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## Interview with a middle school math teacher on the Common Core

Today’s post is an email interview with Fawn Nguyen, who teaches math at Mesa Union Junior High in southern California. Fawn is on the leadership team for UCSB Mathematics Project that provides professional development for teachers in the Tri-County area. She is a co-founder of the Thousand Oaks Math Teachers’ Circle. In an effort to share and learn from other math teachers, Fawn blogs at Finding Ways to Nguyen Students Over. She also started VisualPatterns.org to help students develop algebraic thinking, and more recently, she shares her students’ daily math talks to promote number sense. When Fawn is not teaching or writing, she is reading posts on mathblogging.org as one of the editors. She sleeps occasionally and dreams of becoming an architect when all this is done.

Importantly for the below interview, Fawn is not being measured via a value-added model. My questions are italicized.

——

I’ve been studying the rhetoric around the mathematics Common Core State Standard (CCSS). So far I’ve listened to Diane Ravitch stuff, I’ve interviewed Bill McCallum, the lead writer of the math CCSS, and I’ve also interviewed Kiri Soares, a New York City high school principal. They have very different views. Interestingly, McCallum distinguished three things: standards, curriculum, and testing.

What do you think? Do teachers see those as three different things? Or is it a package deal, where all three things rolled into one in terms of how they’re presented?

I can’t speak for other teachers. I understand that the standards are not meant to be the curriculum, but the two are not mutually exclusive either. They can’t be. Standards inform the curriculum. This might be a terrible analogy, but I love food and cooking, so maybe the standards are the major ingredients, and the curriculum is the entrée that contains those ingredients. In the show Chopped on Food Network, the competing chefs must use all 4 ingredients to make a dish – and the prepared foods that end up on the plates differ widely in taste and presentation. We can’t blame the ingredients when the dish is blandly prepared any more than we can blame the standards when the curriculum is poorly written.

Similary, the standards inform testing. Test items for a certain grade level cover the standards of that grade level. I’m not against testing. I’m against bad tests and a lot of it. By bad, I mean multiple-choice items that require more memorization than actual problem solving. But I’m confident we can create good multiple-choice tests because realistically a portion of the test needs to be of this type due to costs.

The three – standards, curriculum, and testing – are not a “package deal” in the sense that the same people are not delivering them to us. But they go together, otherwise what is school mathematics? Funny thing is we have always had the three operating in schools, but somehow the Common Core State Standands (CCSS) seem to get the all the blame for the anxieties and costs connected to testing and curriculum development.

I see a lot of good in the CCSS. This set of standards is not perfect, but it’s much better than our state standards. We can examine the standards and see for ourselves that the integrity of the standards holds up to their claims of being embedded with mathematical focus, rigor, and coherence.

Implementation of CCSS means that students and teachers can expect consistency in what is being in taught at each grade level across state boundaries. This is a nontrivial effort in addressing equity. This consistency also helps teachers collaborate nationwide, and professional development for teachers will improve and be more relevant and effective.

I can only hope that textbooks will be much better because of the inherent focus and coherence in CCSS. A kid can move from Maine to California and not have to see different state outlines on their textbooks as if he’d taken on a new kind of mathematics in his new school. I went to a textbook publishers fair recently at our district, and I remain optimistic that better products are already on their way.

We had every state create its own assessment, now we have two consortia, PARCC and Smarter Balanced. I’ve gone through the sample assessments from the latter, and they are far better than the old multiple-choice items of the CST. Kids will have to process the question at a deeper level to show understanding. This is a good thing.

What is potentially bad about the CCSS is the improper or lack of implementation. So, this boils down to the most important element of the Common Core equation – the teacher. There is no doubt that many teachers, myself included, need sustained professional development to do the job right. And I don’t mean just PD in making math more relevant and engaging, and in how many ways we can use technology, I mean more importantly, we need PD in content knowledge.

It is a perverse notion to think that anyone with a college education can teach elementary mathematics. Teaching mathematics requires knowing mathematics. To know a concept is to understand it backward and forward, inside and outside, to recognize it in different forms and structures, to put it into context, to ask questions about it that leads to more questions, to know the mathematics beyond this concept. That reminds me just recently a 6th grader said to me as we were working on our unit of dividing by a fraction. She said, “My elementary teacher lied to me! She said we always get a smaller number when we divide two numbers.”

Just because one can make tuna casserole does not make one a chef. (Sorry, I’m hungry.)

Testing is only good for kids when it helps them learn and become more successful – that the feedback from testing should inform the teacher of next moves. Testing has become such a dirty word because we over test our kids. I’m still in the classroom after 23 years, yet I don’t have the answers. I struggle with telling my kids that I value them and their learning, yet at the end of each quarter, the narrative sum of their learning is a letter grade.

Then, in the absence of helping kids learn, testing is bad.

What are the good/bad things for the teachers with all these tests?

Ideally, a good test that measures what it’s supposed to measure should help the teacher and his students. Testing must be done in moderation. Do we really need to test kids at the start of the school year? Don’t we have the results from a few months ago, right before they left for summer vacation? Every test takes time away from learning.

I’m not sure I understand why testing is bad for teachers aside from lost instructional minutes. Again, I can’t speak for other teachers. But I do sense heightened anxiety among some teachers because CCSS is new – and newness causes us to squirm in our seats and doubt our abilities. I don’t necessarily see this as a bad thing. I see it as an opportunity to learn content at a deeper conceptual level and to implement better teaching strategies.

If we look at anything long and hard enough, we are bound to find the good and the bad. I choose to focus on the positives because I can’t make the day any longer and I can’t have fewer than 4 hours of sleep a night. I want to spend my energies working with my administrators, my colleagues, my parents to bring the best I can bring into my classroom.

Is there anything else you’d like to add?

The best things about CCSS for me are not even the standards – they are the 8 Mathematical Practices. These are life-long habits that will serve students well, in all disciplines. They’re equivalent to the essential cooking techniques, like making roux and roasting garlic and braising kale and shucking oysters. Okay, maybe not that last one, but I just got back from New Orleans, and raw oysters are awesome.

I’m excited to continue to share and collaborate with my colleagues locally and online because we now have a common language! We teachers do this very hard work – day in and day out, late into the nights and into the weekends – because we love our kids and we love teaching. But we need to be mathematically competent first and foremost to teach mathematics. I want the focus to always be about the kids and their learning. We start with them; we end with them.

Categories: math, math education

## A simple mathematical model of congressional geriatric penis pumps

This is a guest post written by Stephanie Yang and reposted from her blogStephanie and I went to graduate school at Harvard together. She is now a quantitative analyst living in New York City, and will be joining the data science team at Foursquare next month.

Last week’s hysterical report by the Daily Show’s Samantha Bee on federally funded penis pumps contained a quote which piqued our quantitative interest. Listen carefully at the 4:00 mark, when Ilyse Hogue proclaims authoritatively:

“Statistics show that probably some our members of congress have a vested interested in having penis pumps covered by Medicare!”

Ilya’s wording is vague, and intentionally so. Statistically, a lot of things are “probably” true, and many details are contained in the word “probably”. In this post we present a simple statistical model to clarify what Ilya means.

First we state our assumptions. We assume that penis pumps are uniformly distributed among male Medicare recipients and that no man has received two pumps. These are relatively mild assumptions. We also assume that what Ilya refers to as “members of Congress [with] a vested interested in having penis pumps covered by Medicare,” specifically means male member of congress who received a penis pump covered by federal funds. Of course, one could argue that female members congress could also have a vested interested in penis pumps as well, but we do not want to go there.

Now the number crunching. According to the report, Medicare has spent a total of \$172 million supplying penis pumps to recipients, at “360 bucks a pop.” This means a total of 478,000 penis pumps bought from 2006 to 2011.

45% of the current 49,435,610 Medicare recipients are male. In other words, Medicare bought one penis pump for every 46.5 eligible men. Inverting this, we can say that 2.15% of male Medicare recipients received a penis pump.

There are currently 128 members of congress (32 senators plus 96 representatives) who are males over the age of 65 and therefore Medicare-eligible. The probability that none of them received a federally funded penis pump is:

$(1-0.0215)^{128} \approx 6.19\%$

In other words, the chances of at least one member of congress having said penis pumps is 93.8%, which is just shy of the 95% confidence that most statisticians agree on as significant. In order to get to 95% confidence, we need a total of 138 male members of congress who are over the age of 65, and this has not happened yet as of 2014. Nevertheless, the estimate is close enough for us to agree with Ilya that there is probably someone member of congress who has one.

Is it possible that there two or more penis pump recipients in congress? We did notice that Ilya’s quote refers to plural members of congress. Under the assumptions laid out above, the probability of having at least two federally funded penis pumps in congress is:

$1- {128 \choose 0} (1- 0.0215)^{128} - {128 \choose 1}(1-0.0215)^{127} (0.0215)^1 \approx 76.3\%$

Again, we would say this is probably true, though not nearly with the same amount of confidence as before. In order to reach 95% confidence that there are two or moreq congressional federally funded penis pump, we would need 200 or more Medicare-eligible males in congress, which is unlikely to happen anytime soon.

Note: As a corollary to these calculations, I became the first developer in the history of mankind to type the following command: git merge --squash penispump.

## Guest rant about rude kids

Today’s guest post was written by Amie, who describes herself as a mom of a 9 and a 14-year-old, mathematician, and bigmouth.

Nota bene: this was originally posted on Facebook as a spontaneous rant. Please don’t miscontrue it as an academic argument.

Time for a rant. I’ll preface this by saying that while my kids are creative, beautiful souls, so are many (perhaps all) children I’ve met, and it would be the height of arrogance to take credit for that as a parent. But one thing my husband and I can take credit for are their good manners, because that took work to develop.

The first phrase I taught me daughter was “thank you,” and it’s been put to good use over the years. I’m also loathe to tell other parents what to do, but this is an exception: teach your fucking kids to say “please” and “thank you”. If you are fortunate to visit another country, teach them to say “please” and “thank you” in the native language.

After a week in paradise at a Club Med in Mexico, I’m at some kind of breaking point with rude rich people and their spoiled kids. And that includes the Europeans. Maybe especially the Europeans. What is it that when you’re in France everyone’s all “thank you and have a nice day” but when these petit bourgeois assholes come to Cancun they treat Mexicans like nonhumans? My son held the door for a face-lifted Russian lady today who didn’t even say thank you.

Anyway, back to kids: I’m not saying that you should suppress your kids’ nature joie de vivre and boisterous, rambunctious energy (though if that’s what they’re like, please keep them away from adults who are not in the mood for it). Just teach them to treat other people with basic respect and courtesy. That means prompting them to say “please,” “thank you,” and “nice to meet you” when they interact with other people.

Jordan Ellenberg just posted how a huge number of people accepted to the math Ph.D. program at the University of Wisconsin never wrote to tell him that they had accepted other offers. When other people are on a wait list!

Whose fault is this? THE PARENTS’ FAULT. Damn parents. Come on!!

P.S. Those of you who have put in the effort to raise polite kids: believe me, I’ve noticed. So has everyone else.

Categories: guest post, rant

Great to be here, and glad you came.

Please hop on the nerd advice column bus for another week of ridiculous if not damaging guidance from yours truly, Aunt Pythia.

And please, after enjoying today’s counsel to other poor, unsuspecting fools:

think of something to ask Aunt Pythia at the bottom of the page!

By the way, if you don’t know what the hell Aunt Pythia is talking about, go here for past advice columns and here for an explanation of the name Pythia.

——

Aunt Pythia,

I’m so SORRY. When I asked your opinion about conference sex the other week I thought I had pasted the link to the article: Will you still medal in the morning? It’s all about the behind the scenes sex at the Olympics.

ORR

Dear ORR,

Let me remind readers of your original question (after giving them a chance to read the article). You (thought you) gave me the link and then made the comment:

If conferences like JMM were to have bowls of condoms at the end of the tables where you pick up your badge do you think people would get the idea and pocket a hand full, then use them?

Answer: no. And that’s a good thing. As much as I’m sex positive, and I truly am, I don’t think what’s going on there in the Olympic Village is really about sex. I mean, that’s super dumb for me to say, because obviously tons of sex is happening, but it’s really about freedom and control and conquests.

So, for example, when you go to college, you notice that the kids who had controlling parents and no freedom in high school are particularly prone to spending too much money, drinking too hard, and fucking anything that moves, even relative to the kids whose upbringing was more relaxed and free. It’s about asserting control over their destiny and their freedom, and it lasts a couple of months and then calms down, hopefully in time for them to pass their classes. I’m absolutely sure of this phenomenon but come to think of it I’ve never seen statistics, which is probably a good thing.

Now think about Olympic athletes. They have lives utterly controlled by their coaches and parents and practices, and between you and me it’s often a neglectful if not abusive situation for those kids. Don’t get me wrong, I’m not saying they hate every minute of it, especially the ones we hear from who win the medals, but if you account for the selection bias for a minute and just think about the childhoods these kids have, your heart breaks for them.

So when they finally get to be somewhere away from their coaches, and this ridiculous pressure is off after their events, they need to somehow assert their freedom to themselves, and the most obvious way to do that is to party hard and fuck anything that moves. Plus it’s even better if they have a long list of conquests, because they’ve been trained to be super competitive.

Now think about math people to contrast this. I’m not saying no condoms are used at a math conference, but generally speaking math people have agency over their own lives, time to have sex when they want and so on, so when they get to a math conference, they just have more of the same. There’s some tension built up before one’s talk, and so in that sense there’s some blowing off of steam, but it’s not Olympic level.

Aunt Pythia

——

Dear Aunt Pythia,

I wrote to you a while ago ranting about how my boss sold me a middle-office IT job as a front-office quant role. Anyways, just wanted to let you know that I did manage to land a quant research role at another buy-side firm, and even though you can’t tell if it’s a scam without being there, it seems promising (I’ll start in about a month). I asked smarter questions this time and the pieces seem to be lining up. I want to thank you for providing a voice of sanity, which is always welcome but especially crucial in trying times where one needs to cut through cognitive dissonance.

Perennial Employee

Dear PE,

Awesome!

Auntie P

——

Dear Aunt Pythia,

Reading about people who are trying to find love or a companion at various places (via online dating or in physical environments) at Aunt Pythia’s Saturday column, why do not you consider creating an online dating site for these high-profile intellectuals?

I considered doing this myself during graduate school when I had many smart and single people around who wanted to have a girl or boy friend but for some reason could not, I did not have the courage or time to invest in this.

How about Aunt Pythia, who has so many followers? I personally checked out some online dating environments, but it was so hard to find really sophisticated people and I did not feel like talking to all those people with the hope that one of them was the good looking nerd I was after.

Idea

Dear Idea,

I like your idea (har har!), but here’s the thing. If I started an online matchmaking thing, I’d first of all not restrict to “high-profile intellectuals,” and second of all it would be very very different from the stuff that already exists.

So for example I’m definitely of the opinion that knowing someone’s age, race, height, and weight and seeing a picture of them makes 99% of all people unfairly unattractive.

It’s a case of knowing the wrong thing about someone. I mean, I’m not saying that you don’t eventually know those things about someone you’re into, but you don’t focus on them if it’s an organic meeting. It’s just the wrong information to provide and makes things less sexy and painfully judgmental.

Let me say it this way. Some woman going online for a date might think she needs the man to be taller than she is, and filter stuff out that way, but in real life she’ll meet someone at a party and be really into him and later realize he’s probably a couple inches shorter than she is and she won’t care at all.

So what information would I ask people to provide instead? That’s a toughie, and I’d love your help, readers, but here’s a start:

1. How sexual are you? (super important question)
2. How much fun are you? (people are surprisingly honest when asked this)
3. How awesome do you smell? (might need to invent technology for this one)
4. What bothers you more: the big bank bailout or the idea of increasing the minimum wage?
5. Do you like strong personalities or would you rather things stay polite?
6. What do you love arguing about more: politics or aesthetics?
7. Where would you love to visit if you could go anywhere?
8. Do you want kids?
9. Dog person or cat person?
10. Do you sometimes wish the girl could be the hero, and not always fall for the hapless dude at the end?

That’s a start. Again, looking for more. I think there should be about 20. Also people should answer in sentences.

Aunt Pythia

——

Dear Aunt Pythia,

I’m a graduate student in a STEM field who happens to also be of the male inclination….oh, and I’m gay.

Now, this normally isn’t so much of an issue (though it has been!) with my social circle/friends outside of the academia, but I’m a bit concerned about disclosing this sort of information to the department, my advisor, and other STEM graduate peers, because, well

1. in my humble opinion, STEM departments are not the most gay friendly. I mean, if you’re a gay male, surely you should be studying creative writing or writing your thesis about continental philosophy’s role in post-WWII imperialism (honestly, the former has been said to me….though the latter sounds like an interesting blog post or two), and
2. my main concern is the future of my academic career. Who are we kidding here, people making hiring decisions in the future may not be that cool with the whole “I’m here and queer!” thing. It could affect my career, negatively. (Ever been to Princeton btw?)

Please feel free to straightsplain to me that I’m being paranoid (though I won’t think I was being paranoid, would I?)

Best,

Sardonic Albeit Distinctly Fearful About Gay-friendliness

First of all, I have been to Princeton, thanks for asking.

Second of all, I don’t think you’re being at all paranoid. In fact it warms the cockles of my heart that you might even feel like you’re being paranoid asking about this, and it’s a testament to how far this country has come since I was a graduate student. Progress!

But only partial progress, to be sure. One of the weirdest things about the academic math market is how you’re expected to up and move to just about anywhere for a one year post-doc or what have you. And many people, especially outside of math, don’t actually want to do that, even if those places are nice places. When you add to that the fact that many of those places aren’t particularly nice, and are still crazy backwards when it comes to accepting gay people, then your job search is getting narrower.

Even so, you might want to take that job in that no-so-nice place, especially if it’s only for one year. I get that.

My short-term advice to you is to do what you need to do to keep your options wide, and if that includes coming out because you’re fed up with this bullshit, then definitely keep that in mind. My long-term advice is to end up eventually in a nice place where you can be accepted. Such places exist and the great news is they are increasing in number.

Love,

Aunt Pythia

——

Please submit your well-specified, fun-loving, cleverly-abbreviated question to Aunt Pythia!

Categories: Aunt Pythia

## The Lede Program has awesome faculty

A few weeks ago I mentioned that I’m the Program Director for the new Lede Program at the Columbia Graduate School of Journalism. I’m super excited to announce that I’ve found amazing faculty for the summer part of the program, including:

1. Jonathan Soma, who will be the primary instructor for Basic Computing and for Algorithms
2. Dennis Tenen, who will be helping Soma in the first half of the summer with Basic Computing
3. Chris Wiggins, who will be helping Soma in the second half of the summer with Algorithms
4. An amazing primary instructor for Databases who I will announce soon,
5. Matthew Jones, who will help that amazing yet-to-be-announced instructor in Data and Databases
6. Three amazing TA’s: Charles Berret, Sophie Chou, and Josh Vekhter (who doesn’t have a website!).

I’m planning to teach The Platform with the help of a bunch of generous guest lecturers (please make suggestions or offer your services!).

Applications are open now, and we’re hoping to get amazing students to enjoy these amazing faculty and the truly innovative plan they have for the summer (and I don’t use the word “innovative” lightly!). We’ve already gotten some super strong applications and made a couple offers of admission.

Also, I was very pleased yesterday to see a blogpost I wrote about the genesis and the goals of the program be published in PBS’s MediaShift.

Finally, it turns out I’m a key influencer, according to The Big Roundtable.

## The US political system serves special interests and the rich

A paper written by Martin Gilens and Benjamin Page and entitled Testing Theories of American Politics: Elites, Interest Groups, and Average Citizens has been recently released and reported on (h/t Michael Crimmins) that studies who has influence on policy in the United States.

Here’s an excerpt from the abstract of the paper:

Multivariate analysis indicates that economic elites and organized groups representing business interests have substantial independent impacts on U.S. government policy, while average citizens and mass-based interest groups have little or no independent influence.

A word about “little or no independent influence”: the above should be interpreted to mean that average citizens and mass-based groups only win when their interests align with economic elites, which happens sometimes, or business interests, which rarely happens. It doesn’t mean that average citizens and mass-based interest groups never ever get what they want.

There’s actually a lot more to the abstract, about abstract concepts of political influence, but I’m ignoring that to get to the data and the model.

The data

The found lots of polls on specific issues that were yes/no and included information about income to determine what poor people (10th percentile) thought about a specific issue, what an average (median income) person thought, and what a wealthy (90th percentile) person thought. They independently corroborated that their definition of wealthy was highly correlated, in terms of opinion, to other stronger (98th percentile) definitions. In fact they make the case that using 90th percentile instead of 98th actually underestimates the influence of wealthy people.

For the sake of interest groups and their opinions on public policy, they had a list of 43 interest groups (consisting of 29 business groups, 11 mass-based groups, and 3 others) that they considered “powerful” and they used domain expertise to estimate how many would oppose or be in favor of a given issue, and more or less took the difference, although they actually did something a bit fancier to reduce the influence of outliers:

Net Interest Group Alignment = ln(# Strongly Favor + [0.5 * # Somewhat Favor] + 1) – ln(#
Strongly Oppose + [0.5 * # Somewhat Oppose] + 1).

Finally, they pored over records to see what policy changes were actually made in the 4 year period after the polls.

Statistics

The different groups had opinions that were sometimes highly correlated:

Note the low correlation between mass public interest groups (like unions, pro-life, NRA, etc) and average citizens’ preferences and the negative correlation between business interests and elites’ preferences.

Next they did three bivariate regressions, measuring the influence of each of the groups separately, as well as one including all three, and got the following:

This is where we get our conclusion that average citizens don’t have independent influence, because of this near-zero coefficient in Model 4. But note that if we ignore elites and interest groups, we do have 0.64 in Model 1, which indicates that preferences of the average citizens are correlated with outcomes.

The overall conclusion is that policy changes are determined by the elites and the interest groups.

We can divide the interest groups into business versus mass-based and check out how the influence is divided between the four defined groups:

Caveats

This stuff might depend a lot on various choices the modelers made as well as their proxies. It doesn’t pick up on smaller special interest groups. It doesn’t account for all possible sources of influence and so on. I’d love to see it redone with other choices. But I’m impressed anyway with all the work they put into this.

I’ll let the authors have the last word:

What do our findings say about democracy in America? They certainly constitute troubling news for advocates of “populistic” democracy, who want governments to respond primarily or exclusively to the policy preferences of their citizens. In the United States, our findings indicate, the majority does not rule — at least not in the causal sense of actually determining policy outcomes. When a majority of citizens disagrees with economic elites and/or with organized interests, they generally lose. Moreover, because of the strong status quo bias built into the U.S. political system, even when fairly large majorities of Americans favor policy change, they generally do not get it.

Categories: #OWS, modeling

## How recently have you experienced democracy? #OWS

A few weeks ago Omar Freilla came to talk to my Occupy group. Omar is a founder of the Green Worker Cooperatives and shared his experience as an organizer.

He is a well-spoken guy and talked passionately about forming community cooperatives, where workers “have a direct role in decision-making and a share of all profits, build community wealth and help make a democratic economy real.”

At one point in his presentation, Omar asked us was how recently we’d “experienced democracy.”

On the face of it I didn’t think it was a fair question, especially when he compared it to experiencing anger or happiness. After all, democracy isn’t an emotion and I can’t experience democracy, say, by myself in a room, but of course I can conjure up emotions by myself in a room, especially if I have a laptop, wifi, and Netflix to help me.

But since his visit, I have to admit I have dwelled on that question and it’s become more and more reasonable in my mind, although I made two decisions on how to interpret it.

First of all, I chose to interpret it not as a formal gesture of democracy, like asking how recently have you voted in a formal election. Instead, it’s a local decision-making process question: how recently has your vote mattered in a local decision that affects a group?

Second, it’s not really about me. It’s about looking around and deciding who around me gets to participate in democratic decisions and who doesn’t.

For example, it might be at work. Although I personally get to make a lot of decisions at work, that fact clearly separates me from tons of people who simply get told what to do by some kind of authority. And there is an important distinction between people who have a manager but get to make decisions internal to their projects and people who have every decision laid out for them.

And that latter workplace anti-democratic situation is, I imagine, maximally soul-crushing, and is the audience that Omar is worried about and is reaching out to. And that’s why his question turns out to be a really good question after all.

I also consider democracy inside my own family. Since I’m the mom of the family, I tend to make more decisions that affect my little group than other people, but now I’m more sensitive to sharing that power there when I can. Turns out my kids love making decisions, it makes them gleeful in fact, even if it’s just what to eat for dinner. And they make good decisions too, which I’m consistently proud of.

My final example is Occupy, which is by construction a direct democracy, and I know how good participating and experiencing democracy actually feels there, and it’s a big part of why it works.

What about you? How recently have you experienced democracy?

Categories: #OWS


text:
3 years ago

# Generalized coupled mode formalism in reciprocal waveguides with gain/loss, anisotropy or bianisotropy.

Zhongfei Xiong, Jing Xu, Weijin Chen, Yuntian Chen

In anisotropic or bianisotropic waveguides, the standard coupled mode theory fails due to the broken link between the forward and backward propagating modes, which together form the dual mode sets that are crucial in constructing couple mode equations. We generalize the coupled mode theory by treating the forward and backward propagating modes on the same footing via a generalized eigenvalue problem that is exactly equivalent to the waveguide Hamiltonian. The generalized eigenvalue problem is fully characterized by two operators, i.e., $( \bar{\bm{L}},\bar{\bm{B}})$, wherein $\bar{\bm{L}}$ is a self-adjoint differential operator, while $\bar{\bm{B}}$ is a constant antisymmetric operator. From the properties of $\bar{\bm{L}}$ and $\bar{\bm{B}}$, we establish the relation between the dual mode sets that are essential in constructing coupled mode equations in terms of forward and backward propagating modes. By perturbation, the generalized coupled mode equation can be derived in a natural way. Our generalized coupled mode formalism can be used to study the mode coupling in waveguides that may contain gain/loss, anisotropy or bianisotropy. We further illustrate how the generalized coupled theory can be used to study the modal coupling in anisotropy and bianisotropy waveguides through a few concrete examples.

Publisher URL: http://arxiv.org/abs/1801.06673

DOI: arXiv:1801.06673v1

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text:
How do you solve |7r-4|=24?

Jan 28, 2017

The solutions are $S = \left\{4 , - \frac{20}{7}\right\}$

Explanation:

We have 2 equations to solve

First equation :

$7 r - 4 = 24$

$7 r = 24 + 4 = 28$

$r = \frac{28}{7} = 4$

Second equation :

$- 7 r + 4 = 24$

$7 r = - 24 + 4 = - 20$

$r = - \frac{20}{7}$


text:
# Strengthening neighbourhood substitution

Domain reduction is an essential tool for solving the constraint satisfaction problem (CSP). In the binary CSP, neighbourhood substitution consists in eliminating a value if there exists another value which can be substituted for it in each constraint. We show that the notion of neighbourhood substitution can be strengthened in two distinct ways without increasing time complexity. We also show the theoretical result that, unlike neighbourhood substitution, finding an optimal sequence of these new operations is NP-hard.

## Authors

• 13 publications
03/26/2020

### No-Rainbow Problem is NP-Hard

Surjective Constraint Satisfaction Problem (SCSP) is the problem of deci...
03/26/2020

### No-Rainbow Problem and the Surjective Constraint Satisfaction Problem

Surjective Constraint Satisfaction Problem (SCSP) is the problem of deci...
04/21/2021

### QCSP on Reflexive Tournaments

We give a complexity dichotomy for the Quantified Constraint Satisfactio...
03/08/2011

### On Minimal Constraint Networks

In a minimal binary constraint network, every tuple of a constraint rela...
01/04/2012

### Complexity Classification in Infinite-Domain Constraint Satisfaction

A constraint satisfaction problem (CSP) is a computational problem where...
04/15/2010

### Propagating Conjunctions of AllDifferent Constraints

We study propagation algorithms for the conjunction of two AllDifferent ...
08/27/2017

### Plain stopping time and conditional complexities revisited

In this paper we analyze the notion of "stopping time complexity", infor...
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## 1 Introduction

Domain reduction is classical in constraint satisfaction. Indeed, eliminating inconsistent values by what is now known as arc consistency [Waltz] predates the first formulation of the constraint satisfaction problem [DBLP:journals/ai/Mackworth77]. Maintaining arc consistency, which consists in eliminating values that can be proved inconsistent by examining a single constraint together with the current domains of the other variables, is ubiquitous in constraint solvers [DBLP:reference/fai/Bessiere06]. In binary CSPs, various algorithms have been proposed for enforcing arc consistency in time, where denotes maximum domain size and the number of constraints [DBLP:journals/ai/MohrH86, ac]. Generic constraints on a number of variables which is unbounded are known as global constraints. Arc consistency can be efficiently enforced for many types of global constraints [DBLP:reference/fai/HoeveK06]. This has led to the development of efficient solvers providing a rich modelling language. Stronger notions of consistency have been proposed for domain reduction which lead to more eliminations but at greater computational cost [DBLP:reference/fai/Bessiere06, DBLP:conf/ijcai/BessiereD05, DBLP:conf/cp/WoodwardKCB12].

In parallel, other research has explored methods that preserve satisfiability of the CSP instance but do not preserve the set of solutions. When searching for a single solution, all but one branch of the explored search tree leads to a dead-end, and so any method for faster detection of unsatisfiability is clearly useful. An important example of such methods is the addition of symmetry-breaking constraints [DBLP:journals/constraints/ChuS15, DBLP:reference/fai/GentPP06]. In this paper we concentrate on domain-reduction methods. One family of satisfiability-preserving domain-reduction operations is value merging. For example, two values can be merged if the so-called broken triangle (BT) pattern does not occur on these two values [DBLP:journals/ai/CooperDMETZ16]. Other value-merging rules have been proposed which allow less merging than BT-merging but at a lower cost [vi] or more merging at a greater cost [wBTP, DMTCS:Naanaa]. Another family of satisfiability-preserving domain-reduction operations are based on the elimination of values that are not essential to obtain a solution [DBLP:conf/gcai/FreuderW17]. The basic operation in this family which corresponds most closely to arc consistency is neighbourhood substitution: a value can be eliminated from a domain if there is another value in the same domain such that can be replaced by in each tuple in each constraint relation (reduced to the current domains of the other variables) [DBLP:conf/aaai/Freuder91]. In binary CSPs, neighbourhood substitution can be applied until convergence in time [ns]. In this paper, we study notions of substitutability which are strictly stronger than neighbourhood substitutability but which can be applied in the same time complexity. We say that one elimination rule is stronger than (subsumes) another rule if any value in a non-trivial instance (an instance with more than one variable) that can be eliminated by can also be eliminated by , and is strictly stronger (strictly subsumes) if there is also at least one non-trivial instance in which can eliminate a value that cannot. Two rules are incomparable if neither is stronger than the other.

To illustrate the strength of the new notions of substitutability that we introduce in this paper, consider the instances shown in Figure 1. These instances are all globally consistent (each variable-value assignment occurs in a solution) and neighbourhood substitution is not powerful enough to eliminate any values. In this paper, we introduce three novel value-elimination rules, defined in Section 2: SS, CNS and SCSS. We will show that snake substitution (SS) allows us to reduce all domains to singletons in the instance in Figure 1(a). Using the notation for the domain of the variable , conditioned neighbourhood-substitution (CNS), allows us to eliminate value 0 from and value 2 from in the instance shown in Figure 1(b), reducing the constraint between and to a null constraint (the complete relation ). Snake-conditioned snake-substitution (SCSS) subsumes both SS and CNS and allows us to reduce all domains to singletons in the instance in Figure 1(c) (as well as in the instances in Figure 1(a),(b)).

In Section 2 we define the substitution operations SS, CNS and SCSS. In Section 3 we prove the validity of these three substitution operations, in the sense that they define satisfiability-preserving value-elimination rules. In Section 4 we explain in detail the examples in Figure 1 and we give other examples from the semantic labelling of line drawings. Section 5 discusses the complexity of applying these value-elimination rules until convergence: the time complexity of SS and CNS is no greater than neighbourhood substitution (NS) even though these rules are stictly stronger. However, unlike NS, finding an optimal sequence of value eliminations by SS or CNS is NP-hard: this is shown in Section 6.

## 2 Definitions

We study binary constraint satisfaction problems.

A binary CSP instance comprises

• a set of variables ,

• a domain for each variable (), and

• a binary constraint relation for each pair of distinct variables ()

For notational convenience, we assume that there is exactly one binary relation for each pair of variables. Thus, if and do not constrain each other, then we consider that there is a trivial constraint between them with . Furthermore, (viewed as a boolean matrix) is always the transpose of . A solution to is an -tuple such that , and for each distinct , .

We say that has a support at variable if such that . A binary CSP instance is arc consistent (AC) if for all pairs of distinct variables , each has a support at  [lecoutre].

In the following we assume that we have a binary CSP instance over variables and, for clarity of presentation, we write as a shorthand for . We use the notation for

 ∀c∈D(xj), (b,c)∈Rij⇒(a,c)∈Rij

(i.e. can be substituted for in any tuple ).

###### Definition 1

[DBLP:conf/aaai/Freuder91]   Given two values , is neighbourhood substitutable (NS) by if , .

It is well known and indeed fairly obvious that eliminating a neighbourhood substitutable value does not change the satisfiability of a binary CSP instance. We will now define stronger notions of substitutability. The proofs that these are indeed valid value-elimination rules are not directly obvious and hence are delayed until Section 3. We use the notation for

 ∀d∈D(xk), (b,d)∈Rik ⇒ ∃e∈D(xk)((a,e)∈Rik ∧ ∀ℓ∉{i,k},dkℓ−→e).

This is illustrated in Figure 2, in which ovals represent domains, bullets represent values, a line joining two values means that these two values are compatible (so, for example, ), and the means that . Since in this definition is a function of and , if necessary, we will write instead of . In other words, the notation means that can be substituted for in any tuple provided we also replace by . It is clear that implies since it suffices to set since, trivially, for all . In Figure 1(a), the value is snake substitutable by : we have by taking (where the arguments of are as shown in Figure 2), since and ; and since . Indeed, by a similar argument, the value is snake substitutable by in each domain.

###### Definition 2

Given two values , is snake substitutable (SS) by if , .

In the following two definitions, can be eliminated from because it can be substituted by some other value in , but this value is a function of the value assigned to another variable . Definition 3 is illustrated in Figure 3.

###### Definition 3

Given , is conditioned neighbourhood-substitutable (CNS) if for some , with , such that .

A CNS value is substitutable by a value where is a function of the value assigned to some other variable . In Figure 1(b), the value is conditioned neighbourhood-substitutable (CNS) with as the conditioning variable (i.e. in Definition 3): for the assignments of or to , we can take since , and for the assignment to , we can take since . By a symmetrical argument, the value is CNS, again with as the conditioning variable. We can note that in the resulting CSP instance, after eliminating from and from , all domains can be reduced to singletons by applying snake substitutability.

Observe that CNS subsumes arc consistency; if a value has no support in , then is trivially CNS (conditioned by the variable ). It is easy to see from their definitions that SS and CNS both subsume NS (in instances with more than one variable), but that neither NS nor SS subsume arc consistency.

We now integrate the notion of snake substitutability in two ways in the definition of CNS: the value (see Figure 3) assigned to a variable may be replaced by a value (as in the definition of , above), but the value (see Figure 3) assigned to the conditioning variable may also be replaced by a value . This is illustrated in Figure 4.

###### Definition 4

A value is snake-conditioned snake-substitutable (SCSS) if for some , with , such that .

In Figure 1(c), the value is snake-conditioned snake-substitutable (SCSS) with as the conditioning variable: for the assignment of or to , we can take since (taking for ) and (taking for ), and for the assignment of to , we can take since (again taking for ) and (again taking for ). By similar arguments, all domains can be reduced to singletons following the SCSS elimination of values in the following order: from , , and from , , and from , , and from and from .

We can see that SCSS subsumes CNS by setting in Definition 4 and by recalling that implies that . It is a bit more subtle to see that SCSS subsumes SS: if is snake substitutable by some value , it suffices to choose in Definition 4 to be this value (which is thus constant, i.e. not dependent on the value of ), then the snake substitutability of by implies that for all and , which in turn implies that for ; thus is snake-conditioned snake-substitutable.

## 3 Value elimination

It is well-known that NS is a valid value-elimination property, in the sense that if is neighbourhood substitutable by then can be eliminated from without changing the satisfiability of the CSP instance [DBLP:conf/aaai/Freuder91]. In this section we show that SCSS is a valid value-elimination property. Since SS and CNS are subsumed by SCSS, it follows immediately that SS and CNS are also valid value-elimination properties.

###### Theorem 3.1

In a binary CSP instance , if is snake-conditioned snake-substitutable then can be eliminated from without changing the satisfiability of the instance.

###### Proof

By Definition 4, for some , with , such that

 ∀k∉{i,j},bik⇝a (1) ∧ ∃g∈D(xj)((a,g)∈Rij ∧ ∀m∉{i,j},cjm−→g). (2)

We will only apply this definition for fixed , and for fixed values and , so we can consider as a constant (even though it is actually a function of ). Let be a solution to with . It suffices to show that there is another solution with . Consider . Since is a solution, we know that . Thus, according to the above definition of SCSS, there is a value that can replace (conditioned by the assignment ) in the sense that (1) and (2) are satisfied. Now, for each , , i.e.

 ∀d∈D(xk), (b,d)∈Rik ⇒ ∃e∈D(xk)((a,e)∈Rik ∧ ∀ℓ∉{i,k},dkℓ−→e).

Recall that is a function of and . But we will only consider fixed and a unique value of dependant on , so we will write for brevity. Indeed, setting we can deduce from (since is a solution) that

 ∀k≠i,j,  ∃e(k)∈D(xk)((a,e(k))∈Rik ∧ ∀ℓ∉{i,k},skkℓ−→e(k)). (3)

Define the -tuple as follows:

 tr=⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩aif r=isrif r≠i ∧ (a,sr)∈Rirgif r=j ∧ (a,sr)∉Rire(r)if r≠i,j ∧ (a,sr)∉Rir

Clearly and for all . To prove that is a solution, it remains to show that all binary constraints are satisfied, i.e. that for all distinct . There are three cases: (1) , , (2) , , (3) .

• There are three subcases: (a) and , (b) and , (c) and . In case (a), and , so from equation 2, we have . In case (b), and and so, trivially, . In case (c), and , so from equation 3, we have .

• There are four subcases: (a) and , (b) and , (c) and , (d) and . In case (a), and , so since is a solution. In case (b), and ; setting , in equation 3, we have since . In case (c), and ; setting and in equation 2 we can deduce that since . In case (d), and . By the same argument as in case 2(b), we know that , and then setting and in equation 2, we can deduce that .

• There are three essentially distinct subcases: (a) and , (b) and , (c) and . In cases (a) and (b) we can deduce by the same arguments as in cases 2(a) and 2(b), above. In case (c), and . Setting in equation 3, we have from which we can deduce that since . Reversing the roles of and in equation 3 (which is possible since they are distinct and both different to and ), we also have that . We can then deduce that since we have just shown that .

We have thus shown that any solution with can be transformed into another solution that does not assign the value to and hence that the elimination of from preserves satisfiability.

###### Corollary 1

In a binary CSP instance , if is snake-substitutable or conditioned neighbourhood substitutable, then can be eliminated from without changing the satisfiability of the instance.

## 4 Examples

We illustrate the potential of SS, CNS and SCSS using the examples given in Figure 1. In Figure 1(a), the value is snake substitutable by : we have by taking (where the arguments of are as shown in Figure 2), since and ; and since . Indeed, by a similar argument, the value is snake substitutable by in each domain. In Figure 1(b), the value is conditioned neighbourhood-substitutable (CNS) with as the conditioning variable (i.e. in Definition 3): for the assignments of or to , we can take since , and for the assignment to , we can take since . By a symmetrical argument, the value is CNS, again with as the conditioning variable. We can note that in the resulting CSP instance, after eliminating from and from , all domains can be reduced to singletons by applying snake substitutability.

In Figure 1(c), the value is snake-conditioned snake-substitutable (SCSS) with as the conditioning variable: for the assignment of or to , we can take since (taking for ) and (taking for ), and for the assignment of to , we can take since (again taking for ) and (again taking for ). By similar arguments, all domains can be reduced to singletons following the SCSS elimination of values in the following order: from , , and from , , and from , , and from and from .

To give a non-numerical example, we considered the impact of SS and CNS in the classic problem of labelling line-drawings of polyhedral scenes composed of objects with trihedral vertices [DBLP:journals/ai/Clowes71, Huff1, Waltz]. There are six types of trihedral vertices: , , , , and , shown in Figure 5. The aim is to assign each line in the drawing a semantic label among four possibilities: convex (), concave () or occluding ( or depending whether the occluding surface is above or below the line). Some lines in the top middle drawing in Figure 5 have been labelled to illustrate the meaning of these labels. This problem can be expressed as a binary CSP by treating the junctions as variables. The domains of variables are given by the catalogue of physically realisable labellings of the corresponding junction according to its type. This catalogue of junction labellings is obtained by considering the six vertex types viewed from all possible viewpoints [DBLP:journals/ai/Clowes71, Huff1]. For example, there are 6 possible labellings of an L-junction, 8 for a T-junction, 5 for a Y-junction and 3 for a W-junction [ldbook]. The complete catalogue of labelled junctions is shown in Figure 6, where a question mark represents any of the four labels and rotationally symmetric labellings are omitted. There is a constraint between any two junctions joined by a line: this line must have the same semantic label at both ends. We can also apply binary constraints between distant junctions: the 2Reg constraint limits the possible labellings of junctions such as and in Figure 7, since two non-colinear lines, such as and , which separate the same two regions cannot both be concave [lddl, ldbook].

The drawings shown in Figure 7 and Figure 8 are ambiguous. For example, in Figure 7, any of lines , or could be projections of concave edges (meaning that the two blocks on the left side of the figure are part of the same object) or all three could be projections of occluding edges (meaning that these two blocks are, in fact, separate objects). Similarly, lines and in Figure 8 could be projections of occluding or concave edges. The drawings shown in Figure 7 and Figure 8 are both examples of families of line drawings. In each of these figures there are four copies of the basic structure, but there is a clear generalisation to drawings containing copies of the basic structure. The ambiguity that we have pointed out above gives rise to an exponential number of valid labellings for these families of drawings. However, after applying arc consistency and snake substitution until convergence, each domain is a singleton for both these families of line drawings. We illustrate this by giving one example of a snake substitution. After arc consistency has been established, the labelling for junction in Figure 7 is snake substitutable by . This can be seen by consulting Figure 9 which shows the domains of variables , , , and with lines joining compatible labellings for adjacent junctions: snake substitutability follows from the fact that the labelling for can be replaced by in any global labelling, provided the labelling for is also replaced by and the labelling for is also replaced by . Purely for clarity of presentation, some constraints have not been shown in Figure 9, notably the 2Reg constraints between distant junctions [lddl, ldbook].

Of course, there are line drawings where snake substution is much less effective than in Figures 7 and 8. Nevertheless, in the six drawings in Figure 5, which are a representative sample of simple line drawings, 22 of the 73 junctions have their domains reduced to singletons by arc consistency alone and a further 20 junctions have their domains reduced to singletons when both arc consistency and snake substitution are applied. This can be compared with neighbourhood substitution which eliminates no domain values in this sample of six drawings. It should be mentioned that we found no examples where conditioned neighbourhood substitution led to the elimination of labellings in the line-drawing labelling problem.

## 5 Complexity

In a binary CSP instance , we say that two variables constrain each other if there is a non-trivial constraint between them (i.e. ). Let denote the set of pairs such that constrain each other. We use to denote the maximum size of the domains and to denote the number of non-trivial binary constraints. In this section we show that it is possible to apply CNS and SS until convergence in time and that it is possible to check SCSS in time. Thus, the complexity of applying the value-elimination rules CNS, SS and SCSS is comparable to the time complexity of applying neighbourhood substitution (NS) [ns]. This is interesting because (in instances with more than one variable) CNS, SS and SCSS all strictly subsume NS.

### 5.1 Substitution and arc consistency

It is well known that arc consistency eliminations can provoke new eliminations by neighbourhood substitution (NS) but that NS eliminations cannot destroy arc consistency [ns]. It follows that arc consistency eliminations can provoke new eliminations by SS, CNS and SCSS (since these notions subsume NS). It is easily seen from Definition 3 that eliminations by CNS cannot destroy arc consistency. We therefore assume in this section that arc consistency has been established before looking for eliminations by any form of substitution. Nonetheless, unlike CNS, eliminations by SS (or SCSS) can provoke new eliminations by arc consistency; however, these eliminations cannot themselves propagate. To see this, suppose that is eliminated since it is snake-substitutable by . If is the only support of at , then can then be eliminated by arc consistency. However, the elimination of cannot provoke any new eliminations by arc consistency. To see this, recall that, by Definition 2 of SS, there is a value such that for all , for all , if was a support for at then so was (as illustrated in Figure 2). Furthermore, since was the only support for at , no other value in can lose its support when is eliminated from . In conclusion, the algorithm for applying SS has to apply this limited form of arc-consistency (without propagation) whereas the algorithm to apply CNS does not need to test for arc consistency since we assume that it has already been established. Furthermore, since AC is, in fact, subsumed by SCSS we do not explicitly need to test for it in the algorithm to apply SCSS.

### 5.2 Applying SS until convergence

We first give an algorithm for applying SS until convergence. The following notions used by the algorithm are best understood by consulting Figure 2. An assignment that contradicts some is known as a block, and the associated variable is known as a block variable. For , the term sub denotes a value that could replace when is assigned (in the sense that and , as illustrated in Figure 2). In the context of a possible snake substitution of by , a value that does not have a sub that could replace it is a stop and the corresponding variable is a stop variable. If there are no stop variables, then is snake substitutable by . The algorithm for applying SS until convergence uses the following data structures:

• For all , for all ,
NbBlocks() .

• For all , for all ,
BlockVars() .
If BlockVars( , for some , then can be eliminated from by neighbourhood substitution.

• For all , for all , for all such that ,
NbSubs() .
Note that if and only if .

• For all , for all , NbStops()
.

• For all , for all ,
NbStopVars() .

• For all , for all ,
NbSnake() .
If NbSnake() , then value can be eliminated from by snake substitution.

• For all , for all ,
Inconsistent() true if has no support at some other variable.

We assume, throughout this section, that each set (such as BlockVars()) which is a subset of a fixed set (in this case the set of variables) is stored using an array data structure whose index ranges over the elements of thus allowing all basic operations, such as insertion and deletion, to be performed in time. The data structures listed above can clearly be initialised in time and require space. We also use a list data structure ElimList to which we add pairs if NbSnake() (indicating that could be eliminated by snake substitition). Note that due to possible eliminations of other values, from the same or other domains, between the detection of the snake substitutability of

and the moment this information is processed,

may no longer be snake substitutable when ElimList is processed. We choose to give precedence to neighbourhood substitutability over snake substitutability by placing neighbourhood-substitutable values at the head of the list ElimList (line (1) in the code below) and snake-substitutable values at the tail. We also add to ElimList values that can be eliminated by arc consistency.

The processing of elements of ElimList involves the updating of all the above data structures which can in turn lead to the detection of new eliminations. The algorithm given in Figure 10 performs eliminations and propagates until convergence, assuming that the above data structures (including ElimList) have all been initialised. When a value is eliminated from we have to update NbBlocks because may correspond to value in Figure 2, NbSubs because may correspond to value in Figure 2, NbStops because may correspond to the value in Figure 2, NbSnake because may correspond to the value in Figure 2, and Inconsistent when corresponds to value in Figure 2.

The subprograms DecStops and IncStops for updating NbStops (and the consequent updating of NbStopVars and NbSnake) are given in Figure 11.


text:
## Wednesday, June 25, 2014

### Easily Count Missing Values in SAS

If counting the number of missing values among groups of variables is the task (something I had to do today), then using PROC FORMAT seems to be the easiest and most efficient method if the variables are character whereas if the variables are numeric, PROC MEANS is a good approach.  The code snippet below was borrowed from "Cody's Data Cleaning Techniques Using SAS Software" by Ron Cody.

Character Variables:

* **format for missing vs. non-missing values;
proc format;
value $misscnt ' ' = 'Missing' OTHER = 'Non-Missing'; run; proc freq data=dsin; tables _character_ / nocum missing; format _character_$misscnt.;
run;

And for numeric variables:

proc means data=dsin n nmiss;
var _numeric_;
run;

It really can't get much simpler than this.


text:
Intended for healthcare professionals

Analysis

# Integrating climate action for health into covid-19 recovery plans

BMJ 2020; 370 (Published 02 September 2020) Cite this as: BMJ 2020;370:m3169
1. Kristine Belesova, assistant professor in environmental epidemiology11,
2. David L Heymann, professor of infectious disease epidemiology2,
3. Andy Haines, professor of environmental change and public health1
1. 1Centre on Climate Change and Planetary Health and Department of Public Health, Environments and Society, London School of Hygiene and Tropical Medicine, London, UK
2. 2Department of Infectious Disease Epidemiology, London School of Hygiene and Tropical Medicine, London, UK
1. Correspondence to: K Belesova kristine.belesova{at}lshtm.ac.uk

Kristine Belesova and colleagues argue that recovery from the covid-19 pandemic must safeguard the health of current and future generations in the face of the climate emergency

The covid-19 pandemic caused over a half a million deaths in its first four months and triggered a global recession that threatens to increase poverty and amplify the health effects of the pandemic. At the same time climate change is adversely affecting health, and the effects are projected to intensify worldwide through a range of direct and indirect pathways, including increased frequency and intensity of heatwaves, floods, and droughts.1

The effects of climate change are emerging over decades and centuries rather than the weeks and months seen for the SARS-CoV-2 virus. However, whereas there is hope for an effective vaccine or treatment for the virus, there are no such prospects for the climate emergency, and, as far as we know, the effects are irreversible. While the covid-19 pandemic is a grave human tragedy, it can be used as an opportunity to implement sustainable economic recovery policies that safeguard the health of the current and future generations including by supporting rapid reductions in greenhouse gas emissions.

## Climate and wider health effects of the pandemic

The implementation of physical distancing or complete lockdown by many countries in response to the covid-19 pandemic has resulted in big reductions in economic activity. This in turn has resulted in large reductions in air pollution and greenhouse gas emissions in many places. Estimates suggest that daily global CO2 emissions decreased by 17% (11% to 25% for ±1 SD) in April 2020 compared with the mean emission levels in 2019.2 Annual emissions could decrease by 4% to 7% (2% to 13%), depending on the lockdown duration.2 Satellite images recorded reductions in PM2.5 and NO2 concentrations in some of the examined areas (eg, China and northern Italy) compared with the seasonal levels observed in previous years, although levels were unchanged in others.3 Latest estimates suggest that the direct effect of the response to the pandemic on climate change will be negligible, with a cooling of approximately 0.01°C (95% confidence interval 0.005 to 0.015) by 2030 compared with the trajectory that follows current national policies.4

Experience of previous economic shocks shows that reductions in emissions are likely to be transient. The fall in greenhouse gas emissions with the 2008 recession, for example, was followed by a resurgence that exceeded pre-recession levels.5 Increased post-lockdown production and lower availability of investment capital for low carbon energy may result in a similar emission pattern. Latest estimates suggest that there is already a partial rebound in emissions with the easing of lockdowns.2

The current economic recession is driving some populations into poverty and, in some countries, increasing health inequities, which in turn could increase the risk of adverse outcomes from covid-19.6 The 2008 recession had pervasive effects on health, particularly among men, with declining self-rated health and increasing morbidity, psychological distress, and suicide, although traffic fatalities and population level alcohol consumption declined.7

National responses to the recession largely determined the magnitude and distribution of health effects. Social safety nets and long term investments in health systems in some European countries seemed to protect populations against adverse effects.8 A 5% contraction in income or consumption due to the covid-19 pandemic could force an estimated additional 85 million people, mostly in developing countries, below the international poverty threshold of $1.90 (£1.50; €1.60) a day; 419 million people would be similarly affected by a 20% contraction.9 The current economic shock emphasises the need for planned equitable transition from economic growth powered by fossil fuels to policies that ensure health and other social priorities within environmental boundaries.10 ## Building a better transition to a net zero carbon economy By May 2020 governments and central banks had committed to a$15tn fiscal stimulus globally in response to the pandemic, equivalent to 17% of the global economy.11 The UN secretary general António Guterres and other leaders called for investment of the recovery funds into “building back better” to support a more sustainable, inclusive, and equitable economy that addresses climate change.12 The EU leaders have proposed a recovery plan based on decarbonisation and digital transformation under the European green deal.13

An expert assessment of an early set of fiscal recovery packages suggests, however, that only 4% have the potential for long term reduction of greenhouse gas emissions; 4% are likely to increase emissions, and 92% would sustain the pre-crisis emissions trajectory.14 The emergency rescue response prioritises the injection of liquidity to prevent economic collapse and meet the immediate needs of saving lives and protecting populations, health systems, and livelihoods. Recapitalising firms that have been badly affected by the recession is an opportunity to integrate health, environmental sustainability, and economic recovery by using criteria that reflect these objectives to prioritise the use of government funds.15

Climate action has been hampered by the complexity of the challenges, indirect and complex attribution of the effects, vested interests in maintaining business as usual, and denialist efforts to influence public opinion. The aspirational target of keeping the global average temperature increase below 1.5°C by 2100 requires 7.6% reduction of global emissions each year between 2020 and 2030.16 That requires the countries to increase their nationally determined contributions to emission reductions under the Paris climate agreement fivefold from their December 2019 levels.16 Recovery packages that facilitate the removal of fossil fuel and other harmful subsidies and invest in transition to a net zero-carbon economy can substantially contribute to this goal, helping to avoid future warming of 0.3 °C by 2050,4 and bring sizeable benefits to human health. Phasing out fossil fuels could, for example, avert about 3.6 million premature deaths related to air pollution annually in the near term, and mitigate climate change effects in the medium to long term.17 Global action required to meet the 1.5°C target is estimated to deliver an economic benefit of $264tn-$610tn by 2100.18

Over the first three months of lockdown in the US, more than 45 million people claimed unemployment benefits and jobs in the oil and gas sector declined by 12%.1920 The oil and gas sector is now experiencing its greatest ever crisis. In the absence of new investment this could accelerate major structural changes away from fossil fuels.21 Investment in oil and gas would have negative public health and climate effects and would be less effective in supporting livelihoods than investment to support the zero-carbon transition. Every $1m spent supporting fossil fuel industries would generate only 2.65 full time jobs compared with an estimated 7.49 and 7.72 full time jobs for the same investment in renewables and energy efficiency.22 It would also contribute to increased health risks for workers and residents in the vicinity of fossil fuel extraction.23 In the UK, transition to a circular economy based on recycling, remanufacturing, reuse, and shared services could create between about 200 000 and 500 000 new jobs and reduce dependency on vulnerable supply chains.24 The government subsidised unemployment and furlough time could be used to invest in human capital by developing job and entrepreneurial skills required for a net zero-carbon circular economy, including through online training. ## Building on behaviour change When habits are temporarily disturbed, people are more sensitive to new information and may adopt a mindset that is more conducive to behaviour change.25 People have drastically changed their lifestyles in response to covid-19. Some of these changes reduce greenhouse gas emissions and have health benefits. City governments in Mexico City, Bogotá, New York, Milan, Paris, Berlin, and London responded by allocating more street space for pedestrians and cyclists to facilitate physical distancing and promote physical activity.2 The reduction in motorised traffic was the largest driver of falls in global emissions during the lockdown.2 The urgency of having to make these changes in response to the pandemic helped overcome some of the barriers to active travel and reduced consumption of non-essential goods and international travel. This could provide momentum to lock in the behaviour changes that benefit health and the environment and might catalyse a shift from a consumerist culture to a more sustainable economy. Translating the temporary behaviour changes into permanent culture change could be supported through the development of new infrastructure, such as converting roads into pedestrian and cycle lanes, and new policies, including incentivising more flexible working from home, virtual meetings and medical consultations, and less long distance business travel.26 Such policies could help compensate for reduced public transport capacity and avoid a rebound in car use in urban areas as a means of physical distancing. For example, cycling (including electric bicycles for longer journeys and for elderly and some disabled people)27 is likely to be the best way of getting around urban centres while maintaining a safe distance between commuters. Although in some cities policies and sustainable infrastructure installed in response to covid-19 were temporary, other cities, such as Milan and Paris, committed to making them permanent.28 A case study of transport policy in Ireland shows that times of financial prudence combined with advocacy can allow sustainability initiatives to flourish without the need for radical institutional transformation.29 There is some evidence that in the right circumstances, disasters can act as a spur to major policy change such as after the 2004 tsunami in Indonesia and Sri Lanka and the 2011 earthquake in Christchurch, New Zealand.3031 Confidence in authority, the high status of science in policy making, inclusive leadership, and a well articulated and coherent vision for a sustainable and healthy society facilitated these positive changes. The covid-19 pandemic differs from these events in scale, scope, extent, and its global context. It will therefore be vital to build a positive case for the zero-carbon transition as we emerge from covid-19, emphasising the health benefits from climate change mitigation policies, particularly through reduced air pollution, increased physical activity, and healthy diets with a low environmental impact.3233 Such policies cross a range of sectors, including energy, transport, housing, urban planning, food and agriculture, industry, and healthcare. Reducing deforestation and tackling the drivers of land degradation and freshwater depletion from unsustainable patterns of food production can benefit health, biodiversity, and the climate.34 Reducing exposure to air pollution, increasing physical activity, and consumption of healthy diets reduce the risks of heart disease and stroke, which in turn influence the risk of adverse outcomes from covid-19.6 ## Advocacy for green recovery Opinion polls in 16 countries have shown that most people expect the environment to be prioritised in recovery packages.35 Large majorities of respondents supported the proposition that we have a responsibility to protect the planet for future generations, and that environmental degradation poses a major threat to health. In the UK, a climate litigation charity warned the government of legal action against inadequate investment in a “green” recovery.36 How can health professionals respond to support the required profound economic changes in the face of entrenched interests, such as fossil fuel industries? They could start by making their voices heard in the lobbying for resources. The recent letter from representatives of 40 million health professionals to the heads of G20 governments urging investments in a zero-carbon, healthy recovery is an example of the leadership needed.37 They could also support the UN and national actions to create a healthy, low carbon economy. WHO, for example, has issued a manifesto calling for a healthy emergence from covid-19 comprising protection and preservation of nature; investment in essential services; rapid decarbonisation of the energy system; promotion of healthy sustainable food systems and cities; and stopping the use of taxpayers’ money to fund pollution, including halting the$400bn direct fossil fuel subsidies globally.38

Health professionals can also act to address the climate emergency in their daily work—for example, by supporting the decarbonisation of health services, reducing waste, encouraging reuse of supplies where feasible, promoting healthy sustainable lifestyles, and leading by example.39 Another role is to work with non-governmental organisations bringing health perspectives to advocacy for climate action.

The next year or so will bring major opportunities to unite global actors in decisive action to protect and promote the health of human populations and natural systems. This imperative should motivate increased ambition at the postponed 26th UN climate change conference in Glasgow, UK, in November 2021 and at the 15th Conference of Parties to the Convention on Biological Diversity in Kunming, China. Health professionals can play important and potentially decisive roles in promoting a healthy and sustainable recovery from covid-19 to safeguard the health of current and future generations.

### Key messages

• Our society has a responsibility to implement a sustainable recovery from covid-19 that safeguards planetary health

• Economic recovery packages should help build more resilient social foundations, including reducing health inequalities, and cut greenhouse gas emissions

• Low carbon recovery strategies will benefit the economy and health

• Lifestyle and employment changes in response to covid-19 must be harnessed to catalyse decisive action on the climate emergency

• Health professionals have an important role in promoting healthy and sustainable recovery and decisive action on climate change

## Footnotes

• Contributors and sources: KB has worked on health impacts of global environmental change and co-benefits of climate change mitigation. DLH has worked on infectious disease epidemiology and directed work at WHO on polio eradication, global response to SARS and other emerging and communicable diseases. AH has studied environmental influences on health, including the potential effects of climate change and the health co-benefits of the low carbon economy. He signed the open letter to the heads of G20 governments in a personal capacity. All authors contributed to the conception and design of the paper and approved the final version. KB prepared the full draft of the manuscript incorporating revisions and comments from all authors. KB is the first author and guarantor of the article.

• Competing interests: We have read and understood BMJ policy on declaration of interests and have no relevant interests to declare.

• Provenance and peer review: Not commissioned; externally peer reviewed.

This article is made freely available for use in accordance with BMJ's website terms and conditions for the duration of the covid-19 pandemic or until otherwise determined by BMJ. You may use, download and print the article for any lawful, non-commercial purpose (including text and data mining) provided that all copyright notices and trade marks are retained.

https://bmj.com/coronavirus/usage


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### Home > AC > Chapter 14 > Lesson 14.3.1.6 > Problem3-84

3-84.

Solve the system of equations at right algebraically.

Try using substitution to solve this system of equations.

$\left. \begin{array} { r } { 2 x - 3 y = 12 } \\ { y + x = - 9 } \end{array} \right.$


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mersenneforum.org Planning & Coordination for 2,2330L
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 2019-06-22, 14:14 #254 pinhodecarlos     "Carlos Pinho" Oct 2011 Milton Keynes, UK 24·3·101 Posts Any option on server side to send a few wus to clients so to have some buffer or the only option would be to increase sieve range once again to 4k or even 8k.
 2019-06-22, 17:01 #255 fivemack (loop (#_fork))     Feb 2006 Cambridge, England 6,379 Posts Is there any way that the http://factoring.cloudygo.com could record 'relations in last 24 hours' as well as 'relations since the project started' - it would make it rather easier to see what happens when configurations change.
2019-06-22, 20:01   #256
VBCurtis

"Curtis"
Feb 2005
Riverside, CA

35·19 Posts

Quote:
 Originally Posted by pinhodecarlos Any option on server side to send a few wus to clients so to have some buffer or the only option would be to increase sieve range once again to 4k or even 8k.
None that I know of for buffering. What does "only option" refer to?

As for workunit size, we'll be changing that to 8k or 10k when we move to I=15, as yield drops from 6.x on I=16 at ~200M to 2.2 on I=15 at ~300M.
So far, we have sieved Q from 8M to 68M for 533M raw relations, for an average yield of 8.9. The yield on current WUs is in the mid-8's.

I'll run remdups shortly to see what our duplicate rate looks like.
EDIT: For Q from 8-60M, 349.5M unique, 118.5 duplicates, 0 bad relations, total 468M relations. Duplicate rate 25%, better than the C206 that Greg ran on 16e/33 for us (that one was 792M uniq, 373M dup).
I'll run it again when Q reaches 70M.

Last fiddled with by VBCurtis on 2019-06-22 at 21:39

2019-06-23, 02:21   #257
VBCurtis

"Curtis"
Feb 2005
Riverside, CA

35×19 Posts

Quote:
 Originally Posted by fivemack Is there any way that the http://factoring.cloudygo.com could record 'relations in last 24 hours' as well as 'relations since the project started' - it would make it rather easier to see what happens when configurations change.
I'd like this too. Perhaps replace the timestamp of last workunit with this stat? We can see the project-difficulty trend as our numbers gently slide, see who is new and rising fast, and as fivemack said easily quantify changes in client configuration.

2019-06-23, 04:07   #258
SethTro

"Seth"
Apr 2019

111010112 Posts

Quote:
 Originally Posted by VBCurtis I'd like this too. Perhaps replace the timestamp of last workunit with this stat? We can see the project-difficulty trend as our numbers gently slide, see who is new and rising fast, and as fivemack said easily quantify changes in client configuration.
I'll try to add it to the top line and graph page tonight.

2019-06-23, 06:57   #259
SethTro

"Seth"
Apr 2019

3538 Posts

Quote:
 Originally Posted by SethTro I'll try to add it to the top line and graph page tonight.
Done

http://factoring.cloudygo.com/progre...L.c207/daily_r
(which is under Charts tab) also a number on the first page.
http://factoring.cloudygo.com/

Last fiddled with by SethTro on 2019-06-23 at 06:57

2019-06-23, 07:27   #260
fivemack
(loop (#_fork))

Feb 2006
Cambridge, England

6,379 Posts

Quote:
 Originally Posted by SethTro Done http://factoring.cloudygo.com/progre...L.c207/daily_r (which is under Charts tab) also a number on the first page. http://factoring.cloudygo.com/
The daily_r curve is useful, because differentiating the total-work-done curve by eye isn't really possible; thank you

I think it would be even more useful to have the last-24-hours figure at a per-client level.

Last fiddled with by fivemack on 2019-06-23 at 07:27

 2019-06-23, 09:14 #261 pinhodecarlos     "Carlos Pinho" Oct 2011 Milton Keynes, UK 113608 Posts What about associating the clients to teams?
 2019-06-23, 09:37 #262 fivemack (loop (#_fork))     Feb 2006 Cambridge, England 6,379 Posts Some thoughts on hyper-threading and -t lots Median runtimes in various configurations, on the same hardware (fortunately I have three identical computers) Code: One job -t32 1090s = 2180s for two Two jobs -t8 2132s/2 Two jobs -t16 taskset 0-15; 16-31 1742s/2 Two jobs -t16 taskset 0-7,16-23; 8-15,24-31 1915s/2 So, on these dual-socket eight-core machines, the right answer is to run two jobs, one across both sockets and the other on the other hyperthread across both sockets ; I think I'd expected two jobs to be better than one but am a bit surprised that having both jobs use both sockets is significantly better. Last fiddled with by fivemack on 2019-06-23 at 09:37
 2019-06-23, 14:32 #263 VBCurtis     "Curtis" Feb 2005 Riverside, CA 35·19 Posts We've reached Q=70M, so I re-ran remdups: Code: Q Unique Dup Total 8-60M 349.5 118.5 468.0 8-70M 402.4 147.7 550.1 Q=60-70M added 82M relations, but only 53M unique.
2019-06-23, 15:26   #264
lukerichards

"Luke Richards"
Jan 2018
Birmingham, UK

12016 Posts

Quote:
 Originally Posted by SethTro Vebis has ~10 clients (vebis.1, vebis.2, ...) so I joined all of those to vebis. I'm going to join all your clients into lukerichards. for the main tab and then add a new tab for all clients
Is there any way to add instance-1, localhost and lrichards-pre2core to this collection as well, so I can see my complete combined stats without mental arithmetic?

Thanks.

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## G = D4×C2×C6order 96 = 25·3

### Direct product of C2×C6 and D4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: D4×C2×C6, C246C6, C124C23, C6.16C24, C4⋊(C22×C6), (C23×C6)⋊2C2, C234(C2×C6), (C22×C4)⋊7C6, (C2×C6)⋊2C23, C2.1(C23×C6), (C22×C12)⋊12C2, (C2×C12)⋊15C22, (C22×C6)⋊6C22, C222(C22×C6), (C2×C4)⋊4(C2×C6), SmallGroup(96,221)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — D4×C2×C6
 Chief series C1 — C2 — C6 — C2×C6 — C3×D4 — C6×D4 — D4×C2×C6
 Lower central C1 — C2 — D4×C2×C6
 Upper central C1 — C22×C6 — D4×C2×C6

Generators and relations for D4×C2×C6
G = < a,b,c,d | a2=b6=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 316 in 236 conjugacy classes, 156 normal (10 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C6, C6, C6, C2×C4, D4, C23, C23, C23, C12, C2×C6, C2×C6, C22×C4, C2×D4, C24, C2×C12, C3×D4, C22×C6, C22×C6, C22×C6, C22×D4, C22×C12, C6×D4, C23×C6, D4×C2×C6
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C24, C3×D4, C22×C6, C22×D4, C6×D4, C23×C6, D4×C2×C6

Smallest permutation representation of D4×C2×C6
On 48 points
Generators in S48
(1 30)(2 25)(3 26)(4 27)(5 28)(6 29)(7 32)(8 33)(9 34)(10 35)(11 36)(12 31)(13 21)(14 22)(15 23)(16 24)(17 19)(18 20)(37 45)(38 46)(39 47)(40 48)(41 43)(42 44)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)(25 26 27 28 29 30)(31 32 33 34 35 36)(37 38 39 40 41 42)(43 44 45 46 47 48)
(1 10 15 44)(2 11 16 45)(3 12 17 46)(4 7 18 47)(5 8 13 48)(6 9 14 43)(19 38 26 31)(20 39 27 32)(21 40 28 33)(22 41 29 34)(23 42 30 35)(24 37 25 36)
(1 44)(2 45)(3 46)(4 47)(5 48)(6 43)(7 18)(8 13)(9 14)(10 15)(11 16)(12 17)(19 31)(20 32)(21 33)(22 34)(23 35)(24 36)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)

G:=sub<Sym(48)| (1,30)(2,25)(3,26)(4,27)(5,28)(6,29)(7,32)(8,33)(9,34)(10,35)(11,36)(12,31)(13,21)(14,22)(15,23)(16,24)(17,19)(18,20)(37,45)(38,46)(39,47)(40,48)(41,43)(42,44), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)(25,26,27,28,29,30)(31,32,33,34,35,36)(37,38,39,40,41,42)(43,44,45,46,47,48), (1,10,15,44)(2,11,16,45)(3,12,17,46)(4,7,18,47)(5,8,13,48)(6,9,14,43)(19,38,26,31)(20,39,27,32)(21,40,28,33)(22,41,29,34)(23,42,30,35)(24,37,25,36), (1,44)(2,45)(3,46)(4,47)(5,48)(6,43)(7,18)(8,13)(9,14)(10,15)(11,16)(12,17)(19,31)(20,32)(21,33)(22,34)(23,35)(24,36)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)>;

G:=Group( (1,30)(2,25)(3,26)(4,27)(5,28)(6,29)(7,32)(8,33)(9,34)(10,35)(11,36)(12,31)(13,21)(14,22)(15,23)(16,24)(17,19)(18,20)(37,45)(38,46)(39,47)(40,48)(41,43)(42,44), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)(25,26,27,28,29,30)(31,32,33,34,35,36)(37,38,39,40,41,42)(43,44,45,46,47,48), (1,10,15,44)(2,11,16,45)(3,12,17,46)(4,7,18,47)(5,8,13,48)(6,9,14,43)(19,38,26,31)(20,39,27,32)(21,40,28,33)(22,41,29,34)(23,42,30,35)(24,37,25,36), (1,44)(2,45)(3,46)(4,47)(5,48)(6,43)(7,18)(8,13)(9,14)(10,15)(11,16)(12,17)(19,31)(20,32)(21,33)(22,34)(23,35)(24,36)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42) );

G=PermutationGroup([[(1,30),(2,25),(3,26),(4,27),(5,28),(6,29),(7,32),(8,33),(9,34),(10,35),(11,36),(12,31),(13,21),(14,22),(15,23),(16,24),(17,19),(18,20),(37,45),(38,46),(39,47),(40,48),(41,43),(42,44)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24),(25,26,27,28,29,30),(31,32,33,34,35,36),(37,38,39,40,41,42),(43,44,45,46,47,48)], [(1,10,15,44),(2,11,16,45),(3,12,17,46),(4,7,18,47),(5,8,13,48),(6,9,14,43),(19,38,26,31),(20,39,27,32),(21,40,28,33),(22,41,29,34),(23,42,30,35),(24,37,25,36)], [(1,44),(2,45),(3,46),(4,47),(5,48),(6,43),(7,18),(8,13),(9,14),(10,15),(11,16),(12,17),(19,31),(20,32),(21,33),(22,34),(23,35),(24,36),(25,37),(26,38),(27,39),(28,40),(29,41),(30,42)]])

D4×C2×C6 is a maximal subgroup of
(C6×D4)⋊6C4  (C2×C6)⋊8D8  (C3×D4).31D4  C24.29D6  C24.30D6  C24.31D6  C24.32D6  C24.49D6  C2412D6  C24.52D6  C24.53D6

60 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 3A 3B 4A 4B 4C 4D 6A ··· 6N 6O ··· 6AD 12A ··· 12H order 1 2 ··· 2 2 ··· 2 3 3 4 4 4 4 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 ··· 1 2 ··· 2 1 1 2 2 2 2 1 ··· 1 2 ··· 2 2 ··· 2

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 type + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 D4 C3×D4 kernel D4×C2×C6 C22×C12 C6×D4 C23×C6 C22×D4 C22×C4 C2×D4 C24 C2×C6 C22 # reps 1 1 12 2 2 2 24 4 4 8

Matrix representation of D4×C2×C6 in GL4(𝔽13) generated by

 12 0 0 0 0 12 0 0 0 0 1 0 0 0 0 1
,
 3 0 0 0 0 12 0 0 0 0 1 0 0 0 0 1
,
 12 0 0 0 0 12 0 0 0 0 8 2 0 0 0 5
,
 1 0 0 0 0 1 0 0 0 0 8 2 0 0 1 5
G:=sub<GL(4,GF(13))| [12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[3,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[12,0,0,0,0,12,0,0,0,0,8,0,0,0,2,5],[1,0,0,0,0,1,0,0,0,0,8,1,0,0,2,5] >;

D4×C2×C6 in GAP, Magma, Sage, TeX

D_4\times C_2\times C_6
% in TeX

G:=Group("D4xC2xC6");
// GroupNames label

G:=SmallGroup(96,221);
// by ID

G=gap.SmallGroup(96,221);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-2,601]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^6=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽


text:
Cloud Essays

High Quality Essays

# Differences in gender roles and gender equality in Norway and Greece

\$60.00

Analytically compare and analyze the Differences in gender roles and gender equality in Norway and Greece.

In the assessment formal matters such as language, structure and bibliography will be important – but even more so, contents, degree of independent work, and analytical depth.

Format:

The paper is to be written as a long, formal report. The paper must have a title page and a table of contents, indicating the outline of your text. Sources should be documented with references in the text (or in endnotes or footnotes). Sources must be listed in a bibliography/list of references.

Length: Text should be minimum 20, maximum 25 A4 pages (title page and pictures, and/or graphs excluded). The text should be typed in Times or Times New Roman, font size 12, line spacing 1.5.

Additionally, you must include an Abstract, approx. half a page (a brief summary of the entire paper).

SKU: differences-in-gender-roles-and-gender-equality-in-norway-and-greece


text:
University Physics Volume 2

# Chapter 6

6.1

Place it so that its unit normal is perpendicular to $E→.E→.$

6.2

$m a b 2 / 2 m a b 2 / 2$

6.3

a. $3.4×105N·m2/C;3.4×105N·m2/C;$ b. $−3.4×105N·m2/C;−3.4×105N·m2/C;$ c. $3.4×105N·m2/C;3.4×105N·m2/C;$ d. 0

6.4

In this case, there is only $E→out.So,yes.E→out.So,yes.$

6.5

$E→=λ02πε01dr^E→=λ02πε01dr^$; This agrees with the calculation of Example 5.5 where we found the electric field by integrating over the charged wire. Notice how much simpler the calculation of this electric field is with Gauss’s law.

6.6

If there are other charged objects around, then the charges on the surface of the sphere will not necessarily be spherically symmetrical; there will be more in certain direction than in other directions.

### Conceptual Questions

1.

a. If the planar surface is perpendicular to the electric field vector, the maximum flux would be obtained. b. If the planar surface were parallel to the electric field vector, the minimum flux would be obtained.

3.

False. The net electric flux crossing a closed surface is always zero if and only if the net charge enclosed is zero.

5.

Since the electric field vector has a $1r21r2$ dependence, the fluxes are the same since $A=4πr2A=4πr2$.

7.

a. no; b. zero

9.

Both fields vary as $1r21r2$. Because the gravitational constant is so much smaller than $14πε014πε0$, the gravitational field is orders of magnitude weaker than the electric field. Also, the gravitational flux through a closed surface is zero or positive; however, the electric flux is positive, negative, or zero, depending on the definition of flux for the given situation.

11.

No, it is produced by all charges both inside and outside the Gaussian surface.

13.

No, since the situation does not have symmetry, making Gauss’s law challenging to simplify.

15.

Any shape of the Gaussian surface can be used. The only restriction is that the Gaussian integral must be calculable; therefore, a box or a cylinder are the most convenient geometrical shapes for the Gaussian surface.

17.

No. If a metal was in a region of zero electric field, all the conduction electrons would be distributed uniformly throughout the metal.

19.

Since the electric field is zero inside a conductor, a charge of $−2.0μC−2.0μC$ is induced on the inside surface of the cavity. This will put a charge of $+2.0μC+2.0μC$ on the outside surface leaving a net charge of $−3.0μC−3.0μC$ on the surface.

### Problems

21.

$Φ=E→·A→→EAcosθ=3.8×104N·m2/CΦ=E→·A→→EAcosθ=3.8×104N·m2/C$ electric field in direction of unit normal; $Φ=E→·A→→EAcosθ=−3.8×104N·m2/CΦ=E→·A→→EAcosθ=−3.8×104N·m2/C$ electric field opposite to unit normal

23.

$3 × 10 −5 N · m 2 / C ( 0.05 m ) 2 = E ⇒ σ = 2.12 × 10 −13 C / m 2 3 × 10 −5 N · m 2 / C ( 0.05 m ) 2 = E ⇒ σ = 2.12 × 10 −13 C / m 2$

25.

a. $Φ=0.17N·m2/C;Φ=0.17N·m2/C;$
b. $Φ=0Φ=0$; c. $Φ=EAcos0°=1.0×103N/C(2.0×10−4m)2cos0°=0.20N·m2/CΦ=EAcos0°=1.0×103N/C(2.0×10−4m)2cos0°=0.20N·m2/C$

27.

$Φ = 3.8 × 10 4 N · m 2 / C Φ = 3.8 × 10 4 N · m 2 / C$

29.

$E → ( z ) = 1 4 π ε 0 2 λ z k ^ , ∫ E → · n ^ d A = λ ε 0 l E → ( z ) = 1 4 π ε 0 2 λ z k ^ , ∫ E → · n ^ d A = λ ε 0 l$

31.

a. $Φ=3.39×103N·m2/CΦ=3.39×103N·m2/C$; b. $Φ=0Φ=0$;
c. $Φ=−2.25×105N·m2/CΦ=−2.25×105N·m2/C$;
d. $Φ=90.4N·m2/CΦ=90.4N·m2/C$

33.

$Φ = 1.13 × 10 6 N · m 2 /C Φ = 1.13 × 10 6 N · m 2 /C$

35.

Make a cube with q at the center, using the cube of side a. This would take four cubes of side a to make one side of the large cube. The shaded side of the small cube would be 1/24th of the total area of the large cube; therefore, the flux through the shaded area would be
$Φ=124qε0Φ=124qε0$.

37.

$q = 3.54 × 10 −7 C q = 3.54 × 10 −7 C$

39.

zero, also because flux in equals flux out

41.

$r > R , E = Q 4 π ε 0 r 2 ; r < R , E = q r 4 π ε 0 R 3 r > R , E = Q 4 π ε 0 r 2 ; r < R , E = q r 4 π ε 0 R 3$

43.

$E A = λ l ε 0 ⇒ E = 4.50 × 10 7 N / C E A = λ l ε 0 ⇒ E = 4.50 × 10 7 N / C$

45.

a. 0; b. 0; c. $E→=6.74×106N/C(−r^)E→=6.74×106N/C(−r^)$

47.

a. 0; b. $E=2.70×106N/CE=2.70×106N/C$

49.

a. Yes, the length of the rod is much greater than the distance to the point in question. b. No, The length of the rod is of the same order of magnitude as the distance to the point in question. c. Yes, the length of the rod is much greater than the distance to the point in question. d. No. The length of the rod is of the same order of magnitude as the distance to the point in question.

51.

a. $E→=Rσ0ε01rr^⇒σ0=5.31×10−11C/m2,E→=Rσ0ε01rr^⇒σ0=5.31×10−11C/m2,$
$λ=3.33×10−12C/mλ=3.33×10−12C/m$;
b. $Φ=qencε0=3.33×10−12C/m(0.05m)ε0=0.019N·m2/CΦ=qencε0=3.33×10−12C/m(0.05m)ε0=0.019N·m2/C$

53.

$E2πrl=ρπr2lε0⇒E=ρr2ε0(r≤R)E2πrl=ρπr2lε0⇒E=ρr2ε0(r≤R)$;
$E2πrl=ρπR2lε0⇒E=ρR22ε0r(r≥R)E2πrl=ρπR2lε0⇒E=ρR22ε0r(r≥R)$

55.

$Φ = q enc ε 0 ⇒ q enc = −1.0 × 10 −9 C Φ = q enc ε 0 ⇒ q enc = −1.0 × 10 −9 C$

57.

$qenc=45παr5,qenc=45παr5,$
$E4πr2=4παr55ε0⇒E=αr35ε0(r≤R),E4πr2=4παr55ε0⇒E=αr35ε0(r≤R),$
$qenc=45παR5,E4πr2=4παR55ε0⇒E=αR55ε0r2(r≥R)qenc=45παR5,E4πr2=4παR55ε0⇒E=αR55ε0r2(r≥R)$

59.

integrate by parts: $qenc=4πρ0[−e−αr((r)2α+2rα2+2α3)+2α3]⇒E=ρ0r2ε0[−e−αr((r)2α+2rα2+2α3)+2α3]qenc=4πρ0[−e−αr((r)2α+2rα2+2α3)+2α3]⇒E=ρ0r2ε0[−e−αr((r)2α+2rα2+2α3)+2α3]$

61.

63.

a. Outside: $E2πrl=λlε0⇒E=3.0C/m2πε0rE2πrl=λlε0⇒E=3.0C/m2πε0r$; Inside $Ein=0Ein=0$; b.

65.

a. $E2πrl=λlε0⇒E=λ2πε0rr≥RE2πrl=λlε0⇒E=λ2πε0rr≥R$ E inside equals 0; b.

67.

$E = 5.65 × 10 4 N / C E = 5.65 × 10 4 N / C$

69.

$λ=λlε0⇒E=aσε0rr≥aλ=λlε0⇒E=aσε0rr≥a$, $E=0E=0$ inside since $qenclosed=0qenclosed=0$

71.

a. $E=0E=0$; b. $E2πrL=Qε0⇒E=Q2πε0rLE2πrL=Qε0⇒E=Q2πε0rL$; c. $E=0E=0$ since r would be either inside the second shell or if outside then q enclosed equals 0.

73.

$∫ E → · n ^ d A = a 4 ∫ E → · n ^ d A = a 4$

75.

a. $∫E→·n^dA=E0r2π∫E→·n^dA=E0r2π$; b. zero, since the flux through the upper half cancels the flux through the lower half of the sphere

77.

$Φ=qencε0Φ=qencε0$; There are two contributions to the surface integral: one at the side of the rectangle at $x=0x=0$ and the other at the side at $x=2.0mx=2.0m$;
$−E(0)[1.5m2]+E(2.0m)[1.5m2]=qencε0=−100Nm2/C−E(0)[1.5m2]+E(2.0m)[1.5m2]=qencε0=−100Nm2/C$
where the minus sign indicates that at $x=0x=0$, the electric field is along positive x and the unit normal is along negative x. At $x=2x=2$, the unit normal and the electric field vector are in the same direction: $qenc=ε0Φ=−8.85×10−10Cqenc=ε0Φ=−8.85×10−10C$.

79.

didn’t keep consistent directions for the area vectors, or the electric fields

81.

a. $σ=3.0×10−3C/m2σ=3.0×10−3C/m2$, $+3×10−3C/m2+3×10−3C/m2$ on one and $−3×10−3C/m2−3×10−3C/m2$ on the other; b. $E=3.39×108N/CE=3.39×108N/C$

83.

Construct a Gaussian cylinder along the z-axis with cross-sectional area A.
$|z|≥a2qenc=ρAa,Φ=ρAaε0⇒E=ρa2ε0|z|≥a2qenc=ρAa,Φ=ρAaε0⇒E=ρa2ε0$,
$|z|≤a2qenc=ρA2z,E(2A)=ρA2zε0⇒E=ρzε0|z|≤a2qenc=ρA2z,E(2A)=ρA2zε0⇒E=ρzε0$

85.

a. $r>b2E4πr2=43π[ρ1(b13−a13)+ρ2(b23−a23)ε0⇒E=ρ1(b13−a13)+ρ2(b23−a23)3ε0r2r>b2E4πr2=43π[ρ1(b13−a13)+ρ2(b23−a23)ε0⇒E=ρ1(b13−a13)+ρ2(b23−a23)3ε0r2$;
b. ;
c. ;
d. ; e. 0

87.

Electric field due to plate without hole: $E=σ2ε0E=σ2ε0$.
Electric field of just hole filled with $−σE=−σ2ε0(1−zR2+z2)−σE=−σ2ε0(1−zR2+z2)$.
Thus, $Enet=σ2ε0hR2+h2Enet=σ2ε0hR2+h2$.

89.

a. $E=0E=0$; b. $E=q14πε0r2E=q14πε0r2$; c. $E=q1+q24πε0r2E=q1+q24πε0r2$; d. $0,q1,−q1,q1+q20,q1,−q1,q1+q2$

### Challenge Problems

91.

Given the referenced link, using a distance to Vega of $237×1015237×1015$ m1 and a diameter of 2.4 m for the primary mirror,2 we find that at a wavelength of 555.6 nm, Vega is emitting $2.44×1024J/s2.44×1024J/s$ at that wavelength. Note that the flux through the mirror is essentially constant.

93.

The symmetry of the system forces $E→E→$ to be perpendicular to the sheet and constant over any plane parallel to the sheet. To calculate the electric field, we choose the cylindrical Gaussian surface shown. The cross-section area and the height of the cylinder are A and 2x, respectively, and the cylinder is positioned so that it is bisected by the plane sheet. Since E is perpendicular to each end and parallel to the side of the cylinder, we have EA as the flux through each end and there is no flux through the side. The charge enclosed by the cylinder is $σA,σA,$ so from Gauss’s law, $2EA=σAε0,2EA=σAε0,$ and the electric field of an infinite sheet of charge is
$E=σ2ε0,E=σ2ε0,$ in agreement with the calculation of in the text.

95.

There is Q/2 on each side of the plate since the net charge is Q: $σ=Q2Aσ=Q2A$,
$∮SE→·n^dA=2σΔAε0⇒EP=σε0=Qε02A∮SE→·n^dA=2σΔAε0⇒EP=σε0=Qε02A$

### Footnotes

• 1http://webviz.u-strasbg.fr/viz-bin/VizieR-5?-source=I/311&HIP=91262
• 2http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19910003124.pdf
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