyn, Microsoft Sta. Q, UC, SB - Majorana zero-energy modes in semiconductor/superconductor heterostructures

Wed, Mar 13, 2013, 1:15 pm to 2:30 pm
The exploration of topological phases of matter is one of the main challenges in condensed matter physics. Among the exciting recent developments in this direction are the discoveries of new phases of matter with many intriguing properties, such as topological insulators and superconductors.

### High Energy Theory Seminar Informal - IAS - Seok Kim, Seoul National University - “M5-brane Indices from 5d Gauge Theories”

Tue, Mar 12, 2013, 1:30 pm to 3:00 pm
I will talk about the index which counts BPS local operators ofthe 6d (2,0) theory, calculated from gauge theories on the 5-sphere.

### High Energy Theory Seminar - IAS - Dieter Luest, Max-Planck-Institute for Physics, Munich - “Closed Strings and Non-Commutative/Non-Associative Ge

Mon, Mar 11, 2013, 2:30 pm to 4:00 pm
In this talk I discuss that non-commutative and non-associative geometry emerges for closed stringsmoving in non-geometric backgrounds. The origin of closed string non-commutativity (non-associativity)is due to new closed string boundary conditions in the presence of non-geometric background fields.

### Henrik Flyvbjerg (Danish Technical University) "Optimal estimation of diffusion coefficients from noisy single-particle trajectories"

Mon, Mar 11, 2013, 12:00 pm to 1:00 pm
Einstein argued in 1905 that the mean squared displacement of a microscopic particle suspended in a fluid at rest is proportional to time with the constant of proportionality giving the particle's
Audience: Physics faculty, post docs, grads

### Towards a Predictive Science of Network-Based Systems - Dani Bassett - UCSB

Fri, Mar 8, 2013, 1:30 pm to 2:30 pm
Abstract: Many real world systems utilize networks of mechanical, electrical, or informational signals to perform complex functions.

### Hamilton Colloquium Series - Ignacio Cirac, Max-Planck Institute for Quantum Optics - "Dissipation as a New Tool in Quantum Information Science"

Thu, Mar 7, 2013, 4:30 pm to 5:30 pm

Quantum entanglement, the most striking feature of quantum mechanics, is also the basic ingredient in most applications in the field of quantum information. Unfortunately, it is very fragile: in all experiments so far the coupling of the systems to the environment has lead to dissipation which either destroys entanglement or prevents its...

### Mathematical Physics Seminar - Antti Knowles, Courant Institute - Quantum diffusion and delocalization for random band matrices

Tue, Mar 5, 2013, 4:30 pm to 6:00 pm
I give a summary of recent progress in establishing the diffusion approximation for random band matrices. We obtain a rigorous derivation of the diffusion profile in the regime W > N^{4/5}, where W is the band width and N the dimension of the matrix. As a corollary, we prove complete delocalization of the eigenvectors.

### Condensed Matter Seminar - Collin Broholm, The Johns Hopkins University - "Continuum Excitations in Crystalline Magnets"

Mon, Mar 4, 2013, 1:15 pm to 2:30 pm
I present inelastic neutron scattering data from one- two- and three-dimensional insulating magnetic materials at low temperatures that do not display a coherent resonant mode of excitation. Instead, momentum resolved spectra take the form of bounded continua. I interpret the spectra as evidence for fractionalization of a spin flip into distinct...

### Sebastian Maerkl, EPFL Lausanne "Large-scale single cell analysis"

Mon, Mar 4, 2013, 12:00 pm to 1:00 pm
Observing cellular responses to perturbations is central to generating and testing hypothesis in biology. Time-lapse microscopy is a powerful tool for studying cellular phenotypes, but has been refractory to proteome-wide investigations.
Audience: Physics faculty, post docs, grads

### High Energy Theory Seminar - Enrico Pajer, Princeton University - The Effective Field Theory of Large Scale Structures

Fri, Mar 1, 2013, 1:30 pm to 3:00 pm
Abstract: An analytical understanding of large-scale matter inhomogeneities is an important cornerstone of our cosmological model an helps us interpreting current and future data.

### Hamilton Colloquium Series - Abhay Pasupathy, Columbia University, "What drives electronic nematicity in the iron-based superconductors?"

Thu, Feb 28, 2013, 4:30 pm to 5:30 pm
The iron arsenides are a recently discovered class of unconventional superconducting materials. This class of materials consists of various families (cryptically called 111, 122, 1111, etc.) each of which has a "parent" compound. These parent compounds (e.g. NaFeAs) are typically not superconducting, but display a spin-density wave phase at low...

### High Energy Theory Seminar - Bertrand Duplantier, Institute for Theoretical Physics, Saclay - Schramm-Loewner Evolution and Liouville Quantum Gravity

Wed, Feb 27, 2013, 3:30 pm to 4:30 pm
Liouville quantum gravity in two dimensions is described by the random Riemannian manifold'' obtained by changing the Lebesgue measure in the plane by a random conformal factor, the exponential of the Gaussian free field.

### Cancelled: Math Physics Seminar - Jakob Yngvason, University of Vienna - Interacting Bosons in a Random Potential

Tue, Feb 26, 2013, 4:30 pm to 6:00 pm
We study the effects of random scatterers on the ground state of the one-dimensional Lieb-Liniger model of interacting bosons in the Gross-Pitaevskii regime.


text:
# Does MCMC overcome the curse of dimensionality?

I want to compute an integral like this

$$\frac{\int_y g(y) e^{-\beta f(y)} \text{d} y } {\int_y e^{-\beta f(y)} \text{d} y}$$

where $f(y)$ is not necessarily convex and the dimension $d$ of $y$ is large.

This problem can be viewed as to integrate a function with respect to a density function whose normalized factor is unknown. It seems MCMC is a good choice? However for this general problem I didn't find any literature showing the convergence rate w.r.t the dimension $d$.

Remark: there are two ways I know to implement MCMC algorithms. One is by Metropolis–Hastings algorithm; another is by Metropolis-adjusted Langevin algorithm, which simulates a SDE $dX(t)=-\nabla f(x(t))dt + \sqrt{2\beta^{-1}} dB(t)$. In the latter case we also need to consider discretization errors of the SDE.

I guess without convexity assumptions on $f(y)$ the convergence rate could be $\mathcal{O}(e^{-d})$ slow. If this is true, it may suffer from the curse of dimensionality?

The reason why I came up with this problem is that we can just use simple Monte Carlo method, i.e. just sample uniformly distributed random variable to compute the integral on numerator and denominator separately. And as we know this time the variance of our estimate is not related to $d$.

I am very confused about the role MCMC plays in high dimension problems. Does someone there can help me figure out this? Any references are much appreciated.

You need a global convexity to enjoy the optimal convergence rate, otherwise even local convexity will almost surely(not in probabilistic sense) lead to the worst rate you pointed out.

MCMC(Markov Chain Monte Carlo) does not overcome the curse of dimensionality. Quite the contrary, Bayesians are working very hard in two directions to solve the problems that caused by the high dimension of the parameter space.

(1) The ABC(Approximate Bayesian Computation) scheme. This originated from Laplace's approximation of an exponential integral and its main idea is to discard those samples from each MH step that is "dissimilar from existing samples". But this method suffers from failure of detecting outliers and over-concentration of the posterior.

On the other hand even if this method works well during sampling, we do not have any consistency guarantee since when nuisance parameters $\theta_2$ consist of the majority of the parameter space $\boldsymbol{\theta}=(\theta_1,\theta_2)$, Neyman-Scott example will nullify the information about the parameter $\theta_1$ of concern brought in by the samples. That is also a reason why regularization methods are so popular in high dimension inference problems, they simply produce a weighted norm that cannot be washed out by samples.

(2) The scalable methods(For example). These are sometimes referred to as divide-and-conquer problems. The Bayesian model, or more precisely the parameter space is decomposed into subspaces, and the high dimension issue is divided into many low dimension issues and the $O(n^d)$ problem becomes $O(n^{d/k}(n/k))$ problems. But the problem is also obvious, that is decomposing and combining parameter spaces will artificially delete and add correlations between parameter spaces, which makes the prior information about $\boldsymbol \theta$ lose.

• You mentioned "the worst polynomial rate" in the first sentence. I thought MCMC needs approximately $\mathcal{O}(e^d)$ to converge, i.e. exponential time w.r.t dimension? – Yifan Jul 31 '17 at 23:57
• @Yifan Yes, typo, corrected. – Henry.L Aug 1 '17 at 0:02
• So generally we cannot find an algorithm which only needs polynomial time to converge. And our efforts are to reduce the number of operations, i.e. reduce the constant $C$ in $\mathcal{O}(e^{d}) \leq Ce^{d}$. Do I understand correctly? However for the simple Monte Carlo sample method, the variance of our estimator looks like $\mathcal{O}(1/N)$ which is not related to $d$ (N is the number of sample points) thus avoid the curse of dimensionality. Why don't we use this method? Is it because that in this case the constant $C$ is too large? ( $\mathcal{O}(1/N) \leq C*1/N$ ) – Yifan Aug 1 '17 at 0:15
• @Yifan That is what I believed, and the problem is that there is not a simple method to tell MCMC stop. For simple MC, you need to derive closed form marginal, which we usually do not have. – Henry.L Aug 1 '17 at 0:19
• I mean, we can just use uniform distribution in simple MC method to compute the integral, not looking for closed form marginal. Though it may be slow, its convergence rate is not related to the dimension. So in this sense simple MC has advantages over MCMC in the dependence on dimension. – Yifan Aug 1 '17 at 0:27


text:
# How much weed is that?

## How Much is an Eighth of Marijuana?

When weighing out cannabis products, both standard and metric systems of measurement are used. The most common metric unit used is grams (we don’t deal in kilos in the weed biz) which is used to measure small amounts of product.

Large quantities of cannabis are measured in pounds (as in Colorado sold 148,238 pounds of marijuana (pdf) in 2014) so breaking it down further using the standard system of measurement makes sense. To do this, we assume that a pound weighs roughly 16 oz and that each of those ounces can further be broken down into eight, easy-to-handle parts (or “eighths” in dealer speak).

Eighths are typically weighed out as 3.5 grams. This assumes that an ounce weighs exactly 28 grams which isn’t exactly accurate (in fact, an ounce actually weighs 28.349523125 grams but rounding is easier to calculate and better for overhead). To make up for this discrepancy, some dispensaries offer “weigh heavy” deals where they weigh eighths out at four grams each, so check local deals before making a purchase to get the most for marijuana for your money.

To figure out how much marijuana that really is, let’s assume that the average joint weighs between half and one gram (which should spread between two to four people comfortably, depending, of course, on the tolerance of the consumers). So, if the average joint weighs .7 grams, an ounce of reefer should net about 40 joints total, or roughly enough to get you through a Phish concert! (J/K, don’t smoke 40 joints at once… Seriously.)

all author posts


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Delay-Adaptive Linear Control by Yang Zhu Miroslav Krstic

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Delay-Adaptive Linear Control by Yang Zhu Miroslav Krstic

Yang Zhu

Miroslav Krstic

Volume: 70

DOI: 10.2307/j.ctvrf8c6w

https://www.jstor.org/stable/j.ctvrf8c6w

Book Description:

Actuator and sensor delays are among the most common dynamic phenomena in engineering practice, and when disregarded, they render controlled systems unstable. Over the past sixty years, predictor feedback has been a key tool for compensating such delays, but conventional predictor feedback algorithms assume that the delays and other parameters of a given system are known. When incorrect parameter values are used in the predictor, the resulting controller may be as destabilizing as without the delay compensation. Delay-Adaptive Linear Control develops adaptive predictor feedback algorithms equipped with online estimators of unknown delays and other parameters. Such estimators are designed as nonlinear differential equations, which dynamically adjust the parameters of the predictor. The design and analysis of the adaptive predictors involves a Lyapunov stability study of systems whose dimension is infinite, because of the delays, and nonlinear, because of the parameter estimators. This comprehensive book solves adaptive delay compensation problems for systems with single and multiple inputs/outputs, unknown and distinct delays in different input channels, unknown delay kernels, unknown plant parameters, unmeasurable finite-dimensional plant states, and unmeasurable infinite-dimensional actuator states. Presenting breakthroughs in adaptive control and control of delay systems, Delay-Adaptive Linear Control offers powerful new tools for the control engineer and the mathematician.

eISBN: 978-0-691-20331-7

Subjects: Mathematics, Engineering

1. Front Matter

(pp. i-iv)

DOI: 10.2307/j.ctvrf8c6w.1

(pp. v-viii)

DOI: 10.2307/j.ctvrf8c6w.2

3. List of Figures and Tables

(pp. ix-xii)

DOI: 10.2307/j.ctvrf8c6w.3

4. Preface

(pp. xiii-xviii)

Yang Zhu and Miroslav Krstic

DOI: 10.2307/j.ctvrf8c6w.4

5. Acknowledgments

(pp. xix-xx)

DOI: 10.2307/j.ctvrf8c6w.5

6. List of Abbreviations

(pp. xxi-xxii)

DOI: 10.2307/j.ctvrf8c6w.6

7. Chapter One Introduction

(pp. 1-16)

DOI: 10.2307/j.ctvrf8c6w.7

Time-delay systems, also called systems with aftereffect or dead-time, hereditary systems, equations with deviating argument, or differential-difference equations [42], are ubiquitous in practice. Some representative examples are found in

chemical industry: rolling mills, milling processes, cooling systems, combustion systems, chemical processes,

electrical and mechanical engineering: networked control systems, teleoperation, robotic manipulators, unmanned aerial vehicles,

biomedical engineering: 3D printing/additive manufacturing, neuromuscular electrical stimulation,

management and traffic science: traffic flow, supply chain, population dynamics.

The most common forms of time delay in dynamic phenomena that arise in engineering practice are actuator and sensor delays [133, 134]. Due to the time it takes...

8. I Single-Input Discrete Delay

• Chapter Two Basic Predictor Feedback for Single-Input Systems

(pp. 19-34)

DOI: 10.2307/j.ctvrf8c6w.8

In this chapter, we introduce the basic idea of a PDE backstepping approach for single-input LTI ordinary differential equation (ODE) systems with discrete input delay. This chapter is the core for all the developments in the rest of the book. The results in this chapter are obtained under the assumption that all information about the considered systems is known to designer, and they easily (or even trivially) extend to systems with multiple inputs when the delay is the same in all the input channels.

By treating the delayed input as a first-order hyperbolic transport PDE, we transform the LTI-ODE systems...

• Chapter Three Basic Idea of Adaptive Control for Single-Input Systems

(pp. 35-57)

DOI: 10.2307/j.ctvrf8c6w.9

In chapter 2, the general class of single-input LTI systems with discrete input delay was taken into account, which is recapped as follows to avoid reading back and forth:

${\text{[}}\stackrel{˙}{X}\left(t\right)=AX\left(t\right)+BU\left(t-D\right).\left(3.1\right){\text{]}}$

Through the first-order hyperbolic transport PDE state,

${\text{[}}u\left(x,t\right)=U\left(t+D\left(x-1\right)\right),\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}x\in \left[0,1\right],\left(3.2\right){\text{]}}$

the LTI-ODE systems (3.1) could be represented by the following ODE-PDE cascades:

${\text{[}}\stackrel{˙}{X}\left(t\right)=AX\left(t\right)+Bu\left(0,t\right),\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}t\ge 0,\left(3.3\right){\text{]}}$ ${\text{[}}D{u}_{t}\left(x,t\right)={u}_{x}\left(x,t\right),\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}x\in \left[0,1\right],\left(3.4\right){\text{]}}$${\text{[}}u\left(1,t\right)=U\left(t\right).\left(3.5\right){\text{]}}$

In the control scheme of chapter 2, the full information of the plants (3.3)–(3.5) is assumed to be available. In other words,

the finite-dimensional ODE plant state vector 𝑋(𝑡) is available for measurement,

the infinite-dimensional PDE actuator state 𝑢(𝑥, 𝑡) that represents the delayed input 𝑈(𝜂)...

• Chapter Four Single-Input Systems with Full Relative Degree

(pp. 58-83)

DOI: 10.2307/j.ctvrf8c6w.10

At the beginning of chapter 3, we introduced four types of basic uncertainties that usually come with single-input LTI systems with discrete actuator delay:

unknown actuator delay 𝐷, which is allowed to be arbitrarily large,

unknown plant parameter vector 𝜃 in system matrix 𝐴(𝜃) and input matrix 𝐵(𝜃),

unmeasurable finite-dimensional ODE plant state 𝑋(𝑡),

unmeasurable infinite-dimensional PDE actuator state 𝑢(𝑥, 𝑡), which represents delayed input 𝑈(𝜂) over the time window 𝜂 ∈ [𝑡 − 𝐷, 𝑡].

In subsequent sections of chapter 3, we successively dealt with adaptive control problems under uncertainty solely in either 𝑋(𝑡) or 𝐷, and coexistent uncertainties...

• Chapter Five Single-Input Systems with Arbitrary Relative Degree

(pp. 84-118)

DOI: 10.2307/j.ctvrf8c6w.11

In chapter 4, we dealt with adaptive control for single-input LTI systems with uncertainty collection (𝐷, 𝑋(𝑡), 𝜃) by assuming that the actuator state 𝑢(𝑥, 𝑡) is measurable. Furthermore, as stated in [88], the relative degree plays an important role in determining the difficulty of the output-feedback adaptive problems, namely, problems involving both unmeasurable ODE state 𝑋(𝑡) and uncertain parameter 𝜃 in system and input matrices 𝐴(𝜃) and 𝐵(𝜃). Thus, in chapter 4, the plants of interest were restricted to a class of systems with its relative degree being identical with its system dimension. In this chapter, we remove this...

• II Multi-Input Discrete Delays

• Chapter Six Exact Predictor Feedback for Multi-Input Systems

(pp. 121-133)

DOI: 10.2307/j.ctvrf8c6w.12

In part I, we investigated adaptive control problems of uncertain single-input LTI systems with discrete input delay, and the results obtained seem to easily (or even trivially) extend to systems with discrete multiple inputs when the delay is the same in all the input channels. In part II, we consider adaptive control problems of multi-input LTI systems with distinct discrete input delays; i.e., each delay length in several multi-input channels is not the same. The problem of distinct delays significantly complicates the prediction design, as it requires one to compute different future state values on time horizons, making this computation...

• Chapter Seven Full-State Feedback of Uncertain Multi-Input Systems

(pp. 134-168)

DOI: 10.2307/j.ctvrf8c6w.13

In chapter 6, under the assumption of a perfect prior knowledge of the system, a PDE-based framework of exact predictor feedback for multi-input LTI systems with distinct discrete input delays was proposed. On the basis of that, in this chapter, we investigate adaptive control for uncertain multi-input LTI systems with distinct discrete actuator delays. In parallel with the single-input case in chapter 3, four types of basic uncertainties come with multi-input LTI time-delay systems:

the finite-dimensional ODE plant state vector 𝑋(𝑡) is unavailable for measurement,

the infinite-dimensional PDE actuator state 𝑢𝑖(𝑥, 𝑡) for 𝑖=1, …, 𝑚, which represents the distinct...

• Chapter Eight Output Feedback of Uncertain Multi-Input Systems

(pp. 169-198)

DOI: 10.2307/j.ctvrf8c6w.14

In chapter 7 we solved the problem of adaptive stabilization in the presence of distinct discrete multiple actuator delays that are long and unknown, under the assumption that the actuator state in each input channel is available for measurement. In this chapter this assumption of measurable actuator state is removed, as shown in table 7.1.

Similar to the case of single-input delay, the result of multi-input delays that we obtain in this chapter is not global, as we do not believe the problem where the actuator state is not measurable and the delay value is unknown at the same time...

• Chapter Nine Output Feedback of Systems with Uncertain Delays, Parameters, and ODE State

(pp. 199-216)

DOI: 10.2307/j.ctvrf8c6w.15

In chapter 8 we solved the problem of adaptive stabilization of multi-input LTI systems under the assumption that distinct discrete multiple actuator delays are long and unknown, and the actuator state in each input channel is unavailable for measurement. A local stabilization result is achieved. In this chapter, as shown in table 7.1, we consider the more challenging case where uncertainties exist not only in the input delay and the PDE actuator state, but also appear in the ODE plant state and the parameters of the system and input matrices. To be specific, the uncertainty collections (𝐷𝑖, 𝑢𝑖(𝑥, 𝑡), 𝑋(𝑡))...

• III Distributed Input Delays

• Cite this Item

• Cite this Item

• Cite this Item

• Chapter Ten Predictor Feedback for Uncertainty-Free Systems

(pp. 219-234)

DOI: 10.2307/j.ctvrf8c6w.16

In parts I and II, we investigated predictor control problems of uncertain single-input or multi-input LTI systems with discrete input delays. In this part, we extend the predictor method to compensate for another big family of delays—distributed input delays.

The control designs for unknown discrete input delays in parts I and II are not applicable to the case of unknown distributed input delays, as the plant and actuator states are not in the strict feedback form. In this part, we presents a new systematic method to stabilize uncertain LTI systems with distributed input delays.

First of all, in order...

• Chapter Eleven Predictor Feedback of Uncertain Single-Input Systems

(pp. 235-266)

DOI: 10.2307/j.ctvrf8c6w.17

The single-input linear system with distributed input delay in section 10.1 is recapped as follows:

${\text{[}}\stackrel{˙}{X}\left(t\right)=AX\left(t\right)+{\int }_{0}^{D}B\left(D-\sigma \right)U\left(t-\sigma \right)d\sigma ,\left(11.1\right){\text{]}}$ ${\text{[}}Y\left(t\right)=CX\left(t\right),\left(11.2\right){\text{]}}$

which can be represented by the following ODE-PDE cascade:

${\text{[}}\stackrel{˙}{X}\left(t\right)=AX\left(t\right)+D{\int }_{0}^{1}B\left(Dx\right)u\left(x,t\right)dx,\left(11.3\right){\text{]}}$ ${\text{[}}Y\left(t\right)=CX\left(t\right),\left(11.4\right){\text{]}}$${\text{[}}D{u}_{t}\left(x,t\right)={u}_{x}\left(x,t\right),\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}x\in \left[0,1\right],\left(11.5\right){\text{]}}$ ${\text{[}}u\left(1,t\right)=U\left(t\right).\left(11.6\right){\text{]}}$

As clarified in the last chapter, if we concentrate on (11.1)–(11.2) and its conversion (11.3)–(11.6), a single-input linear plant with distributed actuator delay comes with the following five types of basic uncertainties:

unknown delay 𝐷,

unknown delay kernel 𝐵(𝐷𝑥),

unknown parameters in the system matrix 𝐴,

unmeasurable finite-dimensional plant state 𝑋(𝑡),

unmeasurable infinite-dimensional actuator state 𝑢(𝑥, 𝑡).

In this chapter, we present the predictor feedback for uncertain single-input...

• Chapter Twelve Predictor Feedback of Uncertain Multi-Input Systems

(pp. 267-292)

DOI: 10.2307/j.ctvrf8c6w.18

The multi-input linear system with different distributed input delays in section 10.2 is recapped as follows:

${\text{[}}\stackrel{˙}{X}\left(t\right)=AX\left(t\right)+\sum _{i=1}^{2}{\int }_{0}^{{D}_{i}}{B}_{i}\left(\sigma \right){U}_{i}\left(t-\sigma \right)d\sigma ,\left(12.1\right){\text{]}}$ ${\text{[}}Y\left(t\right)=CX\left(t\right),\left(12.2\right){\text{]}}$

which can be represented by the following ODE-PDE cascade:

${\text{[}}\stackrel{˙}{X}\left(t\right)=AX\left(t\right)+\sum _{i=1}^{2}{D}_{i}{\int }_{0}^{1}{B}_{i}\left({D}_{i}\left(1-x\right)\right){u}_{i}\left(x,t\right)dx,\left(12.3\right){\text{]}}$ ${\text{[}}Y\left(t\right)=CX\left(t\right),\left(12.4\right){\text{]}}$${\text{[}}{D}_{i}{\mathrm{\partial }}_{t}{u}_{i}\left(x,t\right)={\mathrm{\partial }}_{x}{u}_{i}\left(x,t\right),\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}x\in \left[0,1\right],\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}i=1,2,\left(12.5\right){\text{]}}$ ${\text{[}}{u}_{i}\left(1,t\right)={U}_{i}\left(t\right).\left(12.6\right){\text{]}}$

If we concentrate on the system (12.3)–(12.6), a finite-dimensional multi-input linear system with distributed actuator delays comes with five types of basic uncertainties:

unknown and distinct delays 𝐷𝑖,

unknown delay kernels 𝐵𝑖(𝐷𝑖(1 − 𝑥)),

unknown system matrix 𝐴,

unmeasurable finite-dimensional plant state 𝑋(𝑡),

unmeasurable infinite-dimensional actuator state 𝑢𝑖(𝑥, 𝑡).

Based on the predictor feedback framework for uncertainty-free multi-input systems in section 10.2, in this chapter, we present the...

• Appendix A

(pp. 293-295)

DOI: 10.2307/j.ctvrf8c6w.19

• Appendix B

(pp. 296-302)

DOI: 10.2307/j.ctvrf8c6w.20

• Appendix C

(pp. 303-313)

DOI: 10.2307/j.ctvrf8c6w.21

• Appendix D

(pp. 314-316)

DOI: 10.2307/j.ctvrf8c6w.22

• Bibliography

(pp. 317-330)

DOI: 10.2307/j.ctvrf8c6w.23

• Index

(pp. 331-332)

DOI: 10.2307/j.ctvrf8c6w.24

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text:
Generating the graded S_n-module associated to an operad

Suppose I have a symmetric operad $\mathcal{P}$ defined over $\text{Vect}_{\mathbb{K}}$ with generators and relations in degrees at most $l$. Now suppose I already know $\mathcal{P}(k)$ as an $S_k$-representation for $0 \leq k \leq l$.

How do I determine $\mathcal{P}(l+1)$ from $\mathcal{P}(0), \mathcal{P}(1), \ldots, \mathcal{P}(l)$ ? More generally, how do I determine $\mathcal{P}(k)$ for some large $k$?

My current technique is to find vector space generators for each grade of the operadic ideal of relations by building trees in which some relation appears. Then I use linear algebra and Schur-Weyl duality to determine each grade of the ideal as a representation. This technique is reliable for small cases, but the number of trees grows large and this generating set is usually quite redundant.

I am hoping (perhaps naively) for some combinatorial rule in terms of Young diagrams which can answer the question explicitly.

Edit: Vladimir Dotsenko suggests that I limit the scope, so here's a (more reasonable) question which is also interesting:

Assume further that $\mathcal{P}$ is generated by a single associative binary operation. Now can we determine $\mathcal{P}(k)$ from an initial segment using the combinatorics of Young diagrams?

It seems like the Littlewood-Richardson rule might be of some help here. There also seems to be some plethysm going on. Perhaps these tools are sufficient?

-
In this generality, nothing much better can be done, I think. If you are hoping for some suggestions, you might want to narrow things down a bit. – Vladimir Dotsenko Jun 13 '12 at 7:25
@Vladimir Dostenko Good to know, thanks. I have added a smaller question which seems more likely to have an answer. – John Wiltshire-Gordon Jun 13 '12 at 16:10
In your new setup, do you assume characteristic zero? – Vladimir Dotsenko Jun 14 '12 at 3:19
@Vladimir Dostenko Yes. – John Wiltshire-Gordon Jun 14 '12 at 3:58
@Vladimir Dotsenko Sorry I've been misspelling your name. – John Wiltshire-Gordon Jun 14 '12 at 4:04

Operads that are quotients of the associative operad (which I believe your updated question is aimed at) are, in characteristic zero, fairly extensively studied since the 1950s or so, under the somewhat misleading for geometers name "varieties of associative algebras". (One of the first papers that sort of created this field of study is, to the best of my knowledge, Specht's "Gesetze in Ringen", http://www.ams.org/mathscinet-getitem?mr=35274.) The main bit of knowledge one needs to translate papers on varieties of algebras into the operadic language are

"variety of associative algebras" = "quotient of the associative operad"

and

"codimensions of a variety"="dimensions of components of an operad".

Armed with this knowledge, have a look at recent surveys, for example the book of Giambruno and Zaicev (Google Books link: http://tinyurl.com/giambrunozaicev), and you will see many hints on how to approach this kind of questions. In all fairness, the main question people in this area are interested in is asymptotic growth rather than exact formulas, but on the way lots of things are done, including some representation-theoretic computations, so you might find what you need.

P.S. Feel free to e-mail me if you are interested in some particular operads, maybe I happen to know precisely what you want to find out.

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