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## July 15, 2014

### Math and Mass Surveillance: A Roundup

#### Posted by Tom Leinster

The Notices of the AMS has just published the second in its series “Mathematicians discuss the Snowden revelations”. (The first was here.) The introduction to the second article cites this blog for “a discussion of these issues”, but I realized that the relevant posts might be hard for visitors to find, scattered as they are over the last eight months.

So here, especially for Notices readers, is a roundup of all the posts and discussions we’ve had on the subject. In reverse chronological order (and updated after the original appearance of this post):

Posted at July 15, 2014 11:31 PM UTC

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### Re: Math and Mass Surveillance: A Roundup

The AMS ought to be ashamed of itself for publishing such a deceptive, specious, and self-serving column. Richard George makes vague references to unsupported allegations that the NSA weakened crypto standards–and such purposeful vagueness is his only hope of persuasion, given that the evidence we have at hand overwhelmingly supports the conclusion that the NSA engaged in misconduct.

In particular, the New York Times claimed in 2013 that some of the documents released by Edward Snowden “suggest that the N.S.A. generated one of the random number generators used in a 2006 N.I.S.T. standard — called the Dual EC DRBG standard — which contains a back door for the N.S.A,” a conclusion which has been largely acknowledged as overwhelmingly plausible, given the evidence at hand.

Moreover, the NSA’s 2013 budget request, also published in part by the New York Times, describes an NSA program with a stated aim to “insert vulnerabilities into commercial encryption systems, IT systems, networks, and endpoint communications devices used by targets.” I am no cryptographic expert, but this seems unambiguously to declare that the NSA is in fact in the business of weakening cryptographic systems and protocols.

After his various falsehoods, George ends his column with ad hominem attacks against those who have released this information to the public, beginning with his strange declaration that he and his colleagues are “disappointed” and have “a feeling of betrayal” from Snowden & al. (Why should we care how he and his fellow NSA employees feel when the issue at hand is whether they are participating in injustice?), and ending with the oft-repeated claim that such defectors have done “immense” damage–a claim that Snowden’s allies are fond of making yet in support of which they find themselves unable to cite even a single example.

Finally, in the course of his denunciation of the leakers, George misgenders Chelsea Manning–referring to her as Bradley, a name she has since renounced–, a woman who has made the incredibly brave decision to publicly announce her gender identity while incarcerated in a men’s military prison which offers no support for gender dysphoria. Even those who, like George, believe Manning’s contribution to her country to be traitorous rather than heroic ought to have the basic decency to respect her gender identity and the difficulties she has suffered and continues to suffer as a trans woman.

Posted by: bgg on July 16, 2014 11:47 PM | Permalink | Reply to this

### Re: Math and Mass Surveillance: A Roundup

Moreover, the NSA’s 2013 budget request, also published in part by the New York Times, describes an NSA program with a stated aim to “insert vulnerabilities into commercial encryption systems, IT systems, networks, and endpoint communications devices used by targets.”

Indeed. That’s exactly what I thought of when I read Richard George’s sentence:

I have never heard of any proven weakness in a cryptographic algorithm that’s linked to NSA; just innuendo.

He may not have heard of it, but we now have very strong evidence that this has happened.

Another part of George’s article also jumped out at me:

I’ve heard some knowledgeable people claim that this metadata can be used to create detailed profiles of people. Of course, that’s not true.

There are unambiguous statements from top security officials attesting to the power of metadata. The NSA’s former General Counsel, Stewart Baker, said:

Metadata absolutely tells you everything about somebody’s life. If you have enough metadata, you don’t really need content.

Michael Hayden, the former director of both the NSA and the CIA, said:

We kill people based on metadata.

(He may have been referring to the NSA supplying target locations to the CIA for its drone strike programme. “We track ‘em, you whack ‘em”, as the slogan goes.)

To be fair, George does say “this metadata”, not just “metadata”. He may have been referring to just phone metadata (the subject of the preceding text), not all the other types of data that we now know is collected by the NSA.

bgg, will you be writing a letter to the Notices? That would be much more widely read than a comment on this blog.

Posted by: Tom Leinster on July 17, 2014 1:14 AM | Permalink | Reply to this

### Re: Math and Mass Surveillance: A Roundup

I considered preparing an expanded version of the above comment for a letter to the Notices, but since I am only entering my first year of graduate school this fall, I determined that the task was best left to someone more senior than I.

Posted by: bgg on July 17, 2014 1:57 AM | Permalink | Reply to this

### Re: Math and Mass Surveillance: A Roundup

That’s very understandable.

Posted by: Tom Leinster on July 17, 2014 2:03 AM | Permalink | Reply to this

### Re: Math and Mass Surveillance: A Roundup

Obviously the field of actual mathematics is amenable to a lot more logical rigour than general discussion (in this instance regarding ethics), but it would be interesting to look at if other writings AMS publishes have similar amounts of non-sequiturs and “assertions of facts not in evidence”.

Posted by: davetweed on July 17, 2014 12:17 AM | Permalink | Reply to this

### Re: Math and Mass Surveillance: A Roundup

A common feature of these pro-NSA/GCHQ articles is that they never address any of the actual facts that we’ve learned from the Snowden revelations. Under those circumstances, it’s hard to have a meaningful debate.

Of course, another common feature of these articles is that the people who write them are or were employed by the agencies, which maybe means that they’re legally constrained from referring to any of the vast amount of evidence we now have at our disposal. It says something that hardly anyone not employed by the NSA/GCHQ has stepped up.

If you feel like identifying some specific non-sequiturs and unsupported factual assertions, do feel free…

Posted by: Tom Leinster on July 17, 2014 1:23 AM | Permalink | Reply to this

### Re: Math and Mass Surveillance: A Roundup

It says something that hardly anyone not employed by the NSA/GCHQ has stepped up.

As I’ve said before, I think what it says is that the people who get excited about this sort of thing are the ones willing to invest time and energy in writing letters and columns. The rest of us are too busy with the rest of our lives.

Posted by: Mike Shulman on July 17, 2014 4:19 AM | Permalink | Reply to this

### Re: Math and Mass Surveillance: A Roundup

At least among the computer scientists I know, it’s simply taken for granted that the NSA engages in bad behavior. Even those who don’t say much tend to take the view that getting worked up about their misconduct is like getting excited by the sky being blue. E.g., it’s basically a running joke for crypto researchers to write about systems which are secure against “nation-state adversaries”.

Posted by: Neel Krishnaswami on July 17, 2014 10:49 AM | Permalink | Reply to this

### Re: Math and Mass Surveillance: A Roundup

Mike, I hope you didn’t mean that to sound as belittling as it does. But there’s something about your point that’s certainly true: if you approve of the status quo and have the political establishment on your side, then sure, you’ll be more inclined to do nothing.

Posted by: Tom Leinster on July 17, 2014 12:19 PM | Permalink | Reply to this

### Re: Math and Mass Surveillance: A Roundup

I think the thing that struck me was just the way that it was a lot of “here’s some text about something that sounds like it ought to be related, but when you think about it isn’t relevant”, and I was just wondering if this was below the standard that’s found in writings on other topics. I thought that a lot of the issues with the text were very obvious (bgg cites a couple above), but to cite just two examples:

1. Non-sequitur: The text goes from talking about arguing the NSA’s large-scale collection of metadata is cannot be used for very much profiling to statements about how much people who have decided to work at the NSA have their privacy dramatically reduced in order for the NSA to gain confidence about their reliability. I don’t see why the fact that an entity (in this case it’s employees) deciding to submit itself to some severe constraint in an area X has any relevance to the legitimacy of it imposing weaker constraints in area X upon other people. (The standard example is if I’m on a low-calorie diet, even if for a good reason for me, that doesn’t have any relationship to whether I should be able to determine things about what you eat.)

2. Unsupported assertion: There’s an flat assertion that no-one cares about commerical entities building up the sort of data profile that people are caring about the governmental intelligence agencies building up. This isn’t true: you can find lots of examples of people who are concerned about some aspect of commercial entities data profiling. (Eg, the existence of privacy badger and the failed Do Not Track header along with the people who created them and the articles they’ve written are supporting evidence.)It’s arguable (ideally citing the evidence the arguer is considering!) that there’s less concern than about governmental intelligence agencies, but that’s a different to saying that no-one cares.

Part of the reason that I care about this is that I do think that there are reasonable well-supported uses for big data analysis, profiling, etc, but there are also justifiable concerns people have. If the parts of the argument that the intelligence folks make which, as far as I can see, aren’t being self-censored due to stuff they can’t mention/acknowledge are so badly formed, I worry that we won’t progress to vitally needed discussions of what kind of proportionality applies in this area, when it is and is not reasonable for entities to acquire information for analysis, etc.

Posted by: davetweed on July 17, 2014 9:59 AM | Permalink | Reply to this

### Re: Math and Mass Surveillance: A Roundup

Part of me wishes that some mathematician would write a good article in support of NSA/GCHQ mass surveillance. The ones we’ve seen so far, in both the AMS and LMS publications, are so sloppily argued and fact-free that they can’t possibly represent the most coherent thoughts of what is, after all, a large community of intelligent people.

I agree with both your points 1 and 2. To add to point 1, someone in an earlier thread (I forget who or where) made the interesting suggestion that when new NSA employees are compelled to surrender privacy to the organization, this serves as useful indoctrination: they will then tend to be more willing to make light of other people’s privacy.

Point 2: yes, that’s a glaring non-sequitur. It’s possible to be concerned about loss of privacy to both governments and commercial organizations! Personally, I think government surveillance is much more pernicious, as (i) they collect your interactions with all commercial organizations (and non-commercial ones too), and (ii) they can back it up with legal, police and military force.

Posted by: Tom Leinster on July 17, 2014 12:39 PM | Permalink | Reply to this

### Re: Math and Mass Surveillance: A Roundup

Personally, I think government surveillance is much more pernicious, as (i) they collect your interactions with all commercial organizations (and non-commercial ones too), and (ii) they can back it up with legal, police and military force.

(iii) With commercial organizations, however tiny and well-hidden the fine print is, you’ve generally opted into their data collection at least nominally by using their services, and a committed individual may be able to find the fine print that documents that this is happening.

Posted by: Mark Meckes on July 17, 2014 2:14 PM | Permalink | Reply to this

### Re: Math and Mass Surveillance: A Roundup

At this point I probably ought to just put it out there that I’m someone who has worked on machine learning and data-mining in the past and may do again. (However, unlike Hugh Dennis I don’t have a cool story of someone offering me a position as a spook as an undergrad :-( )

I tend to think that goverments and commercial entities are equally potentially serious (at least when the commercial entities are data collectors on a huge scale). I was primarily just pointing out an asserted fact that had no supporting evidence and I think even the couple of links I provided demonstrated is a false statement.

Sticking to governmental agencies relevant here, one difference from commercial entities is that they are supposed to be more constrained by legal process. My biggest issue is the large area where there are fully consciously created legal procedures and safeguards for obtaining data via agencies demanding the data where a technological means has been found which means that the safeguards no longer constrain things. For example, obtaining a US individuals internet behaviour with a particular provider requires a warrant – that has been signed by a judge who has reviewed coherent legal evidence for suspicion, which hence means they’re only used where there is significant suspicion – whereas when obtained by direct snooping on cross-data-centre cables there is no such restraint. I’m of the opinion that at the very least it ought to be a situation where legal restraints for all these things depend only upon the result acheived, not how. (Eg, a government agency must have X to acquire an individuals internet data from a provider, regardless of whether it’s obtained by asking the company directly, buying the data from a 3rd party who has obtained it, obtaining it by snooping on cross data-centre cables, finding a reliable psychic who can tell them, etc…)

Posted by: davetweed on July 17, 2014 5:27 PM | Permalink | Reply to this

### Re: Math and Mass Surveillance: A Roundup

Just in case there’s any misunderstanding, Dave, I’m fully with you on point 2 (and I appreciate the links: I hadn’t heard of Privacy Badger). As you say, lots of people have written and campaigned about the dangers of Google, Facebook etc. having so much of our private data.

A while back, we discussed big data and mass surveillance on this blog, following a great essay on it by Nils Carqueville and Daniel Murfet.

I have a different interest in big data, too. In my work on quantifying biological diversity, I’m hearing increasingly often about the approach where you take a representative sample of the ecosystem you’re interested in, stick it all in a blender, sequence the DNA of the gloop you’ve just made, and deduce from the resulting huge data-file what organisms the ecosystem contains. This is radically different from the traditional approach of having expert botanists, entomologists, etc. picking laboriously through the sample, and it’s only feasible because of the exponential drop in the cost of DNA sequencing.

Since conserving the planet’s biological diversity probably has to start with cataloguing what’s actually out there, I think this is a very important application of data science. So yes: I also agree that there are very positive uses for data-mining etc., and the fact that it’s also used for whole-population surveillance doesn’t mean that people shouldn’t work on data science.

Posted by: Tom Leinster on July 17, 2014 5:44 PM | Permalink | Reply to this

### Re: Math and Mass Surveillance: A Roundup

Don’t worry, there’s no risk of misunderstanding. I think you’ve got a very well explained and reasonble point of view. I mentioned that part of my work more because it might well not be known by readers and I didn’t want to risk “lying by omission” on a fact that I could be said to be an “interest that may inform my views”.

Posted by: davetweed on July 17, 2014 6:01 PM | Permalink | Reply to this

### Re: Math and Mass Surveillance: A Roundup

Here’s another part of Richard George’s article that really bothered me:

The folks I knew in SIGINT, and I knew them very well, would not dream of violating U.S. citizens’ rights.

First, notice that word “U.S.”? Being part of the 95% of the world that isn’t the U.S., I’m not enormously comforted.

Second, it’s what we always hear: “I’ve worked in the secret services, and my colleagues are a decent bunch — you can trust them”. It doesn’t withstand factual examination. In similar articles for the LMS newsletter, GCHQ insiders Richard Pinch and Malcolm MacCallum invoked the head of GCHQ’s claim that if his staff were asked to “snoop” on the general public, they’d quit. In that case, I contrasted that claim with what we now know about GCHQ’s Optic Nerve programme, which surreptitiously collected images from ordinary people’s webcams.

That’s just one example. Virtually every revelation based on the Snowden documents describes some privacy-violating NSA or GCHQ programme. Even if you want to stick to US citizens only, there’s still masses of documentary evidence of privacy violations by the NSA; here’s a good recent example.

Against all the documentary evidence, warmth of feeling for one’s friends and fellow workers isn’t too persuasive. Maybe George’s colleagues weren’t involved in this kind of thing. Maybe they were, but didn’t consider that their work violated U.S. citizens’ rights, for one reason or another: heartfelt opinion, convenient rationalization, disinclination to question one’s superiors, …. Who knows? Whatever it is, it doesn’t change the documented facts.

Posted by: Tom Leinster on July 17, 2014 1:57 AM | Permalink | Reply to this

### Re: Math and Mass Surveillance: A Roundup

In similar articles for the LMS newsletter, GCHQ insiders Richard Pinch and Malcolm MacCallum invoked the head of GCHQ’s claim that if his staff were asked to “snoop” on the general public, they’d quit. In that case, I contrasted that claim with what we now know about GCHQ’s Optic Nerve programme, which surreptitiously collected images from ordinary people’s webcams.

That claim also doesn’t hold up well against the results of studies like the infamous Milgram experiment.

Posted by: Mark Meckes on July 17, 2014 2:19 PM | Permalink | Reply to this

### Re: Math and Mass Surveillance: A Roundup

Excellent point. For those of us unhappy with what the intelligence agencies are doing, there’s a temptation to think of them a bunch of cat-stroking Bond villains. When those who have worked there (some of whom are my friends) say that it’s a fun atmosphere full of ordinary people who just love doing math and don’t have an evil intention in the world, I don’t have a hard time believing it. But as the Milgram experiments and countless historical episodes make clear, you don’t have to be a hollow-eyed psychopath to inflict enormous harm.

Posted by: Tom Leinster on July 17, 2014 2:58 PM | Permalink | Reply to this

### Re: Math and Mass Surveillance: A Roundup

Literary references for this are Kafka’s Der Process and Philip K. Dick’s The Minority Report.

Jacob Appelbaum documents how the NSA records `details about visits to a popular internet journal for Linux operating system users called “the Linux Journal - the Original Magazine of the Linux Community”, and calls it an “extremist forum”. ‘

Posted by: Bas Spitters on July 17, 2014 9:15 PM | Permalink | Reply to this

### Re: Math and Mass Surveillance: A Roundup

This is evidence of an incompetent editor. A serious editor would have demanded that such a vapid statement be supported by some kind of evidence, particularly in a nominally apolitical forum such as the NAMS. Under the editorship of Steven Krantz the NAMS has published a number of low quality and essentially political pieces. He should be criticized for this piece (and others) because it’s his fault that it was published.

Posted by: Bobito on July 18, 2014 3:36 PM | Permalink | Reply to this

### Re: Math and Mass Surveillance: A Roundup

I agree that Richard George’s piece is poor, but I can understand why it got published. As it says in the intro to the first entry in this series, the series editors (Michael Harris and Allyn Jackson) “made many efforts to seek out authors whom we thought might write in defense of the NSA. However, this proved difficult”. So I can see why they and the overall Notices editor would have accepted an article with many deficiencies.

It could be argued that they should have prompted George to back up some of his statements with evidence, but on the whole I think I’d prefer the Notices to give mathematicians the chance to voice their opinions without editing them. (I don’t know whether they do edit them.) Certainly I’d have been unhappy if the editors of the LMS newsletter had wanted to edit the articles that I wrote.

Posted by: Tom Leinster on July 19, 2014 3:16 PM | Permalink | Reply to this

### Re: Math and Mass Surveillance: A Roundup

Not all opinions are valid. Those that are demonstrably in contradiction with the available evidence do not merit publication. If it is the case that the NAMS sought those who would defend the NSA and could find no one who could do so with any substantial argument, perhaps the conclusion for the editors should be that no defense is available. Treating equally a strongly supported argument and a vapid opinion is dishonest and, in the context of the notices of a professional society, unprofessional.

Posted by: Bobito on July 24, 2014 11:30 AM | Permalink | Reply to this

### Re: Math and Mass Surveillance: A Roundup

The AMS articles are getting a bit of attention on Twitter now. Journalist Glenn Greenwald (who’s reported a lot of the Snowden stories):

CERN physicist Subodh Patil:

Posted by: Tom Leinster on July 17, 2014 2:14 AM | Permalink | Reply to this

### More suspicious elliptic curves, according to Rivest

For those who may not have seen it, Peter Woit has a stack of links to NIST material dealing with analysis of the whole Dual EC DRBG debacle which more than blows what Richard George says out of the water.

More interesting, perhaps, and one for mathematicians to look at in detail is this statement:

One way this goes beyond the now-withdrawn NIST standard is that the committee also looked at other NIST current standards now in wide use, which in at least one other case depend upon a specific choice of elliptic curves made by the NSA, with no explanation provided of how the choice was made. In particular, Rivest recommends changing the ECDSA standard in FIPS186 because of this problem.

You can find the current version of NIST’s FIPS186 here and the ECDSA standard in ANSI X9.62:2005 here, which, scandalously, costs 100 dollars [hmm, the dollar sign killed the rest of the comment]. Thankfully, the mathematical details seem to all be contained in section 6 and appendix D of the NIST document. That appendix, in particular, contains lots of specific elliptic curves and ‘magic numbers’.

It would be great if someone could analyse these and if there are any suspicious features, write it up and plonk it on the arXiv or similar. Or better, in the comments section here :-)

Posted by: David Roberts on July 23, 2014 1:57 AM | Permalink | Reply to this

### Re: Math and Mass Surveillance: A Roundup

Dave Tweed wrote:

The text goes … to statements about how much people who have decided to work at the NSA have their privacy dramatically reduced in order for the NSA to gain confidence about their reliability. I dont see why the fact that an entity (in this case its employees) deciding to submit itself to some severe constraint in an area X has any relevance to the legitimacy of it imposing weaker constraints in area X upon other people. (The standard example is if Im on a low-calorie diet, even if for a good reason for me, that doesnt have any relationship to whether I should be able to determine things about what you eat.)

An important question here is, how much choice employees had when giving up big parts of their privacy (please see also this comment), last but not least this is also a question about working conditions. Like pop stars usually also have to give up big parts of their privacy, however they have eventually more means to compensate for that loss (?). So even if due to “professionality” there should be a distinction between the different “rights for privacy”, there may still be psychological factors play a role.

I do think that the aspects of working conditions, moral and judicial inner conflicts of employees in secret services and similar institutions (that could include some companies) should have more a chance to be voiced.

I had mentioned the following already in a comment on the german blog Netzpolitik, and others had said similar, but may be it should be repeated more often - it could make sense to think about a kind of court together with security options for whistleblowers, similar to the International Criminal Court. Not all whistleblowing is good, the overall damage might be vastly bigger than the overall benefits, this seems to be a case by case decision. The current institutions seem not prepared for such cases. In particular it seems that even mass surveillance is not seen as Crimes against humanity, moreover I am not sure how well the International Criminal Courts is able to protect people who point out such crimes (i.e. whistleblowers of crimes against humanity).

### Re: Math and Mass Surveillance: A Roundup

According to the BBC, GCHQ is now in some kind of partnership to run master’s degrees in “cyber security” at several UK universities: Edinburgh Napier ($\neq$ Edinburgh), Lancaster, Oxford, and Royal Holloway (London). The story says very little about what these courses will consist of or how GCHQ is involved, and doesn’t contain any links to further information about the degrees.

Posted by: Tom Leinster on August 3, 2014 9:30 PM | Permalink | Reply to this

### Re: Math and Mass Surveillance: A Roundup

I complained about the lack of links (which seems to be a habitual problem for old-media outlets doing new-media journalism). But I’ve now found the GCHQ press release of which the BBC article is a subset.

Posted by: Tom Leinster on August 5, 2014 2:48 AM | Permalink | Reply to this

### Re: Math and Mass Surveillance: A Roundup

Not sure it’s the maths+stats department at Lancaster who are involved in that one …

Posted by: Yemon Choi on August 26, 2014 2:42 AM | Permalink | Reply to this

### Re: Math and Mass Surveillance: A Roundup

Hi Yemon. I’d expect any qualification called “cyber security” to be hosted by a computer science department (although that doesn’t stop people in maths departments from being involved as consultants, lecturers, etc).

Regardless of location, I’d also expect a “cyber security” to include a large amount of applied mathematics, both of the cryptographic, number-theoretic type and the systems-analysis, probabilistic type.

(One of the reasons why I get irritated by the customary overly-narrow interpretation of the term “applied mathematics” is that it ignores the enormous amount of applied maths that’s done in computer science departments.)

Do you know anything about the implementation of this degree at Lancaster?

Posted by: Tom Leinster on August 26, 2014 3:57 PM | Permalink | Reply to this

### Re: Math and Mass Surveillance: A Roundup

As far as I can see, these degrees can just be regular computing science taught masters that contain a certain number of security-related courses and a security-related dissertation, and that are taught at an institution that contains several researchers with an interest in security. The term “security” is interpreted very broadly (at least by EPSRC), and covers things like human factors, forensics, business economics, hardware design and networking, so it doesn’t have to include any maths.

This seems to be part of a broader trend in offering effectively the same course with multiple titles, to attract a wider audience and to provide attractive looking CVs. For example, you can now do a masters degree in “data sciences” that doesn’t even have an option to take any statistics courses. It’s similar to the way some universities offer “pure” and “applied” maths degrees which are the same as their regular maths degrees, except for a few additional restrictions on course choices.

Posted by: Ciaran McCreesh on August 26, 2014 8:15 PM | Permalink | Reply to this

### Re: Math and Mass Surveillance: A Roundup

some universities offer “pure” and “applied” maths degrees which are the same as their regular maths degrees, except for a few additional restrictions on course choices.

Right — this is really quite strange. We at Edinburgh have an “applied maths” degree that is exactly the same as the maths degree except with a more restricted set of course choices. The only possible reason to take it is that you think it will look better on your CV.

Posted by: Tom Leinster on August 27, 2014 11:03 PM | Permalink | Reply to this

### Re: Math and Mass Surveillance: A Roundup

Hi Tom, I haven’t heard anything mentioned, which is why I don’t think my department is directly involved. No one in the department seems an obvious fit for that kind of course, and since Lancaster seems to go for the more applied side of computer science my guess is that this is CompSci’s initiative to package some existing modules in an “appealing” way, with more of a focus on implementation than theory.

As an aside: given the number theory that I teach our students, I find it hard to believe that my course will be part of this scheme, even if we do use RSA as an excuse to make the undergraduates get to grips with the Euler phi function…

Posted by: Yemon Choi on August 27, 2014 12:23 AM | Permalink | Reply to this

### Re: Math and Mass Surveillance: A Roundup

The September issue of the Notices contains an article by famous NSA whistleblower William Binney. Binney was a senior cryptanalyst at the NSA up until October 2001, when he quit because of what he saw as the NSA’s unconstitutional and unethical surveillance of American citizens.

Posted by: Tom Leinster on August 14, 2014 5:46 PM | Permalink | Reply to this

### Re: Math and Mass Surveillance: A Roundup

Eric Hobsbawm, is a famous historian, and his book ‘The age of extremes’ is his personal vision of the (short) 20C. In the final chapter ‘towards the new millenium’ he wrote:

“Nevertheless, some features of the global political landscape stood out. The first, as already noted, was the weakening of the nation-state…[by] losing its monolpoly on effective power and historic privileges within its own borders…

…[however] these developments did not make the state either redundant or ineffective. Indeed, in some respects its capacity to monitor and control the affairs of its citizens was reinforced by technology, since virtually all their financial and administrative transactions (other than small cash payments) were now likely to be recorded by computer, and all their communications (except for most face-to-face conversations in the open air) could now be intercepted and recorded”.

Hobsbawm, was writing in 1994: so quite prescient really.

Posted by: Mozibur Ullah on August 25, 2014 1:04 AM | Permalink | Reply to this

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# X-UTIL

|   Source

X-UTIL is a set of utility programs intended for (but not limited to) practising protein crystallographers.

format: plain


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# Comet's distance with and without gravity [closed]

I've got this question that I just can't seem to answer. The question is as follows:

A comet approaches the solar system with velocity v and would, if the sun would not attract the comet, pass the sun at a distance d. What is in reality the shortest distance between comet and sun? (Hint: use conservation of angular momentum and energy).

I've tried a few things, but apparently they are all wrong. I thought about getting a formula for the distance to the sun, and then finding where the derivative is 0. Is that the right approach? I just can't get to a correct formula. I've also tried equating the total energy and momentum with gravity to the total energy and momentum without gravity, but that becomes a complete mess. Any help or pointers would be greatly appreciated!

## closed as off-topic by ACuriousMind♦, Gert, user36790, John Rennie, Rob JeffriesNov 18 '15 at 8:14

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "Homework-like questions should ask about a specific physics concept and show some effort to work through the problem. We want our questions to be useful to the broader community, and to future users. See our meta site for more guidance on how to edit your question to make it better" – ACuriousMind, John Rennie, Rob Jeffries
If this question can be reworded to fit the rules in the help center, please edit the question.

• The Sun will ALWAYS attract the comet, even if the comet doesn't end up hitting the Sun. Your question is really confused: what is it you'd like to know or understand? – Gert Nov 18 '15 at 3:38
• I think you should consider changing the title of your question. Its not so much about having/not having gravity. The parameter, $d$, is the impact parameter, which defines the initial angular momentum to be $L=mv_0d$. You can also consider your system's total energy at this point to be $E_{tot} = 1/2mv_0^2$. – tmwilson26 Nov 18 '15 at 4:30
• Hi and welcome to the Physics SE! Please note that this is not a homework help site. Please see this Meta post on asking homework questions and this Meta post for "check my work" problems. – John Rennie Nov 18 '15 at 7:24
• – BowlOfRed Nov 18 '15 at 7:51


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# reStructuredText guidelines¶

These guidelines regulate the format of our reST (reStructuredText) documentation and provide tips for writing them:

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Paragraphs
^^^^^^^^^^

• The main thing to keep in mind as you write and edit docs is that the more semantic markup you can add the better. So:

To remove README.md execute rm README.md...

To remove :file:README.md execute :command:rm README.md...

This is because Sphinx will generate proper links for the latter, which greatly helps readers. There’s basically no limit to the amount of useful markup you can add.

• The file ending of the reST files should be *.txt instead of *.rst this will force github to display them as raw text files. The downside is that you have to tell your editor that you are editing a reST file – :setf rst in vim.

A figure should always start with a label that can be later used to reference to the figure. This label should always start with fig-. After that the figure with the actual image file is added. The image should be provided as a png-file and stored in a img/ folder:

.. _fig-super-result:

.. figure:: img/super-results.png
:align: center

Figure caption explaining the results.

Text in which the :numref:fig-super-result is referenced in.

Like a figure a table should always start with a label for referencing and gets a caption before the actual table:

.. _tab-super-result:

.. table:: Caption of the super result table.

=============== =============== ===============
=============== =============== ===============
1               2               3
1               2               3
1               2               3
=============== =============== ===============

Text in which the Table :ref:tab-super-result is referenced in.

There is another way to define the same table:

.. _tab-super-result:

.. table:: Caption of the super result table.

+-----------------+-----------------+-----------------+
+=================+=================+=================+
| 1               | 2               | 3               |
+-----------------+-----------------+-----------------+
| 1               | 2               | 3               |
+-----------------+-----------------+-----------------+
| 1               | 2               | 3               |
+-----------------+-----------------+-----------------+

This is especially interesting for more complex tables.

Line breaking is unfortunately not handled automatically in a table. Instead, the table will be stretched out to fir the content in one line per cell. You can add line breaks manually by inserting blank lines:

.. _tab-super-result:

.. table:: Caption of the super result table.

+-----------------+-----------------+--------------------+
+=================+=================+====================+
| 1               | 2               | text in line1      |
|                 |                 | more text in line1 |
|                 |                 |                    |
|                 |                 | text in line2      |
|                 |                 |                    |
|                 |                 | text in line 3     |
+-----------------+-----------------+--------------------+
| 1               | 2               | 3                  |
+-----------------+-----------------+--------------------+
| 1               | 2               | 3                  |
+-----------------+-----------------+--------------------+

This will result in

Table 50 Caption of the super result table.
1 2

text in line1 more text in line1

text in line2

text in line 3

1 2 3
1 2 3

## Dealing with referencing throughout the document¶

If you want to add a reference to another section, figure, or table you can do it the following way:

See results in :numref:fig-super-result in Section :ref:sec-results.

This requires of course that the labels for the figure and the section is set accordingly:

.. _sec-results

Results
-------

.. _fig-super-results

.. figure:: img/results.png

Super results from the experiment.

If you do the referencing as shown in the first code listing the :ref:label will be replaced by the title of the section or Fig. and the number of the figure.

## Using acronyms¶

If you have an acronym like HRTF, you should define it in the acronym.py file in the following way:

.. |HRTF| replace:: :abbr:HRTF (head-related transfer function)

In the text you then simply write |HRTF|. This will transform to acronym like HRTF in the HTML version and will be written out only the first time in the PDF version of the documentation. If you need the plural form you have to write |HRTF|\ s which will translate into HRTFs.


text:
# Calculating finite sum using only pen and paper

I know this is something that will pop up for my math exam tomorrow. When searching for this the only thing I could find was this: Calculate $\sum\limits_{n=1}^\infty (n-1)/10^n$ using pen and paper, but this is for infinite sums. So I'd like to find something about how to calculate a finite sum, typically from 1 to 100. This is something a computer could easily calculate in Maple or MATLAB etc. but we are not allowed to using math programs, not even a calculator, pen and paper only!

Examples from old exams are: Calculate the sum: $$\sum_{i=1}^{100}(2i+1)$$

Another example: Calculate the sum: $$\sum_{i=1}^{99}(6i^2+2i)$$

The only way to do it, that I can think of would be like this:

$$\sum_{i=1}^{100}(2i+1)=(2\cdot1+1)+(2\cdot2+1)+(2\cdot3+1)+(2\cdot4+1)+\dots+(2\cdot100+1)=10200$$

However, this would take way too long. I am sure there are better ways to solve these kinds of problem, but how?

• take a look here – Masacroso May 28 '17 at 11:05
• Most people seem to prefer stackexchange to pen and paper. – Professor Vector May 28 '17 at 11:12

For the first use the triangular number formula: $$\sum_{i=1}^{100}(2i+1) = 2\sum_{i=1}^{100}i+\sum_{i=1}^{100}1$$$$= 2\frac{100(100+1)}{2}+100 = 10100 + 100$$$$= 10200$$

For the second use the formula for square pyramidal numbers as well:

$$\sum_{i=1}^{99}(6i^2+2i) = 6\sum_{i=1}^{99}i^2+2\sum_{i=1}^{99}i$$ $$= 6\frac 1 6 99(99+1)(198+1) + 2\frac{99(99+1)}{2}$$ $$= (198+1+1)99(99+1) = 200\cdot100\cdot 99$$ $$= 1980000$$

hint

For $|x|<1$,

$$\sum_0^\infty x^n=1/(1-x)$$

by differentiation $$\sum_1^\infty nx^{n-1}=1/(1-x)^2$$

multilying by $x$

$$\sum_1^\infty nx^n=x/(1-x)^2$$

replace $x=1/10$.

The first one is just an Arithmetic Progression!

Use $\frac n2 (a+\ell)$.

$$\sum_{i=1}^{100}(2i+1)=\frac {100}2 \big(3+201\big)=\color{red}{10200}$$

For the second one, the coefficients give a useful clue to a shortcut.

Note that $\displaystyle\sum_{i=1}^n i^2=\frac {n(n+1)(2n+1)}6$ and $\displaystyle\sum_{i=1}^n i=\frac {n(n+1)}2$.

Hence

$$\sum_{i=1}^{n} 6i^2+2i=n(n+1)(2n+1)+n(n+1)=n(n+1)(2n+2)=2n(n+1)^2$$

Putting $n=99$ gives $$\sum_{i=1}^{99}=2\cdot 99\cdot 100^2=\color{red}{1980000}$$


text:
# Equation Derivation: Help with intermediate steps

As per the attached image above, I don't know how the equation in the first line transformed to the form in second line.

I tried solving it by multiplication but the result was different.

\begin{align}e^{-n/D}\frac{1-e^{(n+1)/D}}{1-e^{1/D}}&=\frac{e^{-n/D}-e^{1/D}}{1-e^{1/D}}\\&=\frac{e^{1/D}}{e^{1/D}}\cdot\frac{e^{-(n+1)/D}-1}{e^{-1/D}-1}\\&=\frac{1-e^{-(n+1)/D}}{1-e^{-1/D}}\end{align}


text:
# What is a silicate tetrahedron?

The silicate ion, ${\text{SiO}}_{4}^{2 -}$, has a silicon atom surrounded by four oxygen atoms at the corners.


text:
# Elitmus previous questions set - 5

1. Two identical cubes if one of them is painted pink on its 4 sides and blue on the remaining two side then how many faces painted pink to other cube so that probability of getting the same color is 1/3 when we roll both the cubes.
Explanation:
First cube has got 4 pink sides and 2 black sides.
Let the other cube got x sides pink and (6 - x) sides black.
Now when we roll both the dice, we can either pink on both cubes or black on both cubes.
Probability = $\dfrac{4}{6} \times \dfrac{x}{6} + \dfrac{2}{6} \times \dfrac{{6 - x}}{6} = \dfrac{1}{3}$
$= \dfrac{{4x + 12 - 2x}}{36} = \dfrac{1}{3}$
${ \Rightarrow x = 0}$
So second cube should not have any pink faces at all.

2. In a right angled triangle, two sides are consecutive whole number in which one side is hypotenuse. what could be the possible length of third side?
1. 360
2. 361
3. 362
4. none of these
Explanation:
Pythagorean triplets are generated with each "odd number" greater than 1 by using a formula.
If n is an odd number, then Pythagorean triplet = $n,\dfrac{{{n^2} - 1}}{2},\dfrac{{{n^2} + 1}}{2}$.
Here 361 is an odd number.  So the triplet is 361, 65160, 65161.

3. Heinz produces tomato puree by boiling tomato juice. The tomato puree has only 20% water while the tomato juice has 90% water. How many liters of tomato puree will be obtained from 20 litres of tomato juice?
a. 2 liters
b. 2.4 liters
c. 2.5 liters
d. 6 liters
Explanation:
In each of the solutions, there is a pure tomato component and some water.  So while boiling, water evaporates but tomato not. So we equate tomato part in the both equations.
$\Rightarrow$ 10%(20) = 80%(x)
$\Rightarrow$ x = 2.5 liters.

4. x,y are odd and z is even then ((x^2+y^2)z^2)/8 is
a. even
b. odd
c. either even or odd
d. fraction
Explanation: c
As x, y are odd ${x^2} + {y^2}$ is always even. Now if z is a multiple of 4, then ${z^2}$ is divisible by 8, then the equation is even.  if z is a not a multiple of 4, but only a multiple of 2, then ${z^2}$ is not completely divisible as it contains only two 2's and other two is cancelled in ${x^2} + {y^2}$ which results in an odd number.
$\dfrac{{\left( {{3^2} + {5^2}} \right){4^2}}}{8} = \dfrac{{34 \times 16}}{8} = 34 \times 2$
$\dfrac{{\left( {{3^2} + {5^2}} \right){6^2}}}{8}$ = $\dfrac{{34 \times 36}}{8} = 17 \times 9$

5. In the formula of converting temperature from Celsius to Fahrenheit F=9/5C+32, How many integer values(not fractional) of F will be there that lies between 100 to 200 for integer values of C.
Explanation:
$F = \dfrac{9}{5}C + 32$
As F needs to be integer, then C should be a multiple of 5. First integer value of F for C = 5 is 41, next value for C =10 is 50 and so on.
The values of F are in A.P with common difference of 9. They are in the format of 41 + 9n.
The first value of F which is greater than 100 is for n = 7 which is 104.
The last value of F which is less than 200 is for n = 17, which is 194.
Total values are $\dfrac{{194 - 104}}{9} + 1$ = 11

6. The product of digit is a Factor of a two digit number.  Number of such digit are:
a. 3
b. 5
c. 9
d. 27
Explanation:
Let the number be xy. So $\dfrac{{10x + y}}{{xy}} = k$.  Here k is some interzer.
$\Rightarrow$ 10x + y = kxy
$\Rightarrow$ x(10 - ky) = - y
$\Rightarrow$ x(ky - 10) = y
So x is a factor of y. The possibilities are,
11, 1×1=1; 12 , 1×2=2; 15 ,1×5=5; 24 ,2×4=8; 36 , 3×6=18

7. Data sufficiency question.
is x>y ?
1. 5x+15y=40
2.7x+21y=56
Explanation:
Statement 1 has three solutions,(8, 0), (5, 1), (2, 2) but we cannot say precisely about the relationship
Statement 2 has three solutions, (8, 0), (5, 1), (2, 2) but we cannot say about the relationship.
So data insufficient.


text:
Some relations between zeros and universality of the Riemann zeta-function
 作者： R. Macaitienė 刊名： St. Petersburg Mathematical Journal, 2019, Vol.31 (1), pp.53-58 来源数据库： American Mathematical Society Journal DOI： 10.1090/spmj/1583 关键词： Riemann zeta-function;  Discrete universality;  Nontrivial zeros; 原始语种摘要： In the paper, some remarks are given on the density of shifts of the Riemann zeta-function $\zeta (s+i\gamma _k)$, which approximate a wide class of analytic functions. Here the $\gamma _k$ denote the imaginary parts of nontrivial zeros of $\zeta (s)$.

• universality　通用性
• function　函数
• approximate　近似的
• relations　性行为
• between　在中间
• gamma　伽马
• class
• analytic　解析的
• imaginary　假想的
• denote　指示


text:
Question

# If L, M, N are physical quantities with different dimensions, then which of the following combinations can never be a meaningful quantity?\begin{align} & \text{A}\text{. }\dfrac{LM}{N} \\ & \text{B}\text{. LN-M} \\ & \text{C}\text{. }\dfrac{LN-{{M}^{2}}}{N} \\ & \text{D}\text{. }\dfrac{L-M}{N} \\ \end{align}

Hint: We know that pure quantity doesn’t have units and therefore doesn’t have a dimension that’s why we can add or subtract it to any type of unit less quantity. In addition to this concept, we can only add or subtract only those quantities which have the same unit and therefore dimension. Find out the quantity in the given options which are violating the principle of homogeneity of dimensions.

Complete step-by-step solution:
We know that an equation contains several terms which can be separated from each other by using the symbols of equality like plus or minus and also the dimensions of all the terms in an equation must be identical to do mathematical operation. This is another way of saying that one can add or subtract similar physical quantities. Thus, addition of velocity to a force or subtraction of electric current from the thermodynamic temperature. This is nothing but the principle of homogeneity of dimensions in an equation and is an extremely useful method to check whether an equation may be correct or not. The equation must be wrong if the dimensions of all the terms are not the same. Let us check the equation $~x=ut+\dfrac{1}{2}a{{t}^{2}}$, for the dimensional homogeneity. Here $x$ is the distance travelled by a particle in time $t$ which starts at a speed $u$ and has an acceleration $a$ along the direction of motion.
\begin{align} & \left[ x \right]=L \\ & \left[ ut \right]=\text{velocity}\times \text{time=}\dfrac{\text{length}}{\text{time}}\text{ }\!\!\times\!\!\text{ time}=L \\ & \left[ \dfrac{1}{2}a{{t}^{2}} \right]=\left[ a{{t}^{2}} \right]=\text{acceleration}\times {{\left( \text{time} \right)}^{2}} \\ & \left[ \dfrac{1}{2}a{{t}^{2}} \right]=\dfrac{\text{velocity}}{\text{time}}\text{ }\!\!\times\!\!\text{ }{{\left( \text{time} \right)}^{2}}=\dfrac{\text{length/time}}{\text{time}}\times {{\left( \text{time} \right)}^{2}}=L \\ \end{align}
Thus the equation is correct as far as the dimensions are concerned.
We have already stated in the question that L, M, and N are different physical Quantities and they have different dimensions, and also by the principle of homogeneity, we know that only like quantities can be added or subtracted i.e. can do mathematical operatic like plus and minus. So, $\dfrac{L-M}{N}$ this combination is not meaningful because two different physical quantities with different dimensions are subtracted (or added) which is not possible.
Therefore option (D) is the correct option.

Note: The dimension of $\dfrac{1}{2}a{{t}^{2}}$ is same as that of $a{{t}^{2}}$. Pure numbers are dimensionless. Dimension does not depend on the magnitude. Due to this reason the equation $x=ut+\dfrac{1}{2}a{{t}^{2}}$ is also dimensionally correct. Thus, a dimensionally correct equation need not be actually correct but a dimensionally wrong equation must be wrong.


text:
Chapter 17, Problem 1PA

### Principles of Microeconomics

7th Edition
N. Gregory Mankiw
ISBN: 9781305156050

Chapter
Section

### Principles of Microeconomics

7th Edition
N. Gregory Mankiw
ISBN: 9781305156050
Textbook Problem

# A large share of the world supply of diamonds comes from Russia and South Africa. Suppose that the marginal cost of mining diamonds is constant at $1,000 per diamond and the demand for diamonds is described by the following schedule: Price Quantity$8,000 5,000 diamonds 7,000 6,000 6,000 7,000 5,000 8,000 4,000 9,000 3,000 10,000 2,000 11,000 1,000 12,000 a. If there were many suppliers of diamonds, what would be the price and quantity?b. If there were only one supplier of diamonds, what would be the price and quantity?c. If Russia and South Africa formed a cartel, what would be the price and quantity? If the countries split the market evenly, what would be South Africa’s production and profit? What would happen to South Africa’s profit if it increased its production by 1,000 while Russia stuck to the cartel agreement?d. Use your answers to part (c) to explain why cartel agreements are often not successful.

Subpart (a):

To determine
Equilibrium price.

Explanation

The profit maximizing firm would produce output at the point where the marginal revenue is equal to or just exceeds marginal cost. Marginal revenue is equal to price...

Subpart (b):

To determine
Calculation of marginal revenue.

Sub part (c):

To determine
Calculation of profit.

Subpart (d):

To determine
What would be the price and quantity of diamond in Russia and South Africa after formation of cartel.

### Still sussing out bartleby?

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text:
# Calcuating Open-Circuit emf for Alternators

1. Jan 9, 2010

### patjk

My question is not for a specific problem, but in general. I don't quite understand how to calcuate emf for alternators. I will give an example or two to give you an idea.

1. The problem statement, all variables and given/known data
Example 1) A star-connected alternator on infinite bus vars supplies 500MW at 23kV at a power factor of .95 lagging. The synchronous reactance is 2.8 ohms. Calculate a)line current, b) reactive VA, c) the corresponding open-circuit emf, d) the load angle

Example 2) The alternator described in 1) is operated at the same current a a power factor of .95 leading. Calculate a) the power output, b) the reactive VA, c) the open-circuit phase emf, d) the load angle

Please refer only to the bolded portions of those 2 examples.

3. The attempt at a solution
For 1), I don't quite understand how to get the open-circuit emf. I have the solution to this, and for 1c., it has:
Eph^2 = (Vph + Xs*Iph*sin(angle))^2+(Xs*Iph*cos(angle))^2
where Eph is emf, Vph is phase voltage, Iph is phase current, Xs is reactance, and angle is cos^-1(power factor).

I am not sure how to determine this with the given info in the problem. If a phasor diagram was given, I'd be able to know this, but without the diagram, how can determine this?

For 2c., the solution is:
Eph^2 = (Xs*Iph*cos(angle))^2+(Vph-Xs*Iph*sin(angle))^2

And I don't see how that came about.
-----------
In general (for a leading power factor), I think Eph^2 = (Iph*R)^2 + (Iph*X)^2, hence why the above is confusing me.

Load angle is just the angle between Eph and Vph (Iph*R), so if I can figure out Eph, that won't be a problem.

Any ideas/tips/suggestions on this matter is greatly appreciated. Thanks

Last edited: Jan 9, 2010


text:
## WeBWorK Main Forum

### unfactored low degree polynomial can slip through PolynomialFactors-Strict context undetected

by Christian Seberino -
Number of replies: 8
Factoring for quadractics of 1 or 2 variables seems to be correctly enforced
by PolynomialFactors-Strict context.

However, polynomials with 4 variables seems to not have factoring enforced.
See below.

If you submit your answer as a*c + a*d + b*c + b*d
it is accepted.

DOCUMENT();
"PGstandard.pl",
"PGML.pl",
"MathObjects.pl",
"PGcourse.pl",
"parserNumberWithUnits.pl",
"contextArbitraryString.pl",
"contextPolynomialFactors.pl",
"contextLimitedPowers.pl",
"parserPopUp.pl",
);
TEXT(beginproblem());
$showPartialCorrectAnswers = 1; ###################################################################### Context("PolynomialFactors-Strict"); Context()->flags->set(singleFactors=>0); LimitedPowers::OnlyIntegers(minPower => 0, maxPower => 1); Context()->variables->add(a => "Real"); Context()->variables->add(b => "Real"); Context()->variables->add(c => "Real"); Context()->variables->add(d => "Real"); BEGIN_PGML Write [ac + ad + bc + bd] in completely factored form. [________________________]{"(a + b)*(c + d)"} END_PGML ###################################################################### ENDDOCUMENT(); In reply to Christian Seberino ### Re: unfactored low degree polynomial can slip through PolynomialFactors-Strict context undetected by Danny Glin - The reason for this is that it is only checking that the power of each variable is at most 1, which is also true of the original expression. You would run into the same problem in the two-variable case with something like xy+2x+3y+6. This type of answer checker is only effective if the solution is linear in each variable, and the original expression is not, so there are also many other cases which it won't handle. Unfortunately, I don't know a good way of checking such factoring in general. I still use questions with two answer blanks in brackets, and then use a multi-answer checker so students can enter the factors in either order. Danny In reply to Danny Glin ### Re: unfactored low degree polynomial can slip through PolynomialFactors-Strict context undetected by Christian Seberino - > I still use questions with two answer blanks in brackets, and then use a multi-> answer checker so students can enter the factors in either order. That's a *great* idea! cs In reply to Christian Seberino ### Re: unfactored low degree polynomial can slip through PolynomialFactors-Strict context undetected by Paul Pearson - Hi, Even easier: use unorderedAnswer.pl instead of parserMultiAnswer.pl. For example, see http://webwork.maa.org/wiki/UnorderedAnswers1 Have a good weekend! Paul Pearson In reply to Paul Pearson ### Re: unfactored low degree polynomial can slip through PolynomialFactors-Strict context undetected by Alex Jordan - The problem I'd have with unorderedAnswers arises with say, factoring 2x^2-3x+1. Probably the expected factors would be (2x-1) and (x-1). But should a student's answer be incorrect if they give (1-2x) and (1-x)? What about (x-1/2) and (2x-2)? multiAnswer could at lease deal with these by multiplying student factors and comparing to the known correct answer. But these approaches are unsatisfying. First, they give away the number of factors to the student. Worse, they just don't feel true to what the pen-and-paper experience would be. Even the method using PolynomialFactors that limits exponents feels artificial, because why should a student a priori know that the given polynomial factors at all? Or doesn't have quadratic factors? How does one ask a student to factor x^4-1? We can't limit the degrees to 2 and below, or (x^2-1)(x^2+1) is legal. But more importantly, why should a student a be told a priori that cubic factors are not allowed in such a problem? It's a tough nut to crack! At least with my degree of pickiness :) In reply to Danny Glin ### Re: unfactored low degree polynomial can slip through PolynomialFactors-Strict context undetected by Christian Seberino - Even with a ONE variable case the problem persists.... e.g. 2y + 6 :) In reply to Danny Glin ### Re: unfactored low degree polynomial can slip through PolynomialFactors-Strict context undetected by Alex Jordan - We're working on something that may be relevant. There are a lot of WeBWorK projects happening here and when we have the time, the plan is: • Use PolynomialFactors for pre-filter checking of the student answer. • Have the checker compare the correct answer and student answer as functions, and of course if they are not equal, the answer is incorrect. • Process the student answer as a Math Object Formula, and break down its parse tree by its top-level operators. The idea is to create an array of the student's primitive factors, so say 3(x+1)(x+2)^2 gives (3,x+1,x+2). • Because we may want factoring over Z, checking the gcd of coefficients within each factor. • Pass each of these things to SAGE and ask if the nonconstant factors are reducible over Z or Q. Also ask if they are monic. These things at least we learned how to do at the Vancouver code camp. The end goal is to count the following forms as correct, possibly controlled by flags: n \prod (factor)^power, where each factor is irreducible in Z[X], n in Z r \prod (factor)^power, where each factor is irreducible and monic in Q[X], r in Q I suppose on the last one the monic requirement could be dropped with a flag. I have no plans to check that the form is fully condensed, e.g. forcing (x+1)^2 and rejecting (x+1)(1+x), but that could be done in SAGE too somehow looking at the gcd of the student's factors. In reply to Alex Jordan ### Re: unfactored low degree polynomial can slip through PolynomialFactors-Strict context undetected by Hedley Pinsent - Seems like we have too much "division of labour" in the system. The Problem (Author) is best qualified to determine the characteristics of an answer. However it seems we are expecting the "Answer Checkers" to go away and, in isolation, sort it all out. Maybe these "answer checkers" (very clever algorithms) should simply make recommendations back to the Problem, the Problem would then make the last call. ---------------------------------------------------------- Factor: a*c + a*d + b*c + b*d$Recommendation1 = Is_NumericallyEqualTo_($StudentAnswer, a*c + a*d + b*c + b*d) //check points$Recommendation2 = AreThere_OccurancesOf_In_(2,"+",$StudentAnswer) // check for 2 "+"s$FinalRecommendation = $Recommendation1 AND$Recommendation2
Submit($FinalRecommendation,$StudentResponse,$CorrectResponse)//only$FinalRecommendation counts

------------------------------------------------------------

Factor: 2x^2 + 10x +12 ................. 2(x+2)(x+3)

$Recommendation1 = Is_NumericallyEqualTo_($StudentAnswer, 2x^2 + 10x +12) //check points
@List = (2,2,3)
$Recommendation2 = Are_In_(@List,$StudentAnswer)
$FinalRecommendation =$Recommendation1 AND $Recommendation2 Submit($FinalRecommendation,$StudentResponse,$CorrectResponse)//only $FinalRecommendation counts ------------------------------------------------------------- Add 12.3 +45.7$Recommendation1 = Is_Equal_($StudentAnswer,58) //check numerical equality$Recommendation2 = AreThere_DigitsIn_(3,$StudentAnswer) //checking for trailing zero$FinalRecommendation = $Recommendation1 AND$Recommendation2

Submit($FinalRecommendation,$StudentResponse,$CorrectResponse)//only$FinalRecommendation counts
In reply to Hedley Pinsent

### Re: unfactored low degree polynomial can slip through PolynomialFactors-Strict context undetected

by Danny Glin -
Alex beat me to most of my comments on "what is considered factored".

In our department alone, there is disagreement on what the factored form of something like 2x^2+3x+1 should be.  One camp expects (2x+1)(x+1), while another expects 2(x+1/2)(x+1).

Regarding not giving away the number of factors in the correct answer, the solution I had in mind for that is something I'd like to see more generally:

For list-type answers, start the student with a single answer blank, but beside it have a (probably javascript) button that adds another blank.  Behind the scenes, compile these blanks into a list object, and pass it to the answer checker.

I have several uses for this:
1. I think it would be a nicer interface for any question where the answer is an unordered list of objects.
2. It could be used for factoring questions (where the button reads "add another factor").
3. The more advanced use I'd have for this is to handle unordered, paired answers.  For example, give the students a matrix and ask for the eigenvalues and corresponding eigenvectors.  Start the student with one pair of blanks: one for an eigenvalue and the next for the corresponding eigenvector(s).  The students can then add another set of eigenvector/eigenvalue blanks if there are additional eigenvectors.
Of course this is a solution to the student input challenges, which doesn't address the (much harder) problem of what to accept as an answer in our factoring discussion.


text:
# Effect of sample size in t test? [duplicate]

Possible Duplicate:
Why does t statistic increase with the sample size?

In t test,the test statistic is $$t = \frac{\overline{x} - \mu_0}{s/\sqrt{n}}$$

I wonder what effect of the sample size $n$ has on the t test?

For example, as $n$ increases, I thought $t$ will increases at first, but later I realized $s$ is at the scale of $1/\sqrt{n-1}$. So $s/\sqrt{n}$ seems not to decrease as $n$ increase?

What other effect does $n$ have?

Thanks!

-
Possibly related: Why does t statistic increase with the sample size?. –  caracal Oct 11 '12 at 21:20

## marked as duplicate by whuber♦Oct 11 '12 at 21:24

If data is Normal($\mu$,$\sigma^2$), the distribution of the the $t$ statistic is the same for every $n$ (namely, it follows a $t$ distribution). What changes is the number of degrees of freedom. See this plot to see how it changes. If the original data is not $iid$ normal, other assumptions are needed, and the results are asymptotic.
On the contrary, the distribution of $t$ is different for every $n$ because the degrees of freedom matter! –  whuber Oct 11 '12 at 21:25


text:
# Probability Density Function (Integration)

A probability density function is given by

$$f(x)=\left\{\begin{matrix} ke^{-2x} &x\geq0 \\ 0&otherwise \end{matrix}\right.$$

Find $k$

My attempt,

$k\int_{0}^{\infty }e^{-2x}dx=1$

But I don't know how to solve this integral because of the infinity.

$$k \int_{0}^{\infty}e^{-2x}dx = k \left[\frac{e^{-2x}}{-2} \right]_{0}^{\infty} = k\left[-\frac{1}{e^{\infty}} + \frac{e^{0}}{2} \right] = \frac{k}{2}$$
• How $-\frac{1}{2e^{2x}}$ becomes $\frac{1}{e^{\infty}}$ ? – Mathxx Nov 4 '15 at 17:33
• So, we are here: $$\left[ -\frac{1}{2e^{2x}} \right]_{0}^{\infty}$$ We simply plug in the top ($\infty$) and the bottom ($0$). If we plug in infinity, we get: $-\frac{1}{2e^{2\infty}}$ However, $2\infty = \infty$, $e^{\infty} = \infty$, $-\frac{1}{2\infty}=-\frac{1}{\infty}=0.$ To be a bit more precise, I should have more limits here, but this is a good way to show what's going on in this problem. – user43395 Nov 4 '15 at 17:34


text:
# How to place several images on top of each other and center them on the page?

I can't figure out how to center the images. The aim is to center the images and make them large enough to stretch over the cropping line.

Optionally: Is it possible to control how close the images are (in other words how big is the gap between the bottom border of one image and the top border of another image).

\documentclass[12pt]{extbook}
% ############################## geometry
\usepackage%[showframe]
{geometry}
\geometry
{ hmarginratio =  1:1
, vmarginratio = 1:1
, bindingoffset = 0cm
, onecolumn
, a4paper
, layoutwidth = 50 mm
, layoutheight = 180 mm
, layouthoffset=\dimexpr(\paperwidth-\csname Gm@layoutwidth\endcsname)/2\relax
, layoutvoffset=\dimexpr(\paperheight-\csname Gm@layoutheight\endcsname)/2\relax
, showcrop
}
\usepackage{fancyhdr}
\usepackage{mwe}
\usepackage{graphicx}
% ############################### Document
\begin{document}
\pagenumbering{gobble}
\begin{figure}
\centering
\begin{minipage}[c]{\textwidth}
\centering
\noindent\includegraphics[width=5cm]{example-image}
\end{minipage}
\begin{minipage}[c]{\textwidth}
\centering
\noindent\includegraphics[width=5cm]{example-image}
\end{minipage}
\begin{minipage}[c]{\textwidth}
\centering
\noindent\includegraphics[width=5cm]{example-image}
\end{minipage}
\begin{minipage}[c]{\textwidth}
\centering
\noindent\includegraphics[width=5cm]{example-image}
\end{minipage}
\begin{minipage}[c]{\textwidth}
\centering
\noindent\includegraphics[width=5cm]{example-image}
\end{minipage}
\end{figure}
\end{document}

• Why all those minipages? I can't see the use. Try putting \vfill between the images. – Johannes_B Feb 17 '16 at 10:48
• Add package (not option) showframe additionally to geometry. – Johannes_B Feb 17 '16 at 10:52
• you do not want minipage or figure here, just \includegraphics and possibly some spacing commands. – David Carlisle Feb 17 '16 at 10:57
• see tex.stackexchange.com/questions/213286/christmas-list-catalog/… for explicit control over position of images and text – David Carlisle Feb 17 '16 at 11:01

The statement of the problem is not very clear, but perhaps this is what you want.

\documentclass[12pt]{extbook}

% ############################## geometry
\usepackage{geometry}
\geometry{
hmarginratio =  1:1,
vmarginratio = 1:1,
bindingoffset = 0cm,
onecolumn,
a4paper,
layoutwidth = 50 mm,
layoutheight = 180 mm,
layouthoffset=\dimexpr(\paperwidth-\csname Gm@layoutwidth\endcsname)/2\relax,
layoutvoffset=\dimexpr(\paperheight-\csname Gm@layoutheight\endcsname)/2\relax,
showcrop
}

\usepackage{graphicx}

% ############################### Document

\begin{document}
\thispagestyle{empty}

{\parindent=0pt
\newcommand{\shiftleft}{\hspace*{-0.5\dimexpr\csname Gm@layoutwidth\endcsname-\textwidth\relax}}

\shiftleft\includegraphics[width=5cm]{example-image}\shiftleft\vfill

\shiftleft\includegraphics[width=5cm]{example-image}\shiftleft\vfill

\shiftleft\includegraphics[width=5cm]{example-image}\shiftleft

}
\end{document}

• Excellent! How can I control the vertical position? (the gap between them). I want to place all 5 images close to each other. This is meant to be bookmark :) – chejnik Feb 17 '16 at 12:20
• @chejnik Put \vspace*{\fill} at the top and at the bottom, and \vspace{<whatever} between the images. – egreg Feb 17 '16 at 13:50
• Thank you, I am using \vspace and I have added also command to shift it up \newcommand{\shiftup}{\vspace*{-0.15\dimexpr\csname Gm@layoutheight\endcsname-\textwidth\relax}} (regards to your code). – chejnik Feb 17 '16 at 14:40


text:
Nigerian Scholars » » The energy, E stored in an inductor of inductance L when current I passes throug...

The energy, E stored in an inductor of inductance L when current I passes throug...

Question

The energy, E stored in an inductor of inductance L when current I passes through it is given by the equation

Options

A) E = $$\frac{1}{2} LI^2$$

B) E = LI2

C) E $$\frac{1}{2}LI$$

D) E = $$\frac{1}{2} L^2I$$


text:
HAL will be down for maintenance from Friday, June 10 at 4pm through Monday, June 13 at 9am. More information

# A bijection proving the Aztec diamond theorem by combing lattice paths

* Corresponding author
Abstract : We give a bijective proof of the Aztec diamond theorem, stating that there are $2^{n(n+1)/2}$ domino tilings of the Aztec diamond of order~$n$. The proof in fact establishes a similar result for non-intersecting families of $n+1$ Schröder paths, with horizontal, diagonal or vertical steps, linking the grid points of two adjacent sides of an $n\times n$ square grid; these families are well known to be in bijection with tilings of the Aztec diamond. Our bijection is produced by an invertible ''combing'' algorithm, operating on families of paths without non-intersection condition, but instead with the requirement that any vertical steps come at the end of a path, and which are clearly $2^{n(n+1)/2}$ in number; it transforms them into non-intersecting families.
Keywords :
Document type :
Reports
Domain :

Cited literature [5 references]

https://hal.archives-ouvertes.fr/hal-00734890
Contributor : Marc van Leeuwen Connect in order to contact the contributor
Submitted on : Monday, September 24, 2012 - 5:26:49 PM
Last modification on : Wednesday, October 20, 2021 - 3:22:17 AM
Long-term archiving on: : Tuesday, December 25, 2012 - 6:35:09 AM

### Files

Aztec.pdf
Files produced by the author(s)

### Identifiers

• HAL Id : hal-00734890, version 1
• ARXIV : 1209.5373

### Citation

Frédéric Bosio, Marc van Leeuwen. A bijection proving the Aztec diamond theorem by combing lattice paths. 2012. ⟨hal-00734890⟩

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text:
Simplicial sets of categories as models for $(\infty,1)$-categories [duplicate]

Disclaimer: I am not knowledgeable at all about the subject of this question, so my apologies if I say something incorrect or too imprecise.

In my understanding, there are several models for $$(\infty,1)$$-categories: Joyal-Lurie's quasi-categories, simplicially enriched categories, Segal catgories, etc. . They are all, for the most part, models for the same mathematical objects and can be used interchangeably to develop higher category theory.

Here is something that sounds like it might be a model. Assume all categories below are as small as necessary.

Definition 1: A (special) simplicial set of categories $$C_\bullet$$ is a functor $$C:\Delta^{\text{op}} \to \bf{Cat}$$, denoted by $$[k] \mapsto C_{[k]}$$, satisfying the following pair of properties.

• The map $$\text{Mor}(C_\bullet) \to \text{Obj}(C_\bullet) \times \text{Obj}(C_\bullet)$$ given by sending a morphism $$f:x \to y$$ to the pair $$(x,y)$$ is a map of simplicial sets. (Not sure if this is automatic)
• The simplicial set $$\text{Map}(x,y)$$, defined as the union over all $$k$$ of the morphisms $$f_{[k]}$$ whose image under any degeneration map $$C_{[k]} \to C_{[0]}$$ is a morphism $$x \to y$$, is a Kan complex.

My question is just the following one.

Question: Does Definition 1 (or a corrected/similar definition) provide a known model for $$(\infty,1)$$-categories? If so, where can I find a reference?

It would be great if this reference related this model to one of the more popular ones, such as quasi-categories.

• Your 'simplicial set of categories' is better known as 'a simplicial object in $\mathbf{Cat}$'. The first part of the definition is automatic, since $\mathrm{Mor},\mathrm{Obj}\colon \mathbf{Cat} \to \mathbf{Set}$ are functors and $(s,t)\colon \mathrm{Mor} \Rightarrow \mathrm{Obj}\times \mathrm{Obj}$ is a natural transformation. – David Roberts Jul 24 '19 at 0:45
• Added [ct.category-theory] tag (and removed the AMScd command, not sure why that was there) – David Roberts Jul 24 '19 at 0:51
• ...and a simplicial object in Cat is also equivalently an internal category in simplicial sets. That version of the question is what was answered by the one I linked in the duplicate mark. – Mike Shulman Jul 24 '19 at 0:56
• Thanks, Mike. I was sure this was known. – David Roberts Jul 24 '19 at 3:21
• Great, this is very helpful. – Julian Chaidez Jul 24 '19 at 7:42


text:
# How do you graph the inequality x<0?

Jul 3, 2016

The region to the left of 0.

#### Explanation:

The type of graph will depend on whether is it a simple number line graph, or on a set of x-y- axes.

On a number line graph, we would start with an open circle on 0, because 0 IS NOT included in the solution. Draw a line extending to the left, indicating that x can be any value to the left of 0.

On ax-y- grid, we would have a vertical line to represent x = 0, but as 0 IS NOT included, the line would be dotted. This would be the same line as the y-axis.

The required region is then anywhere to the left of they-axis, ie the second and third quadrants.


text:
# An injection $F$ from the set of real numbers to itself with $F(x)-F(y)\not\in\mathbb Q$ for $x\neq y.$

Please help me find an injection, $F$, from the set of real numbers into itself such that $F(x) - F(y)$ is an irrational number for any two distinct real numbers $x$ and $y$.

Thank you.

-
This is impossible: If $x = y$, you have $F(x) - F(y) = F(x) - F(x) = 0$, which is rational. Are there any additonal conditions? –  Johannes Kloos May 16 '12 at 15:07
Do you mean that $F(x)-F(y) \notin \mathbb{Q}$ for all distinct pairs of real numbers $x$ and $y$? –  mjqxxxx May 16 '12 at 15:12
Hint: use the axiom of choice to select one element from each set of the form $x+\mathbb Q$. You'll have to find support for the claim that this gives you a set of cardinality continuum. –  user31373 May 16 '12 at 15:18
If the range of $F$ is measurable with positive measure, then the set of differences contains an interval about zero, so contains some rationals. So for your example you need either something non-measurable (as Leonid did) or measure zero. For the latter, think of some things like Cantor sets. –  GEdgar May 16 '12 at 15:26
What do you mean by "find"? You can prove that such an $F$ must exist, in ZFC, but the word "find" might mean you want a definition. Given that it ultimately involves the axiom of choice, there is probably no "constructive" definition. –  Thomas Andrews May 16 '12 at 15:41

Consider $\mathbb R$ as vector space over $\mathbb Q$. Complete the set $\{1\}$ to a basis, call it $B$. There is a bijection $f \colon B \to B - \{1\}$ which extends to a linear transformation $F \colon \mathbb R \to \mathbb R$ that satisfies the requirement, because it is irrational everywhere except 0.

Lemma:

There is an absolutely convergent sequence of real numbers $x_i$ such that $\sum a_i x_i$ is irrational for every $a_i \in \{-1,0,1\}$ except all $a_i = 0$.

Proof assuming the lemma:

Since $\mathbb R$ and $\{0,1\}^{\mathbb N}$ have equal cardinality, it is enough to show $f \colon \{0,1\}^{\mathbb N} \to \mathbb R$.

Define $f(a_1 a_2 \dots) = \sum a_i x_i$. The difference $f(x)-f(y)$ is always irrational by the lemma.

Proof of the lemma:

Parition $\mathbb N$ into infinitely many infinite subsets $A_i$ and let $x_i = \sum_{k \in A_i} 10^{-k!}$. The series $\sum a_i x_i$ is, except when all $a_i = 0$, a Liouville number and therefore irrational.

(Alternatively, the lemma can be shown with precise but boring diagonalization.)

-
Very cool! I thought this result actually required choice, but I can't spot any hidden uses in your proof at least at first glance; I could easily be wrong... –  Steven Stadnicki Jun 23 '12 at 7:27

Let us pull this question off from unanwered questions' hell, following Leonid's idea. First, taking additive groups, we look at the quotient $\,\mathbb R/\mathbb Q\,$ . By Lagrange's theorem, $$|x+\mathbb Q|=|\mathbb Q|=\aleph_0\,\,,\,\,\forall x\in \mathbb R,$$and since the different cosets are in fact a partition of the set $\,\mathbb R\,$ , their (disjoint, of course) union is $\,\mathbb R\,$ . It follows that there are uncountable many different cosets as above (why?).

Now, we can make use of the Axiom of Choice in its form called Zermelo's Well Ordering Theorem, so we can introduce a well ordering in $\,\mathbb R\,$ which in turn induces a well ordering in $\,\mathbb R/\mathbb Q\,$, and then form any index (out of an uncoutable set) $\,\alpha\,$ , we cand define $$f:\mathbb R\longrightarrow \mathbb R\,\,,\,\,f(x_\alpha):=y_\alpha$$where $\,y_\alpha\,$ belongs to the $\,\alpha$-th coset in $\,\mathbb R/\mathbb Q\,$ (also here we use AC...)

-
I think it's not quite enough to show the number of cosets is uncountable; you either have to assume the Continuum Hypothesis, or you have to show that the cosets are in one-one correspondence with the reals. –  Gerry Myerson Jun 23 '12 at 0:28
Ok, fair enough...I think there exists somewhere a proof that $\,\mathbb R/\mathbb Q=2^{\aleph_0}\,$, but if nothing better can be gotten then I'd be willing to accept the C.H...:) –  DonAntonio Jun 23 '12 at 0:58
Certainly $|\mathbb{R}/\mathbb{Q}|=2^{\aleph_0}$ assuming AC, since $|\mathbb{R}/\mathbb{Q}|\aleph_0=2^{\aleph_0}$, and with AC, the product of two infinite cardinals is their maximum. –  Harry Altman Jun 23 '12 at 7:31

Under AC there is a well-order of $\mathbb R$ with each element having $\lt \mathfrak c$ predecessors. So just go along the well-order assigning an allowable value to each real. Each previous element excludes $(\lt \mathfrak c) \times \aleph_0$ values, so you will have at least one available.

-


text:
So in my last question, @joshphysics showed me how to prove $$K_\pm$$ were ladder operators. Now I need to show that there is a lowest state, i.e $$\langle m_0|K_+=K_-|m_0\rangle=0$$ I am not completely sure how I should approach this. I saw that $$J_3(K_-|m\rangle)=(m-1)(K_-|m\rangle$$ But I do not see how this shows that there has to be a minimum eigenvalue.

EDIT::: So I think I may have figured something out

$$\langle m_0|K_+K_-|m_0\rangle = (K_-|m_0\rangle)^\dagger K_-|m_0\rangle$$ This last quantity should be greater than 0, so we have a minimum state m_0 that we act on and get 0. Does this make sense?

• Essentially a duplicate of physics.stackexchange.com/q/23028/2451 , physics.stackexchange.com/q/54691/2451 and links therein. Feb 14, 2014 at 0:43
• I'm not sure that they duplicates... The links talk about non-degenerate vacua and non-integer quantum numbers. Feb 14, 2014 at 1:02
• Your edit is exactly the right idea. There is more argumentation necessary to get to the point where you can assert that there is a state annihilated by $K_-$, but that's the start. Feb 14, 2014 at 3:27
• What else is necessary to show that? Feb 14, 2014 at 3:34
• @yankeefan11 I don't want to spoil it for you; think about it some more! Also, I'd recommend reading a good treatment of the one-dimensional harmonic oscillator that uses ladder operators. Feb 14, 2014 at 3:36

1. The potential of the system is bounded from below. Thus there is no way to have a particle with energy below this point. This would require negative kinetic energy. The way to see that is $$E_{tot}= E_kin + V \ge V$$ Thus the total energy must always be greater than or equal to the potential energy.
2. To see the same effect using Quantum Mechanics you can solve the eigenvalue equation using brute force. You will find that the eigenstates are only normalizable for integer $n\ge 0$.
3. Using ladder operators one can look for the expectation value of the Hamiltonian, $\hbar \omega \left(K _+ K_-+ \frac{1 }{2} \right)$, and show that it's greater than zero.


text:
시간 제한메모리 제한제출정답맞힌 사람정답 비율
5 초 512 MB111100.000%

## 문제

Tree stands are elevated wooden platforms attached to trees. Typically, huntsmen use tree stands to watch or to shoot their prey.

In our county, huntsmen have built a remarkable system of tree stands. The tree stands are connected by narrow straight paths which form a kind of maze on the hunting grounds. The builders wanted to minimize the impact on the environment and so they built the minimum possible number of paths which ensure that there is a connection between any two tree stands. Also, a tree stand is visible from another stand if and only if the two are connected by a path.

A group of local huntsmen wants to find out which particular tree stands will serve the best their hunting interests. Each day they climb a different set of tree stands and watch the wildlife.

There are a few more important circumstances to consider:

• Security rules dictate that any occupied tree stand must be visible from at least one other occupied tree stand so that in case of an emergency the huntsman in the neighbour tree stand can come to help the colleague.
• A tree stand is always occupied by at most one huntsman.
• It does not matter which huntsman is in which tree stand. It only matters which tree stands are occupied and which are not.
• The size of the group does not change.

How many days will the group spend in the tree stands before they investigate all possible choices of tree stands available to them?

## 입력

There are more test cases. Each case starts with a line containing two integers N and K (2 ≤ K ≤ N ≤ 200) separated by space. N is the number of tree stands, K is the size of the group of huntsmen. The tree stands are labeled 1, 2, 3, . . . , N.

Next, there are N − 1 lines, each line specifies one path between two tree stands. The line contains the labels of the stands separated by a space. The order of the labels on a line and the order of the paths in the input is arbitrary.

## 출력

For each test case, print on a separate line the number of days which the group will spend in the tree stands. Express the result modulo 1 000 000 007.

## 예제 입력 1

4 3
1 2
1 4
1 3
5 4
1 2
2 3
3 4
4 5

## 예제 출력 1

3
3


text:
# American Institute of Mathematical Sciences

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Derivation of effective transmission conditions for domains separated by a membrane for different scaling of membrane diffusivity
August  2017, 10(4): 799-813. doi: 10.3934/dcdss.2017040

## On the geometry of the p-Laplacian operator

 1 Mathematisches Institut, Universität zu Köln, 50923 Köln, Germany 2 Fakultät Maschinenbau, TH Ingolstadt, Postfach 21 04 54,85019 Ingolstadt, Germany

* Corresponding author: Bernd Kawohl

Received  April 2016 Revised  August 2016 Published  April 2017

The
 $p$
-Laplacian operator
 $\Delta_pu={\rm div }\left(|\nabla u|^{p-2}\nabla u\right)$
is not uniformly elliptic for any
 $p\in(1,2)\cup(2,\infty)$
and degenerates even more when
 $p\to \infty$
or
 $p\to 1$
. In those two cases the Dirichlet and eigenvalue problems associated with the
 $p$
-Laplacian lead to intriguing geometric questions, because their limits for
 $p\to\infty$
or
 $p\to 1$
can be characterized by the geometry of
 $\Omega$
. In this little survey we recall some well-known results on eigenfunctions of the classical 2-Laplacian and elaborate on their extensions to general
 $p\in[1,\infty]$
. We report also on results concerning the normalized or game-theoretic
 $p$
-Laplacian
 $\Delta_p^Nu:=\tfrac{1}{p}|\nabla u|^{2-p}\Delta_pu=\tfrac{1}{p}\Delta_1^Nu+\tfrac{p-1}{p}\Delta_\infty^Nu$
and its parabolic counterpart
 $u_t-\Delta_p^N u=0$
. These equations are homogeneous of degree 1 and
 $\Delta_p^N$
is uniformly elliptic for any
 $p\in (1,\infty)$
. In this respect it is more benign than the
 $p$
-Laplacian, but it is not of divergence type.
Citation: Bernd Kawohl, Jiří Horák. On the geometry of the p-Laplacian operator. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 799-813. doi: 10.3934/dcdss.2017040
##### References:

show all references

##### References:
The positive viscosity solution of (4.4)
Conceivable nodal lines of the second eigenfunction for $p=\infty$ in the disc
Illustration of (5.4) and (5.5)
Numerical simulation of $u_{15}$ and side view in diagonal direction
Numerical simulation of $u_p$: normalized values along half of the diagonal for $p=2, 3, 4, 6, 8, 10, 15$ (left), and for $p=15$ compared to the line $y=x$ (right)
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2019 Impact Factor: 1.233

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text:
(redirected from InputMethod)

# InputMethods

Input Methods allow you to enter characters that are not supported by your keyboard.

• ‘C-x C-m C-\’ or ‘M-x set-input-method’ or switch to a new input method.
• ‘C-h C-\’ or ‘C-h I’ to describe the current input method.
• ‘C-\’ to toggle the input method on and off

Emacs should also work correctly with appropriate XIM input methods under X.

Possibly the most convenient casual method for inputting unicode characters is the TeXInputMethod, which permits you to enter Unicode characters by typing a \ prefixed sequence reminiscent of TeX. For example, the greek character λ may be entered by typing \lambda.

An obsolete way of entering accented characters is iso-accents-mode. These days, you should use the input-method ‘latin-1-prefix’ instead. Both will make accent characters combine with the following characters, so that you can type ” u’ to achieve ü’, for example.

An even stranger way of entering accented characters is the C-x 8 keymap. See WritingIsoLatinOne.

Another way is to use QuotedInsert with ‘C-q’.

To activate the input method for a specific mode put the following line in your config file:

   (add-hook 'text-mode-hook 'toggle-input-method)

The input method is then automaticaly activated for files which open in text mode.

To activate the input method on a per-buffer basis. First put the following lines in the .emacs:

  (set input-activate nil)
(lambda ()(if (eq input-activate t) (toggle-input-method))))

Then put the following line at the beginning of a file:

  -*- input-activate: t -*-

The input method will be activated for this file as soon as it is loaded.

These are some additional input methods that do not come with Emacs:

It’s convenient for people who speak in korean or japanese or etc… to use only the internal emacs input method (mule) without scim, nabi, Microsoft IME, etc…

For X Windows, add the following line to your ~/.Xresources file (if not exist, make it)

  Emacs.useXIM: off

For MS Windows, you may need to use DisableImeForEmacs.

Question: Is there a way to synchronize OS keyboard layout (ubuntu in my case) with Emacs input method?

It is not convenient to use different bindings for switching languages inside and outside of Emacs…

Question: How do I allow input methods such as SCIM to override Emacs keybindings in a desktop environment such as GNOME (instead of using the version of SCIM internal to Emacs)?

Set LC_CTYPE=ja_JP.UTF-8 before launching Emacs, and Emacs will recognize your SCIM keys, such as C-SPC to change input method.

See Also: ChangingCursorDynamically to automatically change the TextCursor color when you are using an input method. This makes it easy to tell if an input method is in use.

CategoryInternationalization


text:
# Circle A has a center at (5 ,1 ) and an area of 16 pi. Circle B has a center at (2 ,8 ) and an area of 67 pi. Do the circles overlap?

Aug 26, 2016

Yes they overlap

#### Explanation:

First circle
${\left(x - 5\right)}^{2} + {\left(y - 1\right)}^{2} = {r}^{2}$

This is the general form of the equation of the circle
The area of the circle is $\pi {r}^{2}$which we are told equals$16 \pi$
So the radius of the first circle is 4
Like wise the area of the second circle is $\sqrt{67}$

The distance between the centres of the 2 circles is
$\sqrt{{\left(8 - 1\right)}^{2} + {\left(5 - 2\right)}^{2}}$=$\sqrt{58}$
This from Pythagoras.if you are not sure then draw the centre points on a graph and draw the right triangle.

If the circles did not overlap then the distance between the centres would be greater than the sum of the two radii. It is not.


text:
type: bool
[contents]

#### Syntax

The syntax for bool definitions is:

f++:
bool definitions
bool(definitions)
:=(bool, definitions)

n++:
@bool definitions
@bool(definitions)
@:=(bool, definitions)

Note: If you are using the first syntax for variable definitions and want to have more code and/or text following on the same line then simply end the definition with ';'.

#### Description

The bool type is used for boolean values, though note that ExprTk will happily assign any floating point value to them (one limitation of ExprTk is that it only really works with doubles and strings).

Note: So that ExprTk has direct access to booleans (for the sake of convenience), underneath they are actually doubles. If you are defining large volumes of bools where memory usage is of concern then std::bool uses proper bools underneath, which you can still access from all of f++, ExprTk, Lua and n++, however accessing and setting them with ExprTk is not quite as nice syntactically and not as efficient.

Note: If you need to define thousands of variables then := is faster, plus it has useful error messages for unrecognised types.

#### Options

The following options are available for bool definitions:

option description
const definition of a constant
!exprtk do not register variable with ExprTk
layer="x" define variable at layer x
private definition of a private
scope+="x" add x to scopes variable can be accessed from
option description

#### f++ example

Examples of bool being defined with f++:

bool a=0, b=1
bool(x, y, z)
:=(bool, finished = 0)

#### n++ example

Examples of int being defined with n++:

@bool a=0, b=1
@bool(x, y, z)
@:=(bool, finished = 0)