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# Statkat – find the right statistical technique for your data and research question

### tl;dr

Finding a statistical technique that is suitable for your data and research question can be challenging. The Statkat module helps you to find a statistical method that suits your needs!

## Using the Statkat module

To get started, first make sure that you are working with jamovi version 0.9.1.1 or higher. Then install the Statkat module from the jamovi library and open a data set. We are going to use the Tooth Growth data set as an example data set (see Menu > Open > Examples). The Tooth Growth data set contains three variables: a numeric variable len (tooth length), a dichotomous variable supp (supplement type: vitamin C or orange juice), and a nominal variable dose (dose in milligrams per day). One question of interest could be whether there is a relationship between tooth length and supplement type. But which analysis should we use to answer this question? Let’s find out using the Statkat module. The Statkat module contains three options: Single Variable, Correlational Research, and Repeated Measurements.

Since we are interested in whether there is a relationship between tooth length and supplement type, we choose the Correlational Research option. You should also choose this option if you want to predict a variable from other variables or if you are interested in the difference between independent (unrelated) groups on a certain variable (whether there is a difference in tooth length between the two unrelated supplement groups, is just a different formulation of the example question).

• Choose the Single Variable option if you are interested in only a single variable, for instance, the mean tooth length.
• Choose the Repeated Measurements option if you are interested in differences between related variables, like differences between repeated measurements over time, or differences between scores of women and their husbands.

As long as there are no variables defined for Variable 1/Dependent Variable and Variable 2/Independent Variables, the output field contains general instructions on how to use the Statkat module. The most important instruction is that the measurement levels of the variables should be set correctly, since otherwise you will end up with wrong method recommendations.

For a method recommendation given our research question, we drop the tooth length variable len in the box below Variable 1/Dependent Variable field, and the supplement type variable supp in the box below Variable 2/Independent Variables.

Based on the measurement level of the two variables, Statkat recommends to use a two sample t test, or, if the normality assumption is violated, the Mann-Whitney U test. If you need a quick refreshment of what the recommended methods are about, click on the relevant links provided in the output. The links will lead you to neat method-overview tables at the official Statkat website.

## Variable 1/Dependent Variable or Variable 2/Independent Variables: Which field to use?

If your research question does not make a clear distinction between an independent variable and a dependent variable, the decision of which variable to define as Variable 1/Dependent Variable and which as Variable 2/Independent Variables can be arbitrary. But doesn’t this decision affect the recommended method? Well, in some cases it does affect the primary method recommendation, but if a simpler method can be performed by flipping the two variables, this is usually mentioned. It is then up to you which of the recommended methods you prefer. It is important to keep in mind here that none of the correlational statistical techniques can say anything about causality anyway (not even a method like regression analysis), so even if you do make a distinction between an independent and dependent variable, the statistical method will only say something about association, not causation.

## Conclusion

We hope that the Statkat module will make selecting an appropriate statistical method easier. If you don’t have any data yet or if you prefer a decision tree to find an appropriate method, then Statkat’s method selection tool can be of help.


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## Algebra 2 (1st Edition)

$\displaystyle \sqrt{\frac{18}{11}}=\frac{3\sqrt{22}}{11}$
$\sqrt{\frac{18}{11}}\qquad$ ...apply the Quotient Property:$\displaystyle \sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$ $=\displaystyle \frac{\sqrt{18}}{\sqrt{11}}\qquad$ ...rewrite 18 as a product of two factors so that one factor is a perfect square. ($18=9\cdot 2$) $=\displaystyle \frac{\sqrt{9\cdot 2}}{\sqrt{11}}\qquad$ ...use the Product Property of square roots in the numerator:$\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}$ $=\displaystyle \frac{\sqrt{9}\cdot\sqrt{2}}{\sqrt{11}}\qquad$ ...evaluate part of the numerator ($\sqrt{9}=3$) $=\displaystyle \frac{3\sqrt{2}}{\sqrt{11}}\qquad$ ...rationalize the denominator by multyplying both the numerator and the denominator with $\sqrt{11}$. $=\displaystyle \frac{3\sqrt{2}\cdot\sqrt{11}}{\sqrt{11}\cdot\sqrt{11}}\qquad$ ...use the Product Property $\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}$ $=\displaystyle \frac{3\sqrt{2\cdot 11}}{\sqrt{11\cdot 11}}\qquad$ ...simplify.($\sqrt{11\cdot 11}=11$). $=\displaystyle \frac{3\sqrt{22}}{11}$

format: plain


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I've tried to follow your hedge fund thinking, and at this point, it seems like you are needing a handful of sorta institutional-grade algos that you can fund at $5M or so each, versus an innovative crowd-sourced fund with widespread participation (but at lower levels of capital per algo). If you are "looking for very consistent positive returns month over month and year over year" then in my mind, you are looking for effectively high-yield bank CDs (return like (1+r)^n). One of your goals, as I understand, is to have uncorrelated return streams, but in the limit you are talking about, all of the strategies will be highly correlated in their returns--they'll all just go straight up! Yes, it'll be crowd-sourced, but in name only. If you could figure out a way to get perhaps 10,000 of your 45,000+ users on board, and get cash flowing to them, then it'd actually be something. There would be a relatively low threshold to get$5K-$10K of working capital, that would grow as real-money results flow in (or decrease as the algo falls apart). By the way, in the spirit of transparency and crowd-sourcedness, do you have any outside investors lined up? Or are your VCs willing to supply an initial round? And I've never seen any potential investors post to this site? Where are they? How do we know that you are looking at the right market of investors? Maybe there is a cohort of investors that actually doesn't want consistent returns, and would trade risk for potential reward? Do you ever get inquiries along these lines, or do all of your candidate investors want CD-like returns? I understand that the "pure alpha" scalable institutional money is what you are aiming for, but I have to figure that there are 1%-ers who would like something more exciting for the speculative portion of their holdings. 26 responses I'd also be curious if you've gotten any feedback from potential institutional investors on the black-box nature of the algos that would be in the fund? Are they comfortable with this approach? Grant, you've got several questions bound up in those two paragraphs. i'm going to try to tease them apart. First, I want to again say that I don't think that it's productive to think of the hedge fund as a high-yield bank CD. I've tried to point out before that the expected returns of those two instruments are different by an order of magnitude. I'll expound on that: No one would ever borrow money to put it into a high-yield CD. The structure of the market is that the borrow rate is higher than the CD rate; it's a losing proposition. But if we're building the hedge fund right, then it does make sense to borrow money, in the form of leverage, and invest in the hedge fund. We expect to have institutions that, for instance, put$100M in capital to work with our hedge fund, and choose to use 3X leverage so that the fund is actually managing 300M in investments for them. I don't see that happening with ETFs. The next question is whether, at limit, will the strategies all be highly correlated. I suppose, ad extremum, we could be confronted with a host of algorithms that each go up exactly .05% per day. If that happened we'd have to develop other tools to figure out the degree of correlation. In practice, though, the world is a lot messier. Algos go up a a few points one day, down the next, and up again. An algorithm can be quite predictable over a long period, but that doesn't mean it's going up every day, exactly like every other desireable algo. The fact that they go up differently means we can evaluate their correlation. Next I want to tackle is crowdsourcing. Your first sentence characterizes the choice as one of "institutional grade algos versus an innovative fund." I think that's a false choice. We're actually doing both. We are building a quant hedge fund from crowd-sourced algorithms. That is innovative, despite my lack of modesty in saying it so baldly. I agree, it would be both neat and awesome if we could get 10,000 of our community members allocations from the hedge fund. Unfortunately, we don't have that many members writing high-quality algorithms that we can invest in (yet). You can download the contest CSV and look at the entries. There aren't 1000 good algos, let alone 10,000. If we get there, we'll fund them, and that will be the thrill of a lifetime for us all. You don't quite say it explicitly, but I think you're saying that it doesn't count as crowd-sourcing if we use algorithms from a few dozen people rather than a few thousand. Does Kaggle count as a crowd-sourcing solution? I think that's a semantic argument, and probably not a very interesting one. We have not signed any institutional investors. (Also, to be clear, we are not soliciting nor advertising for them either.) We've had informal talks with a number of people in the industry, and the response has been very positive. When you get down to it, a lot of these deals boil down to "what's the price?" We're doing a lot of work to lay the foundation for the fund so that when it comes to sign the deal, we get a price for the money that we like. That day isn't here yet. I think there are a lot of parallels between the institutional money and VC money. Any of the questions you ask apply to VC and institutional investors alike. Why aren't they members of the community? I can't speak for them, but my guess is that because it's not their community. We're a community of quants, not institutional investors. How can you (or I, for that matter) know that we're building a product they will invest in? That's not something that is provable beforehand. We are, I assert, building something that has never been built before. You can never be sure it will work until it already has worked. What you can count on, though, is that our interests are aligned. We only make money if the community makes money, and vice versa. Are there investors out there who are looking for something like us, but not exactly us? There certainly are. However, we can't be everything. One of the best ways to make a startup fail is to lose focus and to try to do too much. We've talked to a lot of smart people, collected a lot of advice and experience, and we've made a choice. The response we've gotten supports that choice. If the data starts running the other way, we'll evaluate it when we get there. Finally, your question about the black-box algos. It hasn't been an issue with investors yet. It's weird, but they are used to investing in black boxes. I'm not too worried about it, either way. As we negotiate contracts with algorithm authors, I see granting the right for someone to review the algorithm code as one of the potential terms of the contract. We will see how that one shakes out. Disclaimer The material on this website is provided for informational purposes only and does not constitute an offer to sell, a solicitation to buy, or a recommendation or endorsement for any security or strategy, nor does it constitute an offer to provide investment advisory services by Quantopian. In addition, the material offers no opinion with respect to the suitability of any security or specific investment. No information contained herein should be regarded as a suggestion to engage in or refrain from any investment-related course of action as none of Quantopian nor any of its affiliates is undertaking to provide investment advice, act as an adviser to any plan or entity subject to the Employee Retirement Income Security Act of 1974, as amended, individual retirement account or individual retirement annuity, or give advice in a fiduciary capacity with respect to the materials presented herein. If you are an individual retirement or other investor, contact your financial advisor or other fiduciary unrelated to Quantopian about whether any given investment idea, strategy, product or service described herein may be appropriate for your circumstances. All investments involve risk, including loss of principal. Quantopian makes no guarantees as to the accuracy or completeness of the views expressed in the website. The views are subject to change, and may have become unreliable for various reasons, including changes in market conditions or economic circumstances. Investors trust fund managers. Well, investors trust fund managers' records. Without a record, you're nothing. Q-Fund won't have a record for a year or more. Not a record that major money will consider adequate for establishing a trust relationship at least. Now, what fund managers trust to create records (the algos) that investors can be sold, that is what is being tested here. Can 100, 1000 or more anonymous quants build algos that a fund manager can assemble into a system which will prove out through a sustainable, viable, vettable record? That is the experiment that Q represents. So far the record is, well, the record is non-existent. Yet so far the diversity of algos seem to have hit an evolutionary bottleneck. A viable ecosystem relies upon diversity to ensure shocks to the system don't destroy the sustainability of the system. Lack of genetic variation spells evolutionary dead ends. The more and more restrictive the algo purity tests become the less and less resilient the system becomes as genetic material gets filtered out. Can the Q-Fund manager take such a rarefied species of algo and build a robust, stalwart profit system? We'll see, in about a year or more. Not been done before? Not that has been publish publicly, that we've all heard of -- so far. Maybe not. I can't imagine the likes of Citadel or its ilk not having serious quant competitions -- internally -- that would have accomplished effectively the same intent that the Q is shooting for. Hundreds of top-notch quants all vying for a spot in the limelight? Yeah, it's been done before. But only the Q will, that is may, provide a window into the process. For all we know though, when the million hit the Q-machine the curtains may fall and all the winning quants become hired guns, the website gets internalize, the forum goes dark and the Q's VCs insist on zipped lips and high walls. Money trumps all good intentions.

(Oh, happy Saturday Jonathan.)

Say Q could offer a product that returned X% year-over-year, with zero volatility, and could almost guarantee that investor's capital would be safe. It'd smell like a CD, right? And it would be very attractive to investors. So, the more the Q product behaves like a CD, the better. In fact, if I look at Pravin's algo on https://www.quantopian.com/posts/the-process-of-naming-the-september-prize-winner, it has some choppiness, but it basically follows a straight line, just like a CD would. It looks like noisy compound interest to me, (1+r)^n, with some variability in the return, r. You are rewarding algos with CD-like returns, and if you build a fund out of a bunch of algos with CD-like returns, it'll look even more like a CD. Am I missing something? Wouldn't the ideal pure-alpha fund look like a high-yield CD (returning >10% per year, adjusted for inflation, etc.)?

I'm not exactly sure what the world's first crowd-sourced hedge fund would look like, but it doesn't feel like you are on the right path. As Market Tech points out, it seems you are asking the crowd to develop algos similar to what hedge funds must ask their R&D staff to be working on all the time. So, there's no innovation there. The innovative part, I suppose, is that you are getting the R&D tools out to the masses, at low cost and at a global scale. My impression (perhaps wrong) is that you are being pretty darn conventional in what you are asking users to develop with those tools. The risk is that you'll end up with the same thing that every other hedge fund produces. But to Market Tech's point, maybe that's what you desperately need at this point. In a year or so, you'll need to put together a slide pack that shows that you turned $25M into$27.5M, and that all indications are that the trend will continue. The easiest way to get there is to put $5M into 5 algos that are low-risk, conventional approaches. The question is how do you engage all 45,000 users and get to$10B? Then you'd be doing something new.

Regarding participation from the money side, you must know where your first injection of capital would come from (sounds like VCs). Why not eliminate the middle man, and just have them tell us what they need to put up $25M (or whatever the number is)? Or at least share the slide packs that you are presenting to them. ...negotiate contracts with algorithm authors, I see granting the right for someone to review the algorithm code as one of the potential terms of the contract. This doesn't sound like a approach that will scale to 10,000 crowd-sourced algos, with "managers" coming and going on a daily/weekly/quarterly/yearly basis. It sounds more like the Q platform is a recruiting tool for a handful of highly skilled quants, who can write high-capacity institutional-grade algos that you can pour money into. In my mind, if you are truly crowd-sourcing, there would be no way to negotiate on an individual basis (although I could imagine a standard form, with the choice to share code or not as a button to be clicked). your relentless skepticism is out of place with the rest of the board Hello Tybalt, Believe it or not, I'd like to see Q succeed wildly (Dan and others know this). If there is something I've said that you think is bogus or if you have a different spin, feel free to comment. Grant As an entrepreneur I can tell you that you'll never get anywhere if you ignore or try to shut down the Grant's of the world who ask the probing questions and apply some rigor to the thought process. That type of thinking is what gets you ahead in big organizations (I did work for the government for some time), but it's the kiss of death to an entrepreneur and the reason why startups can take on and win against big organizations. To add some skepticism of my own, I actually took a several month break from Quantopian because of the exact phenomenon Grant and Market Tech are describing. Tightening the parameters so much that they end up looking for this holy grail of 10% nearly risk free yield in a 1% risk free rate world using nothing but equities. And not even all equities because some are apparently bad. Trading on minute bars. With completely unreasonable commission rates. And a fill model that doesn't match reality. Maybe they can find one or two algorithms that can achieve this, but so far they haven't. Even if they do, as Market Tech points out it will likely be a case of multiple discovery of the same underlying phenomenon, which doesn't give you robustness. I don't see that Quantopian has done any investigation of the portfolio effect of multiple, completely different algorithms that don't necessarily independently meet these crazy, must be long and short, must trade all the time, must have super low beta... type requirements. Not the anti-gaming rules, I'm all for that, the "ensuring CD like returns" rules. Not only has Q never mentioned doing this type of analysis, but there are only probably two people on Q's staff who would have the background to do it. It's still a software company, which is great for me because I get a ton out of the free platform. However the software part will only be around until the VC cash is gone or the company because self-sustaining, so we all have a vested interest in them reaching that goal. The portfolio concept is fundamental to modern finance, and its a shame that Quantopian is shutting down such a promising avenue in search for a much more elusive goal that has a far lower chance of success. It's completely throwing away the concept that it's far more likely that you'll find 10 algo writers whose independent strategies, when combined, produce the holy grail, than the chance that one algo writer will strike on all 10 of those concepts and put them together in one algo. Or the even smaller chance that you'll have tens or hundreds of individual algo writers who can do this. The idea that when you combine an algo with a huge positive beta and one with a huge negative beta, you could end up with very low beta, for example. Even better you combine 5 very positive and 5 very negative beta's. The idea that an all long fund and an all short fund, when combined, could produce the same results as an enforced long/short fund. Bottom line, as I've been saying for about a year, Q really doesn't know what they don't know about finance. They're all super smart, no doubt about it, and it's a trap I recognize that just because you're smart in one area doesn't mean you automatically are in another, even if its related. The danger here is that they'll paint themselves into a corner where a bunch of their promising algo writers leave, or at least take a break from the site, they don't find the holy grail with what remains, and they run out of money before they can pivot. I sincerely hope this doesn't happen. I don't want to be the complainer without suggestions, so here are a couple. Maybe now that we've stepped away from monthly contests, you hold two 6 month contests, once with all the crazy parameters and one with only the anti-gaming rules? Even better, you put together a platform that allows "crowd portfolio managers" to pick from any combination of the contest algo's to put together an algo that itself is in the contest. That way you don't even have to hire someone with portfolio selection experience, you can use the crowd which is in line with the company's DNA. Thanks Kevin, Yeah, my concern is that Q is on a "me too" path with respect to their business strategy, which means that they'll have nothing in the end, if they are looking for strategies that match what is already out there on the market. There needs to be a significant differentiator, which just doesn't leap out to me (but I'm not in the financial field, so maybe the hedge fund industry is shaking in its boots). Grant Hi Dan, So do you have a commitment of$25M or so for the hedge fund, to get it off the ground? And if so, what do you need to show to get the money? Or is this whole thing still a dream? Market Tech is likely correct that for any big money, you'll need something like a 3-year track record, so we are a long ways off to get to any scale. You seem to suggest that you haven't started engaging institutional investors (since they'd say "Let's see your track record" and you'd be lucky to get a free cup of coffee from them). So, I think there is a consistent theme here.

What's the story? You are asking community members to put time into writing super-duper algos. Do you have money lined up? Or are we still in the "pull ourselves up by our own bootstraps" phase?

On a separate note, do you ever get inquiries from individuals (or maybe "family offices") with money they'd like deploy to an algo on Quantopian? If so, how do you respond to them? Why did you decide to go the hedge fund route, versus more of a peer-to-peer model, where Q acts as a facilitator/intermediary rather than "The Man" (in this respect, you've taken a conventional "Wall Street" approach versus something new/innovative/open)?

Grant

I work for a large hedge fund and amongst other things am involved in seeding new managers. so on the condition that I will answer what I can, please feel free to ask me any questions.

Thanks Tybalt,

I guess the main question is does Q have a viable business strategy with their hedge fund concept, as it has been laid out? If there are shortcomings, what are they and how could they be remedied? Will they end up having any advantage in the hedge fund marketplace? Will their product be unique?

This is not to be overly pessimistic, but if they are headed toward a cliff, then they should realize it.

Grant

One thing to consider is the product, the story, and who it will appeal to.

Institutional investors have v different metrics to retail investors. Both are valid ways of attracting capital.

Another is the type of fund they wish to launch, how often will subscriptions be allowed (daily, monthly etc), a 40 act fund, a UCITs fund, an offshore fund?

What they need to do, and I believe they are trying, is putting together a suit of super safe algos that they can show to somebody with capital to deploy and convince them they will not lose al of their money.

This game is about building a real track record for 1 to 3 years, and then growing assets at various intervals that opens you up to different tiers of investors.

Whether I deem their product viable or not is not really the question, the fact they have raised finance a few rounds says there is the appetite to at least TRY!

IMO they will have no problem raising a few million to deploy using the selected Algos. I know of a lot less sensible ideas getting money!

Thanks Tybalt,

Yeah, the "putting together a suit of super safe algos" is the part I'm actually worried about. Does it sound any different from what you would do in the hedge fund you work for? If they end up making and selling the same lemonade as everyone else on the street, then what's the advantage? I thought the safety part would come from pooling together lots of algos, with a little money deployed to each. Or maybe I don't yet grasp the crowd-sourcing part.

Grant

Is the difference not simply that they are drawing their talent from a population which is different from street professionals? Why is that not difference enough?

I feel like you have some expectation of them to be patrons of the quant community (beyond offering free tooling and data), when I think their role might be better described as talent scouts...

Yes, true. I just can't quite put my finger on what will be different in the outcome. Maybe Q's costs will be lower, so they'll be able to offer a better deal? Or the performance will be superior? Or the risk will be lower, due to greater diversification? Or street quants will be more creative that professionals, and uncover unbeknownst gems? In the end, as a start-up, Q has to have a unique edge, right? What will it be?

To understand a new or foreign paradigm I often rely upon finding an analogous situation. For instance, what if we were talking about paintings as art -- Artopian.

Initially, Artopian would offer free easels , paints and brushes to any who care to participate. They claim that painting should be democratized, that Artopian will be the advocate for the underprivileged artist. That they will build a studio where they will take the best impressionist, expressionist, realists, abstract, etc. paintings and display them for the world to see and for art connoisseurs to peruse and to buy. What a great idea!. They will certainly be a unique art studio.

But time goes by and because the funders of Artopian, for whatever reason, feel that they would rather sell only realistically rendered painting that they change the offering to their open studio to only allow realism paintings. Painters of impressionism, expressionism, and abstraction will not be allowed to display their art in the studio. They can still use the free tools and supplies, but they are banned from the studio.

Bummer. And we thought they were going to be the champion of young artists everywhere. Now only artists of Norman Rockwell'esque style will be shown in their studio. So much for a democratic representation of artists.

Ehhh, I'm not much of a realism painter. Forget it.

And we thought they were going to be the champion of young artists everywhere. . . . So much for a democratic representation of artists.

This is the part that I think is a problem. At the end of the day, their purpose is to sell art, and that is what finances the free easels and paints. If realism is what sells, it's churlish and frankly ungrateful for an impressionist to begrudge Artopian for not trying to sell their art. The photographers might appreciate using the free easels as well, yet don't feel so entitled that they are upset Artopian doesn't sell photographs. It's the impressionists and other non-realist painters specifically that feel persecuted or dismissed, arguably because they have unrealistic expectations.

Boring! "Announcing the Quantopian Crowd-Sourced 'Me Too' Pure Alpha Fund!" Doesn't sound very sexy, but with a cheap frame, maybe it'll sell at Wal-Mart and Target.

arguably because they have unrealistic expectations.

Those expectations having been set, originally, as a broad representational spectrum of art styles. And only later narrowing down to an exclusive method.

Expectations change, sure, but when they get set up front, and changed thereby filtering out most participants, one can only expect to see some rebuffed responses.

Oh I am not surprised to see them, I just disagree with them! :)

Reportedly, the bar ain't all that high to get Q hedge fund money. Sharpe > 0.7, 7% annual return minimum, beta +/- 0.3 or less, long-short in some fashion, and maybe a few other constraints. 10 year performance that isn't too ugly. There may be hope for you, Market Tech. Why don't you whip something up? Get out your paint brushes and easel, and slap some splotches on that canvas!

I'm afraid I would only embarrass myself. Besides I only know how to paint impressionist style and as we know, that's not what cuts here here anymore.

Market Tech,

You sell yourself short! Although, if the payout is unclear, it may not be worth the effort. In the next 2-3 years, there could only be 5-10 manager slots for Q hedge fund capital, unless there will also be seed money at $25K-$250K for more speculative algos that aren't institutional grade "stat arb" ones.

Grant

I don't see it as anything to worry about. Safe means you stay in the game, and while you stay in the game you earn your fees and can research more and more algos. :)

Tybalt,

Yes, if there is real money just waiting to put money into Q strategies that meet certain criteria, then by all means, strike while the iron is hot. And maybe there is no path to money for the Market Tech's of the world. Maybe it is obvious to you and others in the field that pure alpha CD-like strategies will win the day. But if it is obvious, then there is a risk, since it'll be obvious to everyone in the industry, and Q won't have a differentiated product (although Q may have more flexibility in fees charged to investors). On the other hand, I guess there is a real potential to very quickly get to 30 or more high-quality strategies that could then be mixed and matched into a much better offering than any other start-up hedge fund could pull off, with only a couple smart guys/gals and a few support staff. In the end, I guess that this is the Q "big idea"--scouting for talent globally, and being able to employ them quickly and at low cost. The threat to Wall Street is that talent scouting and acquisition has been innovatively globalized by Q on a grand scale. Cool. Nice work, Fawce & Co. See, I ain't so negative.

Grant

Honestly the differentiation is the story to begin with, the outperformance, or not, will come later. The story is enough to get people interested I am sure.


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# 'Cases' command does not recognize specified patterns

I am having a problem with the "Cases" command. If I execute the following code

Cases[(32 W[2, 0])/5  + 32 W[2, 0]^2, W[_, _] | W[_, _]^2, {0, 2}]
Coefficient[(32 W[2, 0])/5  + 32 W[2, 0]^2,
Cases[(32 W[2, 0])/5  + 32 W[2, 0]^2, W[_, _] | W[_, _]^2, {0, 2}]]

then the outputs are as expected:

{W[2, 0], W[2, 0]^2}
{32/5, 32}

However, the following similar code

Cases[a/(3 + c)  W[2, 0] W[2, 3], W [__, ___] W[_, __], {0, 1}]
Coefficient[a/(3 + c) W[2, 0] W[2, 3],
Cases[ a/(3 + c) W[2, 0] W[2, 3], W [__, ___] W[_, __], {0, 1}]]

yield nothing in the outputs. Specifically, I got:

{}
{}

The expected outputs should be:

{W[2, 0] W[2, 3]}
{a/(3+c)}

If I get rid of the coefficient 'a/(3 + c)' then this command works

Cases[ W[2, 0] W[2, 3], W [__, ___] W[_, __], {0, 1}]
Coefficient[W[2, 0] W[2, 3],
Cases[ W[2, 0] W[2, 3], W [__, ___] W[_, __], {0, 1}]]

and the output is as expected:

{W[2, 0] W[2, 3]}
{1}

Is there any way to get the expected output for the second code block above? Thanks in advance for any help or suggestions !

• I don't have an immediate solution but you can see why this doesn't work by using FullForm. The FullForm of your expression is Times[a,Power[Plus[3,c],-1],W[2,0],W[2,3]] and that of the pattern you're looking for is Times[W[Blank[],BlankSequence[]],W[BlankSequence[],BlankNullSequence[]]]. This doesn't match since the Times[...] in the expression has more arguments than in the pattern. Oct 15 at 10:19
• change it to Cases[a/(3 + c) W[2, 0] W[2, 3], any_*W[__, ___] W[_, __], {0, 1}] then it will work. But I do not know about your use of Coefficient that follows. Oct 15 at 10:25
• First, you need HoldPattern. But there is still a problem. Consider, if we give a list instead of an expression: Cases[{a/(3 + c), W[2, 0] W[2, 3]}, HoldPattern[ W[_, _] W[_, _]], 1] it works. But if we change the List to Times: Cases[Times @@ {a/(3 + c), W[2, 0] W[2, 3]}, HoldPattern[ W[_, _] W[_, _]], 1]  it does not work. Please report this to : "support@wolfram.com" and post their answer herr. Oct 15 at 10:33
• @DanielHuber I don't think this is a bug. Again, by using FullForm we can see what's going on. The FullForm of the list is List[Times[a,Power[Plus[3,c],-1]],Times[W[2,0],W[2,3]]] and that of the times expression is Times[a,Power[Plus[3,c],-1],W[2,0],W[2,3]]. Only the list-expression matches the pattern. Oct 15 at 11:11
• @banone Hi, according to the help, Casesshould work on list as well as expressions: see Cases[expr,pattern,levelspec] Oct 15 at 12:38


text:
# TEX 10 Pick Hammer Uses a Light Touch

Tue July 03, 2007 - National Edition

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text:
zbMATH — the first resource for mathematics

Averaging of symmetric diffusion in random medium. (English. Russian original) Zbl 0614.60051
Sib. Math. J. 27, 603-613 (1986); translation from Sib. Mat. Zh. 27, No. 4(158), 167-180 (1986).
The accuracy of convergence is studied of the solutions $$u_{\epsilon}$$ of the elliptic equations depending on a random parameter $$L_{\epsilon}u_{\epsilon}=f$$, $$x\in Q\subseteq {\mathbb{R}}^ d$$, $$u_{\epsilon}=g$$, $$x\in \partial Q$$, $$\epsilon >0$$, to the solution of the averaging problem $$\bar LU=f$$, $$x\in Q$$, $$U=g$$, $$x\in \partial Q$$ where $$L_{\epsilon}u=(1/2)D_ i(a_{ij}(y,\omega)D_ ju)$$, $$D_ i=\partial /\partial x_ i$$, $$y=x/\epsilon$$, $$\bar Lu=(1/2)\bar a_{ij}D_ iD_ ju$$, $$\bar a{}_{ij}$$ constants, $$d\geq 3$$, and $$L_{\epsilon}$$ is approaching $$\bar L$$ using the corresponding resolvent operators.
The diffusion coefficients are supposed to be homogeneous, $$a_{ij}(y+z,\omega)=a_{ij}(y,\omega)$$, $$z\in {\mathbb{Z}}^ d$$, strictly elliptic, $$C_ 1| \xi |^ 2\leq a_{ij}(y,\omega)\xi_ i\xi_ j\leq C_ 2| \xi |^ 2$$, $$\xi \in {\mathbb{R}}^ d$$, and to satisfy an ergodic property with respect to the random parameter $$\omega\in \Omega$$. The main theorem states the following estimate $E| u_{\epsilon}-U|^ 2_{\infty}\leq \tilde C\epsilon^{\gamma},\quad \tilde C,\gamma >0$ where $$| v|_{\infty}=\sup \{| v(x)|$$, $$x\in Q\}$$.
Reviewer: C.Vârsan

MSC:
 60H15 Stochastic partial differential equations (aspects of stochastic analysis)
Full Text:
References:
 [1] S. M. Kozlov, ?Averaging of random operators,? Mat. Sb.,109, No. 2, 188-202 (1979). [2] V. V. Yurinskii, ?On averaging of elliptic boundary value problem with random coefficients,? Sib. Mat. Zh.,21, No. 3, 209-223 (1980). [3] V. V. Zhikov, S. M. Kozlov, O. A. Oleinik, and Ha Tien Ngoan, ?Averaging and G-convergence of elliptic operators,? Usp. Mat. Nauk,34, No. 5, 65-133 (1979). · Zbl 0445.35096 [4] V. V. Yurinskii, ?On averaging of diffusion in a random medium,? Teor. Veroyatn. Primen.,29, No. 3, 607 (1984). [5] V. V. Yurinskii, ?On averaging of diffusion in random medium,? Tr. IM SO AN SSSR,5, 75-85 (1985). [6] I. P. Kornfel’d, Ya. G. Sinai, and S. V. Fomin, Ergodic Theory [in Russian], Nauka, Moscow (1980). [7] I. I. Gikhman and A. V. Skorokhod, Introduction to Theory of Random Processes [in Russian], Nauka, Moscow (1977). · Zbl 0429.60002 [8] J. Neveu, Mathematical Foundations of the Calculus of Probability, Holden-Day (1965). · Zbl 0137.11301 [9] D. G. Aronson, ?Nonnegative solutions of linear parabolic equations,? Ann. della Scuola Normale Superiore di Pisa, Ser. III,22, No. 4, 607-694 (1968). · Zbl 0182.13802 [10] Yu. V. Kazachenko and M. I. Yadrenko, ?Local properties of sampling functions of random fields. I,? in: Theory of Probability and Mathematical Statistics [in Russian], No. 14, Vishcha Shkola, Kiev (1976), pp. 53-66. [11] O. A. Ladyzhenskaya and N. N. Ural’tseva, Linear and Quasilinear Elliptic Equations, Academic Press (1968). [12] I. A. Ibragimov and Yu. V. Linnik, Independent and Stationarily Connected Quantities [in Russian], Nauka, Moscow (1965). · Zbl 0154.42201 [13] S. M. Kozlov, ?Conductivity of two-dimensional random media,? Usp. Mat. Nauk,34, No. 4, 193-194 (1979). [14] O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc. (1968).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


text:
Area of a sector IB Maths SL

Algebra, Functions and equations, Circular functions and trigonometry, Vectors, Statistics and probability, Calculus. IB Maths SL Revision Notes

Area of a sector IB Maths SL

IB Mathematics HL– Trigonometry, Mensuration, Area of a sector

How can we find the area of sector with central angle $\frac{\pi}{6}$ in a circle with radius 7 cm?

Thanks
Jane

Posts: 0
Joined: Tue Mar 19, 2013 8:47 pm

Re: Area of a sector IB Maths SL

IB Maths HL – Trigonometry, Mensuration, Area of a sector

Let $\theta$ (in radians) be the central angle in a circle with radius r, then the area $A$ of the sector formed by angle $\theta$ is given by the following formula:

$A=\frac{1}{2}r^2 \theta$

Therefore the arc length in our case is

$A=\frac{1}{2}(7)^2 \frac{\pi}{6} =\frac{49\pi}{12} cm^2$

Hope these help!!
miranda

Posts: 268
Joined: Mon Jan 28, 2013 8:03 pm

Re: Area of a sector IB Maths SL

Thanks a lot!
Jane

Posts: 0
Joined: Tue Mar 19, 2013 8:47 pm


text:
# nLab digraph

Contents

This page is about the concept of directed graphs that is usual in combinatorics. For the notion of directed graph as commonly understood in category theory, see quiver. For some commentary on how the two formalizations relate to one another, see directed graph.

$\,$

#### Graph Theory

graph theory

graph

category of simple graphs

# Contents

## Idea

In combinatorics, a digraph (a shortening of directed graph) consists of a set and a binary relation on that set that is irreflexive. The elements of the set are called nodes or vertices, and elements of the relation are called edges or arcs; the idea is that whenever an ordered pair $(x, y)$ belongs to the relation, then we depict it as an arrow or directed edge going from node $x$ to node $y$. The irreflexivity condition means there is never an edge from a node to itself.

## Definition and basic notions

In this section we collect some definitions that are fundamental to digraph theory, but presented largely from the point of view of category theory.

###### Definition

(digraph)

A digraph is a pair $(V,A)$ of sets, with $A\subseteq (V\times V)\setminus\{ (v,v)\colon v\in V\}$. Here, $V\times V$ denotes the product, and ${}\setminus{}$ the relative complement in the category of sets. The elements of $V$ are called vertices, the elements of $A$ are called arcs. For $a = (v, w)$ an arc, we call $v$ the source of $a$ and $w$ the target of $a$, also denoted $s(a)$ and $t(a)$ respectively.

A morphism from a digraph $(V, A)$ to a digraph $(V', A')$ is a function $f: V \to V'$ such that $(f(v), f(w)) \in A'$ whenever $(v, w) \in A$. Digraphs and their morphisms form a category denoted $\mathsf{Dgr}$.

A digraph $(V, A)$ may be regarded as a quiver in an evident way, with source and target functions $s, t: A \rightrightarrows V$. It is straightforward that $\mathsf{Dgr}$ is (equivalent to) a full subcategory of the category of quivers Quiv. Of course most quivers are not digraphs since quivers can have parallel edges: distinct edges $a, b$ where $s(a) = s(b)$ and $t(a) = t(b)$.

Some useful examples of digraphs are based on ordinals:

###### Example

If $\alpha$ is an ordinal, we may form a digraph $(V, A)$ where $V$ is the underlying set of $\alpha$ and $A$ is the relation $\{(i, i+1): i, i+1 \in \alpha\}$. We call this an ordinal digraph and denote it by the ordinal itself.

###### Definition

(walk in a digraph) Suppose $D=(V,A)$ is a digraph. A walk in $D$ is a digraph-morphism $\alpha \to D$ where $\alpha$ is an ordinal digraph.

Notice that for $\alpha = 1$, the hom-set $\mathsf{Dgr}(1, D)$ may be identified with the set of vertices of $D$. The definition of graph morphism implies that the underlying set functor $U = \mathsf{Dgr}(1, -): \mathsf{Dgr} \to Set$ is faithful, and realizes $\mathsf{Dgr}$ as a concrete category.

For $\alpha = 2$, the hom-set $\mathsf{Dgr}(2, D)$ may be identified with the set of arcs of $D$. However, the arc-set functor $\mathsf{Dgr}(2, -): \mathsf{Dgr} \to Set$ is very far from being faithful.

If we let $C$ denote the full subcategory of $\mathsf{Dgr}$ consisting of the ordinal digraphs $1, 2$, then the inclusion $i: C \hookrightarrow \mathsf{Dgr}$ induces a functor $\mathsf{Dgr} \to Set^{C^{op}} = Quiv$ according to the usual “nerve” construction

$\mathsf{Dgr} \to Set^{C^{op}}$
$D \mapsto (c \mapsto \mathsf{Dgr}(i c, D))$

and this nerve functor is exactly the full embedding $\mathsf{Dgr} \to Quiv$. Since full embeddings reflect and preserve isomorphisms, a digraph morphism $f: D \to D'$ is an isomorphism iff the vertex-function $\mathsf{Dgr}(1, f)$ and arc-function $\mathsf{Dgr}(2, f)$ are isomorphisms (which is in any case obvious from the definition of digraph morphism, but this underscores the general importance of walks).

Generally we will only be interested in walks $\alpha \to D$ such that $\alpha \leq \omega$ in the sense of ordinals; here $\omega$ is of course the first infinite ordinal. Unwinding the definitions, here is an elementary description of such walks:

###### Definition

(walk in a digraph; elementary definition)

Suppose $D=(V,A)$ is a digraph. A walk in $D$ consists of

• a sequence $P\colon\alpha\rightarrow V$, with $\alpha\leq\omega$,

such that

• for all $i\in\alpha$, if $i+1\in\alpha$, then $(P(i),P(i+1))\in A$.

The phrase $\alpha$-vertex path means any path whose domain is $\alpha$.

Some special cases: If $\alpha=0$, then $P$ is called the empty walk. We have already seen that a $1$-walk is tantamount to a vertex, and that a 2-walk is tantamount to an edge. We call a walk $\omega \to D$ a ray in $D$.

###### Example

Another useful example of a digraph is a finite cyclic order. For each $n \geq 2$, there is a digraph $Z_n$ whose vertex set is the underlying set of the finite cyclic group $\mathbb{Z}/n$, and whose arc relation is the set $\{(i, i+1): i \in \mathbb{Z}/n\}$.

## Categorical properties of $\mathsf{Dgr}$

The category $\mathsf{Dgr}$ is not especially well-behaved for a category theorist. For example, it doesn’t have a terminal object (no, the ordinal digraph $1$ is not terminal), and it does not admit all coequalizers. In both cases the culprit is the irreflexivity condition.

It does however have products indexed over nonempty sets, and equalizers. These limits are computed as they are in Quiv, e.g., the quiver product of two digraphs is again a digraph and is their product in $\mathsf{Dgr}$. Such limits are easy to compute because $Quiv$, being a presheaf topos $Set^{C^{op}}$, has limits (and colimits) that are computed pointwise/objectwise.

It follows from these observations that $\mathsf{Dgr}$ has pullbacks which are simply $Quiv$-pullbacks; they are computed objectwise at the objects $c = 1, 2$, i.e., one simply takes pullbacks in $Set$ at each of the vertex and edge levels.

In our development, it will be useful to characterize various classes of monomorphisms in $\mathsf{Dgr}$. We begin with plain monomorphisms.

###### Proposition

A morphism of digraphs $f: D \to D'$ is monic iff its underlying function $U(f)$ is monic (injective).

###### Proof

It is well-known that faithful functors $U$ reflect monomorphisms and epimorphisms. For suppose $U(f)$ is monic. Given a diagram in $\mathsf{Dgr}$ of the form

$C \underoverset{h}{g}{\rightrightarrows} D \stackrel{f}{\to} D'$

if we have $f \circ g = f \circ h$ then $U(f) \circ U(g) = U(f) \circ U(h)$, whence $U(g) = U(h)$ by monicity of $U(f)$. Since $U$ is faithful, it follows that $g = h$; therefore $f$ is monic. The argument for epimorphisms is dual.

For the other direction, use the fact that $U = \mathsf{Dgr}(1, -)$ is representable. It follows immediately from the definition of monomorphism that representable functors preserve monomorphisms, so if $f$ is monic, then so is $U(f)$.

(Still to come: characterization of regular monomorphisms.)

## Paths, trails, and cycles

###### Definition

(path in a digraph)

Suppose $D=(V,A)$ is a digraph. A path in $D$ is a walk $P: \alpha \to D$ whose underlying function is injective.

Or what is the same, that $\mathsf{Dgr}(1, P)$ is a monomorphism in $Set$. In view of Proposition , we may also define a path to be a monic walk. (Notice this digraph concept of path is at variance with the concept of path in topology which is simply a continuous map $[0, 1] \to X$ into a topological space, where there is no assumption that its underlying function is injective.)

###### Definition

(trail in a digraph) Suppose $D=(V,A)$ is a digraph. A trail in $D$ is a walk $\alpha\overset{P}{\longrightarrow}V$ such that the partial function $e_P: \alpha\rightarrow A$ with $e_P(i)=(P(i),P(i+1))$ is injective.

The domain of $e_P$ is canonically identified with the arc set of $\alpha$, and the trail condition is equivalent to saying that $\mathsf{Dgr}(2, P)$ (which we again denote as $e_P$), as a function between edge sets $\mathsf{Dgr}(2, \alpha) \to \mathsf{Dgr}(2, D)$, is a monomorphism in $Set$.

We have the following easy result.

###### Proposition

In a digraph $D$, every path is a trail.

###### Proof

Since we just observed that a path $P$ is the same as a monic walk, and since representable functors like $\mathsf{Dgr}(2, -)$ preserve monomorphisms, it follows that $e_P = \mathsf{Dgr}(2, P)$ is a monomorphism.

Continuing the theme of monic digraph morphisms, we introduce the notion of cycle:

###### Definition

An $n$-cycle in a digraph $D$ is a monic morphism $P: Z_n \to D$ (see Example ). A cycle in $D$ is just an $n$-cycle in $D$ for some $n$. A digraph $D$ is acyclic if it admits no cycles.

There is a natural bijection between digraph morphisms $C: Z_n \to D$ and walks $P: \{0, \ldots, n\} \to D$ such that $P(0) = P(n)$. Indeed we have a digraph quotient map $q: \{0, \ldots, n\} \to Z_n$ defined as the mapping $i \mapsto i \mod n$, and the walk associated with $C: Z_n \to D$ is $P = C \circ q$. The map $q$ induces a bijection on arcs:

$\mathsf{Dgr}(2, q): \mathsf{Dgr}(2, \{0, \ldots, n\}) \stackrel{\sim}{\to} \mathsf{Dgr}(2, Z_n).$

Thus if $C: Z_n \to D$ is monic, we have a composite injective map in $Set$,

$\mathsf{Dgr}(2, \{0, \ldots, n\}) \underoverset{\mathsf{Dgr}(2, q)}{\sim}{\to} \mathsf{Dgr}(2, Z_n) \underset{\mathsf{Dgr}(2, C)}{\to} \mathsf{Dgr}(2, D),$

which implies that the walk $P$ associated with a cycle $C: Z_n \to D$ is a trail.

## Symmetrizing and “weak” notions

For various digraph (directed graph) notions above, it can be useful to consider parallel notions for undirected graphs. A slick method for doing this is by invoking a “symmetrizing” functor construction.

###### Definition

For a digraph $D$, we define $Symm(D)$ to have the same vertices as $D$, and the arc relation to be the symmetric closure of the arc relation of $D$. In other words, $(v, w)$ is an arc of $Symm(D)$ iff $(v, w)$ or $(w, v)$ is an arc of $D$.

Clearly the symmetric closure of an irreflexive relation remains irreflexive, so $Symm(D)$ is again a digraph, and the assignment $D \mapsto Symm(D)$ is the object part of an obvious functor $Symm: \mathsf{Dgr} \to \mathsf{Dgr}$. Also the monomorphism $D \to Symm(D)$ that is the identity on vertices is the component $u_D$ of a natural transformation

$u: 1_\mathsf{Dgr} \to Symm.$
###### Theorem

The functor $Symm$ carries an idempotent monad structure.

###### Proof

It is clear that $u_{Symm(D)}: Symm(D) \to Symm^2(D)$ is an isomorphism (and even an identity map) because the operation of taking the symmetric closure of a relation is itself idempotent. Similarly $Symm(u_D)$ is also an identity map: $u_{Symm(D)} = Symm(u_D)$. The multiplication $m: Symm^2 \to Symm$ of the monad is defined by taking the inverse: $m = u_{Symm}^{-1}$. Then $m$ is natural and the unit law $m \circ u_{Symm} = 1$ is automatic; the other unit law $m \circ Symm(u_D)$ follows. The associativity of $m$ follows by inverting a naturality square for $u$.

Morally we may consider the category of undirected graphs to be the category of algebras over the monad $Symm$. In any case, this is a full subcategory of $\mathsf{Dgr}$ consisting of digraphs of the form $Symm(D)$.

We proceed to “weaken” various of the notions above to take into account the undirected graph context, simply by applying the functor $Symm$. For example:

• If $\alpha$ is an ordinal digraph, a weak walk $\alpha \to D$ is defined to be a walk $\alpha \to Symm(D)$.

(Obviously, a weak walk $\alpha \to D$ need not be literally a walk $\alpha \to D$; cf. red herring principle.)

Alternatively, we could define a weak walk to be a digraph morphism $Symm(\alpha) \to Symm(D)$ where $\alpha$ is an ordinal digraph: the two ways of defining weak walks are (naturally) equivalent.

Similarly,

• A weak trail $\alpha \to D$ is defined to be a trail $\alpha \to Symm(D)$.

• A weak cycle $Z_n \to D$ is defined to be a cycle $Z_n \to Symm(D)$.

## Miscellaneous notions

In contemporary mathematics, ordered pairs $(x,y)$, $(y,x)$ are called antiparallel arcs.

###### Definition

(in-neighborhood, out-neighbourhood)

Suppose $D=(V,A)$ is a digraph and $v\in V$. The in-neighborhood of $v$ in $D$, denoted $N^{\rightarrow}_D(v)$, is the set $\{ u\in V\colon (u,v)\in A\}$. The out-neighborhood of $v$ in $D$, denoted $N^{\leftarrow}_D(v)$, is the set $\{ u\in V\colon (v,u)\in A\}$.

We note that the rationale for the slightly non-standard notation $N^{\leftarrow}_D(v)$ is to have it be intuitively understandable: the arrow points away from the vertex and points to the letter which stands for the set which contains the out-neighbours.

###### Definition

(in-degree, out-degree)

Suppose $D=(V,A)$ is a digraph and $v\in V$. The in-degree of $v$ in $D$, denoted $d^{\rightarrow}_D(v)$, is the cardinality of the in-neighborhood of $v$ in $D$. The out-degree is defined analogously.

The axiom implies—digraphs being irreflexive— that always $\neg \, (P(i)=P(i+1))$. Walks are only allowed to be non-injective every other step, so to speak.

###### Example

In a sense the least injective walk is obtained by considering any digraph containing a 2-cycle (see below) $C$, taking $\alpha=\omega$ and defining $P$ to alternate back and forth on $C$ forever. This walk has a two-element image.

###### Definition

(start vertex, $A$-$B$-walk in a digraph)

Suppose $D=(V,A)$ is a digraph. Suppose $\alpha\overset{P}{\rightarrow} V$ is a non-empty walk in $D$. Then $P(0)$ is called the start vertex of $P$. Suppose that $A,B$ are subsets of $V$, not necessarily disjoint. Then $P$ is an $A$$B$-path if and only if $\alpha\gt 0$ is a finite ordinal and $P(0)\in A$, $P(\alpha-1)\in B$ and $0\lt i\lt \alpha-1$ implies $\neg\, (P(i)\in A\cup B)$. If $A=\{a\}$ and $B=\{b\}$ are singletons, one also writes $a$$b$-path for $\{a\}$$\{b\}$-path.

Discussion of weak trails from a previous revision mentioned Power’s proof of unique interpretability of pasting schemes.

The standard term in topology for a path which is free of self-intersections is arc (eg Kowalksky 1965, Definition 29b). As a saving grace, the standard way to rigorously define the arcs of plane digraphs uses precisely the topological arcs.

Allowing 2-cycles, yet neither loops nor parallel arcs, has become a standard meaning of digraph in combinatorics.

###### Definition

(acyclic digraph (DAG); classical definition)

Suppose $D=(V,A)$ is a digraph. Then $D$ is called acyclic if and only if there does not exist any cycle in $D$.

A usual abbreviation in combinatorics for acyclic directed graph is DAG.

Here, “classical” refers to the negative definition.

The set of all weak cycles of a digraph $D$ is denoted $\mathrm{WeCy}(D)$.

The following non-standard definition is useful in when constructing proofs of Power’s pasting theorem:

###### Definition

(constructive acyclic digraph (cDAG))

A constructive acyclic digraph is a triple $D=(V,A,z)$ of data

• a digraph $(V,A)$
• a function $z\colon\mathrm{WeCy}(D)\rightarrow V$

subject to the axioms

• for each $P\in \mathrm{WeCy}(D)$, $z(P) \in \mathrm{image}(P)$

• for each $P\in \mathrm{WeCy}(D)$, if ( $u\in \mathrm{image}(P)$ and ( $(u,z(P))\in A$ or $(z(P),u)\in A$ ) ) , then $(u,z(P))\in A$.

In words, the axioms mean that $z$ gives us one vertex on any given weak cycle $P$ whose out-degree on $P$ is two. Then $z(P)$ witnesses $P$‘s not being a cycle. Evidently, a (classical) DAG can usually be made a cDAG in many ways: there usually are many functions $z$ one can specify and that satisfy the axioms. We are here not interested in describing or counting these possibilities, rather in doing things with a given cDAG.

There is an apparently inescapable arbitrariness in whether to favor out-neighbors or in-neighbor when defining constructive DAG. The above definition would to all intents and purposes be equivalent the one obtained by replacing the conclusion in the axiom with “then $(z(P),u)\in A$”.

The following concept appears (under another name) in Power’s proof:

###### Definition

(king, coking, $d$-king, $d$-coking, $\omega$-king, $\omega$-coking)

Suppose $D=(V,A)$ is a digraph. A vertex $v$ of $D$ is a king (resp. coking) of $D$ if and only if $(v,w)\in A$ (resp. $(w,v)\in A$) for all $w\in V\setminus\{v\}$.

For any finite cardinal $d$, a vertex $v$ of $D$ is a $d$-king if and only if each vertex of $D$ can be reached along a path, starting with $v$, the path having at most $d$ arcs.

For any finite cardinal $d$, a vertex $v$ of $D$ is a $d$-coking if and only if for each $u\in V$ there exists at least one path starting with $u$ and ending with $v$ and having at most $d$ arcs.

A vertex $v$ of $D$ is an $\omega$-king if and only if each vertex of $D$ can be reached along a path of $D$, starting with $v$, and having any finite number of arcs.

A vertex $v$ of $D$ is an $\omega$-coking if and only if for each $u\in V$ there is at least one path of $D$, starting with $u$, ending with $v$, and having any finite number of arcs.

The term $\omega$-king is a natural variant of the usual term $k$-king in digraph theory. By definition, for each $k\in\omega$, each $k$-(co)king is an $\omega$-(co)king. Of course, a vertex can be a king and a coking simultaneously. (Which is why e.g. in Power’s proof, non-equality of the king and the coking must be made a separate axiom in the definition of what Power calls plane graph with source and sink).

###### Definition

(plane digraph)

A plane digraph consists of the data

• a subset $V\subseteq\mathbb{R}^2$ of the plane
• a set $A$ of arcs (in the usual topological sense), each $a\in A$ of the form $[0,1]\overset{a}{\rightarrow}\mathbb{R}^2$

and the axioms

(v.connect) for each $a\in A$, $a(0)\in V$ and $a(1)\in V$,

(free.interior) if $a\in A$ and $0\lt t\lt 1$, then $\neg\,(a(t)\in V)$,

(not.par) if $a,a'\in A$ and if $a(0) = a'(0)$ and $a(1) = a'(1)$, then $a=a'$,

(disjoint.interiors) if $a,a'\in A$ and ( $\exists t:[0,1]$, $0\lt t \lt 1$ and $\exists t':[0,1]$, $0 \lt t' \lt 1$ and $a(t)=a(t')$ ), then $a=a'$.

Note that plane digraphs are often introduced into a context like (abstract) digraphs are, i.e. by writing $D=(V,A)$, although

• the type of $D=(V,A)$ is invisible from this notation (it may be a digraph or a plane digraph)

• for a plane digraph $D=(V,A)$ we do not in general have $A\subseteq V\times V$.

We make use of the treatment of undirected plane graphs given in (Diestel, Chapter 4), with some adaptations made for the present purposes:

• here, arc means arc in the usual topological sense, without exception, while, since the author focuses on undirected graphs, in Diestel, “arc” is defined to be a subset of the plane, in particular is not parametrized, in particular does not carry directional information (compare def. where said information is used).

• in (Diestel, Chapter 4), both $V$ and $A$ in the definition of plane graphs are required to be finite sets. Here we make no such restriction (cf. ).

• for ease of reference, we calibrate the point-set topological terminology according to Introduction to Topology (example: we will say “component” instead of “region”) when defining faces, and also Munkres 2000.

###### Example

An example of a plane digraph with countably infinite vertex set, and whose satisfying the axioms of plane digraphs is evident, is given by $T:=\{\varphi\in\mathbb{R}\colon 0\le \varphi\lt 2\pi,\quad \varphi\in\mathbb{Q} \}$, $D=(V,A)$ with $V:=\{(0,0)\} \cup \{ ( \cos( \varphi ) , \sin( \varphi ) ) \colon \varphi\in M \}$ and $a(\varphi)\colon [0,1]\rightarrow\mathbb{R}^2$ with $a(\varphi)(t) = ( t\cdot\cos(\varphi), t\cdot \sin(\varphi) )$ and $A:= \{ a(\varphi)\colon \varphi\in M \}$. Note that this example (0) is not a pasting scheme in the sense of Power (already for the strong reason that in it there does not exist any $\omega$-coking), and (1), while large, it would make for a rather dull and vacuously-uniquely-interpretable pasting diagram: there are no arrows that could possibly be composed, let alone is there any 2-cell.

In combinatorics, this is sometimes referred to as an infinite star.

Applications of the following standard concept are e.g. the study of maps on surfaces, and Power’s (proof) of unique interpretability of pasting schemes:

###### Definition

(face of a plane digraph)

Suppose $D=(V,A)$ is a plane digraph. The set of faces of $D$, denoted $\mathrm{Fa}(D)$, is the set of connected components, in the sense of Section 7, of the relative complement $\mathbb{R}^2\setminus\bigcup\{ \mathrm{im}(a) \colon a\in A \}$, where $\mathrm{im}(a)=\{ a(t)\colon 0\leq t\leq 1 \}$.

A face of $D$ is a member of the set $\mathrm{Fa}(D)$. Each face of $D$ is an open connected subspace of the plane.

Of course one needs a map from plane digraphs to (abstract) digraphs:

###### Definition

(the (abstract) digraph of a plane digraph)

Suppose $D=(V,A)$ is a plane digraph. Then $\mathrm{Dig}(D)=(V',A')$ is the digraph defined as follows: it has vertex set $V':=V$ and arc set $A' := \{ ( a(0) , a(1) ) \colon a\in A \}$.

Note that by axiom (v.connect), $A'$ is a subset of $V\times V$, and because each $a$ is an topological arc, and, as such, injective, $A'$ is moreover disjoint from the diagonal $\{(v,v)\colon v\in V\}$, so $A'$ has the type required by def. .

###### Definition

All digraph-theoretic technical terms like out-neighborhood, outdegree which are defined for digraphs are defined for plane digraphs $D$ by first applying the map $\mathrm{Dig}$ to $D$, then applying the digraph-theoretic definition, then applying the inverse of $\mathrm{Dig}$.

A central technical tool in Power’s proof are boundary walks: boundary walks validate a form of the red herring principle: each boundary walk is a weak walk, yet need not be a walk.

The boundary walks one works with when using A.J. Power’s application of plane digraphs to category theory are often weak trails, “often” in the sense that most pasting diagrams one encounters in practice do not have any arc-repetitions. Equivalently, the 1-cells of usual pasting diagrams have an underlying undirected graphs with is bridgeless. However, this is not generally true, and to have both walks and trails is not futile but necessary: a typical example of a pasting diagram occurring naturally in category theory are whiskering diagrams: the boundary trail of the exterior face of the typical whiskering diagram does have arc-repetions, hence is not a trail but only a walk.

## Remarks on the definitions

The convention to have digraph imply that there be no loops and no parallel arcs, and resort to other terms such as directed pseudograph to signal loops or parallel arcs, is widespread in modern combinatorics. For example, see p. 2 of (Gutin and Bang-Jensen), and Section 1.2 of (Csaba et al).

Unsurprisingly, while the convention that digraph implies that there be no parallel arrows has been widely adopted nowadays, there is a generous disregard for whether a digraph may contain loops. In this set-theoretical definition given above, this corresponds to the decision whether to allow $A$ to contain elements of the form $(v,v)$.

The terms walk, trail and path are standard in modern combinatorics. They are in particular compatible with standard usage in the theory of undirected graphs, in that upon forgetting (so to speak) the directions in the various definitions, one obtains the standard notions of walk, trail and path in (undirected) graph theory (for example Figure 1.8 illustrates precisely this convention).

Many (e.g. p. 11 and Chapter 1) modern reference define walks to be sequences whose codomain conains both vertices and arcs. For simplicity we do not do so. Having paths consist of vertices only, and letting the structure of the target space determine the axioms seems cleaner. We can afford to make this simplification only because digraphs cannot have any parallel arcs: for digraphs, our definition and the vertex-arc-alternating definition of p. 11 are equivalent to all intents and purposes. There is, however a good reason why e.g. Bondy and Murty use the latter definition: for them, a directed graph is (cf. 10.1, though Bondy–Murty’s formalization is different from usual quivers and a rare hybrid between purely functional and purely relation-theoretic formalizations: they use maps with codomain set $V\times V$) not a digraph but a quiver, and—evidently—if multiple parallel arcs are permitted, then a vertex-sequence does not contain enough information to define a “walk”. Upon stepping from one vertex to the next, a vertex sequence by itself cannot tell one which parallel arc to pick.

Needless to say (since given by the definitions), in a trail, arc-repetitions are forbidden, and in a path, any repetition is forbidden.

The perhaps oddly hybrid concept of trail has a crucial role in A. J. Power’s (proof) of unique interpretability of pasting schemes: any face of a plane digraph determines a boundary trail which always is a weak trail.

Generically, such a boundary trail has many vertex-repetitions, but it never has any arc-repetition.

The terminology “king” above is standard in modern digraph theory; the term “coking” is not, but in tune with category-theoretic usages. The term “$\omega$-king” appears not to be attested but is in tune with a standard notion of $d$-king in digraph-theory, and also with some usages in mathematics which let $\omega$ signal that some parameter is unconstrained (cf. omega-category omega-groupoid).

The choice of definitions documented here is biased towards (one of) the uses digraphs are put to in category theory, in particular in giving a rigorous justification of pasting diagrams. The most salient example of this is the emphasis given to plane digraphs in this article.

A reason for treating the concept of $\omega$-kings here is A. J. Power’s proof (Power 1990) of a pasting theorem, a rigorous justification of the notational practice of pasting diagrams: therein, both $\omega$-kings and $\omega$-cokings play an important role (Power calls $\omega$-kings “sources” and $\omega$-cokings “sinks”; both these terms clash with two standard technical terms in, respectively, contemporary digraph theory and flow theory, which is why the alternative terms were chosen).

Moreover, Power makes crucial use of a concept of addition of $a$$b$-paths, which is one of the reasons why $A$$B$-paths have been introduced.

## Uses of digraphs in category theory

One example of why digraphs are relevant to category theory are pasting schemes, which are a rigorous justification of the (older) notational practice of pasting diagrams. Pasting diagrams are fundamental to higher category theory: already the axioms for various notions of higher categories are formalized in terms of pasting diagrams. The formal justification of pasting diagrams was achieved in the late 1980. Note that there are more than one definition of pasting scheme in the literature. E.g. there are M. Johnson’s pasting schemes ((Johnson 1987, Chapter 2), (Johnson 1989, Section 2)) and Power’s pasting schemes (Power 1990). The definition which is strictly relevant to the present article is Power’s. He defines defines pasting schemes as a special kind of digraph (Power 1990, Section 3), details will be found at pasting scheme.

## References

• A. Bondy, U.S.R. Murty: Graph Theory with Applications. Fifth Printing. North Holland 1982.
• B. Csaba, D. Kühn, A. Lo, D. Osthus, A. Treglown: Proof of the 1-factorization and Hamilton decomposition conjectures. Memoirs of the American Mathematical Society, 244 (2016), monograph 1154, 170 pages
• R. Diestel, Graph Theory, 4. ed., Springer 2010

• Gregory Gutin?, Jørgen Bang-Jensen?: Digraphs: Theory, Algorithms and Applications. Springer Monographs in Mathematics. Second Edition (2009)

• Michael Johnson?: Pasting Diagrams in $n$-Categories with Applications to Coherence Theorems and Categories of Paths, Doctoral Thesis, University of Sydney, 1987
• Michael Johnson?: The Combinatorics of $n$-Categorical Pasting, Journal of Pure and Applied Algebra 62 (1989)
• Manfred Weis (https://mathoverflow.net/users/31310/manfred-weis), Calculating Cost-Optimal 1-Factors in Digraphs, URL (version: 2017-07-26): https://mathoverflow.net/q/277288
• John Power: A 2-Categorical Pasting Theorem, Journal of Algebra 129 (1990)
• H. J. Kowalksky: Topological Spaces. Academic Press. 1965.

Last revised on September 1, 2017 at 14:51:32. See the history of this page for a list of all contributions to it.


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# Published Articles

2015-07-28
11:25
 Lagrangian coherent structures and plasma transport processes / Falessi, M V ; Pegoraro, F ; Schep, T J A dynamical system framework is used to describe transport processes in plasmas embedded in a magnetic field. For periodic systems with one degree of freedom the Poincar\'e map provides a splitting of the phase space into regions where particles have different kinds of motion: periodic, quasi-periodic or chaotic. [...] arXiv:1507.07464.- 2015 - Published in : October 4958105 (2015) 5 External link: Preprint

2015-07-28
11:25
 Causation entropy from symbolic representations of dynamical systems / Cafaro, Carlo ; Lord, Warren M ; Sun, Jie ; Bollt, Erik M Identification of causal structures and quantification of direct information flows in complex systems is a challenging yet important task, with practical applications in many fields. Data generated by dynamical processes or large-scale systems are often symbolized, either because of the finite resolution of the measurement apparatus, or because of the need of statistical estimation. [...] arXiv:1507.07262.- 2015 - 22 p. - Published in : Chaos 25 (2015) 043106 External link: Preprint

2015-07-28
11:25
 Quantized form factor shift in the presence of free electron laser radiation / Fratini, F ; Safari, L ; Hayrapetyan, A G ; Jänkälä, K ; Amaro, P ; Santos, J P In electron scattering, the target form factors contribute significantly to the diffraction pattern and carry information on the target electromagnetic charge distribution. Here we show that the presence of electromagnetic radiation, as intense as currently available in Free Electron Lasers, shifts the dependence of the target form factors by a quantity that depends on the number of photons absorbed or emitted by the electron as well as on the parameters of the electromagnetic radiation. [...] arXiv:1507.07135.- 2015 - Published in : Europhys. Lett. 107 (2014) 13002 External link: Preprint

2015-07-28
11:25
 Long range correction for multi-site Lennard-Jones models and planar interfaces / Werth, Stephan ; Rutkai, Gábor ; Horsch, Martin ; Vrabec, Jadran ; Hasse, Hans A slab based long range correction approach for multi-site Lennard-Jones models is presented for systems with a planar film geometry that is based on the work by Janecek, J. Phys. [...] arXiv:1507.07130.- 2015 External link: Preprint

2015-07-28
11:25
 Performance metrics in a hybrid MPI-OpenMP based molecular dynamics simulation with short-range interactions / Pal, Anirban ; Agarwala, Abhishek ; Raha, Soumyendu ; Bhattacharya, Baidurya We discuss the computational bottlenecks in molecular dynamics (MD) and describe the challenges in parallelizing the computation intensive tasks. We present a hybrid algorithm using MPI (Message Passing Interface) with OpenMP threads for parallelizing a generalized MD computation scheme for systems with short range interatomic interactions. [...] arXiv:1507.07068.- 2015 - Published in : o. 3 (2014) 2203-2214 External link: Preprint

2015-07-28
11:24
 One-loop QCD correction to top pair production in the littlest Higgs model with T-parity at the LHC / Yang, Bingfang ; Liu, Ning In this work, we investigate the one-loop QCD correction to top pair production in the littlest Higgs model with T-parity at the LHC. We calculate the relative correction of the top pair production cross section and top-antitop spin correlation at the LHC for $\sqrt{s}=8,14$ TeV. [...] arXiv:1507.07104.- 2015 - Published in : Europhys. Lett. 111 (2015) 11001 External link: Preprint

2015-07-28
11:20
 2+1-dimensional wormhole from a doublet of scalar fields / Mazharimousavi, S Habib ; Halilsoy, M We present a class of exact solutions in the framework of $2+1-$dimensional Einstein gravity coupled minimally to a doublet of scalar fields. Our solution can be interpreted upon tuning of parameters as an asymptotically flat wormhole as well as a particle model in $2+1-$dimensions.. arXiv:1507.07257.- 2015 - 6 p. - Published in : Phys. Rev. D 92 (2015) 024040 External link: Preprint

2015-07-28
11:19
 Variation around a Pyramid theme: optical recombination and optimal use of photons / Fauvarque, O (Laboratoire d'Astrophysique de Marseille) ; Neichel, B (Laboratoire d'Astrophysique de Marseille) ; Fusco, T (Laboratoire d'Astrophysique de Marseille ; ONERA) ; Sauvage, J F (Laboratoire d'Astrophysique de Marseille ; ONERA) We propose a new type of Wave Front Sensor (WFS) derived from the Pyramid WFS (PWFS). This new WFS, called the Flattened Pyramid-WFS (FPWFS), has a reduced Pyramid angle in order to optically overlap the four pupil images into an unique intensity. [...] arXiv:1507.07387.- 2015 - 4 p. - Published in : Vol. 40, No. 15, August 1 2015, Optics Letters External link: Preprint

2015-07-28
11:19
 The study of triple systems V949 Cen, V358 Pup, and V1055 Sco / Zasche, P ; Honkova, K ; Jurysek, J ; Masek, M The systems V949 Cen, V358 Pup, and V1055 Sco are triples comprised of an eclipsing binary orbiting with a distant visual component on a much longer orbit. The first detailed photometric analysis of these interesting systems was performed using also the archival data from Hipparcos, ASAS, SuperWASP, OMC, and Pi Of The Sky surveys. [...] arXiv:1507.07325.- 2015 - 19 p. External link: Preprint

2015-07-27
15:53
 New potential for the Leitz Infinity Coordinate Measuring Machine / Sanz, Claude (CERN) ; Cherif, Ahmed (CERN) ; MAINAUD DURAND , Hélène (CERN) ; SCHNEIDER, Jurgen (HEXAGON Metrology) ; STEFFENS, Norbert (HEXAGON Metrology) ; MORANTZ , Paul (Cranfield University) ; SHORE , Paul (Cranfield University) The following study is realised within the frame of the PACMAN project: a study on Particle Accelerator Components Metrology and Alignment to the Nanometre scale, which is a Marie Curie program supported by the European commission and hosted by CERN (European Organisation for Nuclear Research). The aim of this program is to develop and build a pre-alignment bench on which each component is aligned to the required level in one single step using a stretched wire [...] CERN-ACC-2015-0078.- Geneva : CERN, 2015 - Published in : Fulltext: PDF; In : 15th International Conference & Exhibition, Leuven, Belgium, 1 - 5 Jun 2015

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# zbMATH — the first resource for mathematics

A sharp Schrödinger maximal estimate in $$\mathbb{R}^2$$. (English) Zbl 1378.42011
The authors treat the solution of Schrödinger equation, given by $$e^{it\Delta}f(x)=(2\pi)^{-n}\int e^{i(x\dot \xi+t|\xi|^2)}\hat f(\xi)d\xi$$. They consider the problem posed by Carleson: determine the optimal $$s$$ for which $$\lim_{t\to 0}e^{it\Delta}f(x)=f(x)$$ a.e. whenever $$f$$ is in the Sobolev space $$H^s(\mathbb R^n)$$. Their main result is: For every $$f\in H^s(\mathbb R^2)$$ with $$s>1/3$$, $$\lim_{t\to 0}e^{it\Delta}f(x)=f(x)$$ almost everywhere.
On the other hand, Bourgain showed that the convergence can fail if $$s<n/(2(n+1))$$ (in the case $$n=2$$: $$s<1/3$$).
So, they solve Carleson’s problem in the case $$n=2$$. For the proof, they use polynomial partitioning and decoupling, introduced by L. Guth and N. H. Katz [ibid. 181, No. 1, 155–190 (2015; Zbl 1310.52019)].
In the case $$n=1$$ the optimal number is $$1/4$$, which was proved by Carleson and Dahlberg and Kenig.

##### MSC:
 42B25 Maximal functions, Littlewood-Paley theory 42B35 Function spaces arising in harmonic analysis 42B37 Harmonic analysis and PDEs
Full Text:
##### References:
 [1] Bourgain, Jean, Some new estimates on oscillatory integrals. Essays on {F}ourier Analysis in Honor of {E}lias {M}. {S}tein, Princeton Math. Ser., 42, 83-112, (1995) · Zbl 0840.42007 [2] Bourgain, Jean, On the {S}chr\"odinger maximal function in higher dimension, Tr. Mat. Inst. Steklova. Trudy Matematicheskogo Instituta Imeni V. A. Steklova. Rossi\u\i skaya Akademiya Nauk, 280, 53-66, (2013) · Zbl 1291.35253 [3] Bourgain, Jean, A note on the {S}chr\“odinger maximal function, J. Anal. Math.. Journal d”’Analyse Math\'ematique, 130, 393-396, (2016) · Zbl 1361.35151 [4] Bourgain, Jean; Demeter, Ciprian, The proof of the {$$l^2$$} decoupling conjecture, Ann. of Math. (2). Annals of Mathematics. Second Series, 182, 351-389, (2015) · Zbl 1322.42014 [5] Carleson, Lennart, Some analytic problems related to statistical mechanics. Euclidean Harmonic Analysis, Lecture Notes in Math., 779, 5-45, (1980) · Zbl 0425.60091 [6] Dahlberg, Bj\“orn E. J.; Kenig, Carlos E., A note on the almost everywhere behavior of solutions to the {S}chr\'”odinger equation. Harmonic Analysis, Lecture Notes in Math., 908, 205-209, (1982) · Zbl 0519.35022 [7] Demeter, C.; Guo, S., Schr\"odinger maximal function estimates via the pseudoconformal transformation, (2016) [8] Guth, Larry; Katz, Nets Hawk, On the {E}rd{\H o}s distinct distances problem in the plane, Ann. of Math. (2). Annals of Mathematics. Second Series, 181, 155-190, (2015) · Zbl 1310.52019 [9] Guth, Larry, A restriction estimate using polynomial partitioning, J. Amer. Math. Soc.. Journal of the American Mathematical Society, 29, 371-413, (2016) · Zbl 1342.42010 [10] Guth, Larry, Restriction estimates using polynomial partitioning {II}, (2016) · Zbl 1342.42010 [11] Lacey, Michael; Li, Xiaochun, On a conjecture of {E}. {M}. {S}tein on the {H}ilbert transform on vector fields, Mem. Amer. Math. Soc.. Memoirs of the American Mathematical Society, 205, viii+72 pp., (2010) · Zbl 1190.42005 [12] Lacey, Michael; Thiele, Christoph, {$$L^p$$} estimates on the bilinear {H}ilbert transform for {\(2
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


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# Bivariate Poisson

1. Apr 21, 2007

### jimmy1

I have 2 dependent random Poisson distributed variables, $$X$$ and $$Y$$. I have that $$E[X] = mu$$ and $$E[Y] = c*mu$$ where $$c$$ is just a constant.

Now I'm trying to get the joint distribution of $$XY$$. I've found the expression of the bivariate Poisson distribution but the problem is in order to use it I have to define $$X$$ and $$Y$$ as

$$X = X' + Z$$ and $$Y = Y' + Z$$

where $$X', Y', Z'$$ are independent Poisson distributions with $$E[X'] = (mu - d)$$, $$E[Y'] = (c*mu - d)$$ and $$E[Z'] = d$$.

So basically my question is how do I get the parameter $$d$$?? Is there any formal way to get it??

Last edited: Apr 21, 2007
2. Apr 21, 2007

### mathman

You have not been given enough information. X and Y could be independent or else Y=cX or something in between.

3. Apr 21, 2007

### jimmy1

Well, X and Y are definitley dependent, it is always $$E[Y] = cE[X]$$.
Does that help??

In the paper I have about these bivariate Poisson distribution it also states that $$P(X|Y) = d/(c*mu + d)$$ and also $$P(Y|X) = d/(mu + d)$$, if that's any help?

Last edited: Apr 21, 2007
4. Apr 22, 2007

### mathman

Not so, they can be independent and their means happen to obey the equation.

5. Apr 22, 2007

### Hurkyl

Staff Emeritus
You sure you have that right? It doesn't make notational sense. (Incidentally, if you write \mu, LaTeX will convert that into a mu)

6. Apr 22, 2007

### ZioX

Ummm, if P(Y|X) is a function that doesn't depend on X, then Y and X are independent.

7. Apr 23, 2007

### jimmy1

If this is the case, then how to you formally define a dependent variable?

8. Apr 23, 2007

### Hurkyl

Staff Emeritus
Two random variables X and Y are independent if and only if, for all outcomes x for X and y for Y,
P(X = x and Y = y) = P(X = x) * P(Y = y).
(Equivalently, P(X = x | Y = y) = P(X = x))

Two random variables are dependent if and only if they are not independent.

9. May 1, 2009

### ryusukekenji

Any idea to operate with Excel???


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1 Cauchy-Schwartz Inequality & Jensen's Inequality

These two inequalities are widely applied in lots of areas not only probability. For probability, they can be written as follow: (1) Cauchy-Schwartz Inequality, if $X$ and $Y$ have finite variances, then

$E[XY]\leq\sqrt{E[X^2]E[Y^2]}$

(2) Jensen's Inequality: If $g$ is a convex function, then

$Eg(X)\geq g(EX)$

2 Markov Inequalities

Markov's inequality is pretty famous and basic, so firstly recall this inequality as follow: For any random variable $X\geq 0$ and any positive function $f$ for $X$:

There are several pretty interesting yes or no questions about simplex method in linear programming. All of them are chosen from the book in the references(actually it is a part of my homework). Although simplex method is very popular and lots of people know how to solve LP via simplex, we still may forget or omit some details of this theory. That is why I think the following questions are interesting. For these questions, we consider a LP to maximize $\textbf{cx}$ subject to $\textbf{x}\in X=\{\textbf{x}:\textbf{Ax}=\textbf{b},\textbf{x}\geq\textbf{0}\}$, where $\textbf{A}$ is $m\times n$ of rank $m.

1 Let $\bar{\textbf{x}}$ be a feasible solution with exactly $m$ positive component. Then $\bar{\textbf{x}}$ is an extreme point of $X$.

Apart from lasso algorithm, there are also many extensions of lasso regression like adaptive lasso and elastic net.

Adaptive lasso is one of the extensions of lasso. It uses a weighted penalty of the form, that is

$\hat{\beta}=\arg\min||y-\sum\limits_{j=1}^px_j\beta_j||^2+\lambda \sum\limits_{j=1}^pw_j|\beta_j|$

where $w_j=1/|\beta_j|^{\nu}(\nu>0)$, and $\beta_j$ is the OLS estimator. As a matter of fact, it is clear that both lasso and ridge regression are the particular cases of adaptive lasso. Continue reading

Gaussian Processes is a kind of Bayesian methods in machine learning. Actually the most significant difference between classical algorithm is that Bayesian methods always do not have to make the "best guess"  predictions for new test points. Instead, Bayesian methods always just compute a posterior predictive distributions for the new test inputs. That is, Bayesian algorithms are able to provide a good way to quantify the uncertainty in model estimates, but classical methods cannot in most time.

1 Multivariate Gaussian Distributions

Autumn is a beautiful season, so it is a pretty popular choice to go outside and enjoy the scenery at fall break. Before I arrived in Blacksburg, I have heard that Blue Ridge Mountains are very beautiful and it is worthy to go hiking there. Luckily,  it is not very far from Blacksburg. Although the distance is more than 8 miles and all of us felt extremely tired, I was still very happy. Since my English is not good enough, I cannot express the fantastic landscape in English, so I just want to share some photos here, although the quality of this photos is not very good.

Matrix Decomposition (or factorization) is pretty important in many research areas, especially in data analysis, such as using SVD or EVD in PCA. Actually there are more than 10 kinds of matrix decomposition methods.  In general, researchers divide these methods into 4 types, diagonal factorization (like SVD), triangularization factorization (like LU), triangle-diagonal decomposition (like schur decomposition) and tri-diagonal decomposition. Here the triangularization factorization is just discussed first.

1 Cholesky factorization


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# Isometries of $\mathbb{R}^3$

So I'm attempting a proof that isometries of $\mathbb{R}^3$ are the product of at most 4 reflections. Preliminarily, I needed to prove that any point in $\mathbb{R}^3$ is uniquely determined by its distances from 4 non-coplanar points, and then that an isometry sends non-coplanar points to non-coplanar points in $\mathbb{R}^3$. I've done the first preliminary step, and finished the proof assuming the second, but I can't find a simple way to prove the second...

Intuitively it makes a lot of sense that non-coplanar points be sent to non-coplanar points, but every method I've stumbled upon to prove such has been quite heavy computationally...

I know for example that any triangle chosen among the four points, A, B, C, D must be congruent to the triangles of their respective images, but what extra bit of information would allow me to say that the image of the whole configuration can't be contained in a single plane...

-

Would this work? Construct the altitude from $D$ to the plane containing $A$, $B$, and $C$. Call the foot of this altitude $E$ (the point where the altitude meets the plane). Triangles $ADE$, $BDE$, and $CDE$ all have right angles at $E$ and you know that the isometry preserves angles, so triangles $A'D'E'$, $B'D'E'$, and $C'D'E'$ all have right angles at $E'$, which makes $A'$, $B'$, $C'$, and $E'$ coplanar and since $D'E'\ne 0$ and $\overline{D'E'}$ is perpendicular to that plane, $D'$ cannot be coplanar with the other points.

-

Isometries are affine and preserve volume : so if $A$, $B$, $C$, $D$ are non-coplanar, their convex hull $ABCD$ has non-zero volume, and the convex hull $f(A)f(B)f(C)f(D)$, which is the image of the convex hull $ABCD$ under $f$ (since $f$ is affine) has non-zero volume, therefore $f(A)$, $f(B)$, $f(C)$ and $f(D)$ must be non coplanar.

-
Isn't this essentially a stronger restatement of what I'm trying to prove? –  AsinglePANCAKE Dec 11 '11 at 22:50
actually, this is a result of measure theory when you construct Lebesgues measure. but I see what you mean, this is kind of heavy artillery for your problem... one other mean is to use barycentric coordinates consider the spanned vector space –  Glougloubarbaki Dec 11 '11 at 22:58
Hmmm, care to elaborate? There has to be a simpler explanation that I'm not seeing... –  AsinglePANCAKE Dec 12 '11 at 0:42

Well, how do you do it for collinear points? You may be able to generalize the idea.

But, I suspect, the easiest way is to find a condition equivalent to "coplanar" in terms of things you already know are preserved and reflected by isometries. (e.g. define "coplanar" in terms of "collinear")

-
Well for non-collinear points I have the lovely SSS congruence axiom for triangles that ensures non-collinear points remain non-collinear. I don't know an easy way to prove SSSSSS for tetrahedra... –  AsinglePANCAKE Dec 11 '11 at 22:53


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# Observed N-value of an SPT test is 21. The N-value after correcting for dilatancy is

This question was previously asked in
SSC JE CE Previous Paper (Held on 23 Mar 2021 Evening)
View all SSC JE CE Papers >
1. 18
2. 21
3. 19
4. 15

Option 1 : 18
Free
ST 1: Power Systems
8.4 K Users
20 Questions 20 Marks 15 Mins

## Detailed Solution

Concept;

Dilatancy correction:

It is to be applied when No obtained after overburden correction, exceeds 15 in saturated fine sands and silts. IS: 2131-1981 incorporates the Terzaghi and Peck recommended dilatancy correction (when No > 15) using the equation

$$\rm N= 15 + \frac{1}{2}\left( {N_0-\;15} \right)$$

N0 - SPT value after overburden correction

Calculation:

Given: N0 = 21

$$\rm N= 15 + \frac{1}{2}\left( {21-\;15} \right)$$

⇒ N = 18

Last updated on Dec 9, 2022

The Power Grid Corporation of India Limited (PGCIL) has released the notification for PGCIL Diploma Trainee on 9th December 2022. A total of 211 vacancies have been released for various disciplines such as Civil, Electronics and Electrical. The applications are invited from the candidates from 9th December 2022 to 31st December 2022. Also note that the cut off date for meeting the eligibility will also be 31st December 2022. Go through the PGCIL Diploma Trainee Syllabus and Exam Pattern and continue their preparation in the right direction.


text:
# Two waves meet a Two light sources emit waves that are in phase with a wavelength of lambda = 400...

## Question:

Two waves meet a Two light sources emit waves that are in phase with a wavelength of {eq}\lambda {/eq} = 400 nm. The two waves meet at a distant point where Wave #1 has traveled 100 nm. The distance taveled by Wave #2 is variable. Which of the distances for Wave #2, listed below, would result in constructive interference?

a. 450 nm

a. 450 nm

b. 900 nm

c. 1500 nm

d. 1100 nm

e. 700 nm

## Constructive Interference:

Interference is the phenomenon in which two separate wavefronts originated from two coherent sources meet each other and form a new wavefront of higher wavelength. The condition for constructive interference is as follows.

{eq}{d_2} - {d_1} = n\lambda {/eq}

Given Data

• The wavelength of wave is: {eq}\lambda = 400\;{\rm{nm}} {/eq}
• The distance travelled by wave 1 is: {eq}{d_1} = 100\;{\rm{nm}} {/eq}

Use relation for the constructive interference.

{eq}{d_2} - {d_1} = n\lambda {/eq}

Here, the wavelength is {eq}\lambda {/eq} , the distance travelled by the wave 1 is {eq}{d_1} {/eq} and distance travelled by wave 2 is {eq}{d_2} {/eq}.

Substitute all the values in the above equation.

{eq}\begin{align*} {d_2} - {d_1} &= n \times 400\\ {d_2} - 100 &= n \times 400\\ {d_2} &= 400n + 100 \end{align*} {/eq}

Calculate the distance travelled by wave 2 for n=1, 2, 3.

{eq}\begin{align*} {\left( {{d_2}} \right)_{n = 1}} &= 400 \times 1 + 100\\ &= 500\;{\rm{nm}}\\ {\left( {{d_2}} \right)_{n = 2}} &= 400 \times 2 + 100\\ &= 900\;{\rm{nm}}\\ {\left( {{d_2}} \right)_{n = 3}} &= 400 \times 3 + 100\\ &= 1300\;{\rm{nm}} \end{align*} {/eq}

Thus, the correct option is (b).


text:
# nLab complete loop graph

Contents

category theory

## In higher category theory

#### Graph theory

graph theory

graph

category of simple graphs

# Contents

## Definition

A complete loop graph is a loop graph? $G$ where between any two vertices $a:G$ and $b:G$ there is exactly one edge $p:a \to_G b$.

In the context of homotopy type theory, a complete loop graph is a type $G$ with a type family $\to_G$ such that the dependent type $a \to_G b$ is a contractible type for all elements $a:G$ and $b:G$.

A complete loop graph is univalent, a subsingleton, or a proposition if the canonical function

$idtoedge:(a =_G b) \to (a \to_G b)$

is an equivalence of types.


text:
# Newtheoremstyle head aligned with top of tabularx

Here's an example :

\documentclass[12pt]{report}
\usepackage{amsmath, amssymb, amsthm}

\usepackage{tikz}
\newcommand*\squared[1]{\tikz[baseline=(char.base)]{\node[shape=rectangle, draw, inner color = white, drop shadow = {opaque, black}, inner sep=3pt, text justified] (char) {#1};}}

\newtheoremstyle{exercise}
{\topsep}
{\topsep}
{\hangindent=2em}
{}
{}
{}
{.8em}
{\squared{\thmnumber{#2}}}

\theoremstyle{exercise}
\newtheorem{exercise}{}

\renewcommand{\arraystretch}{1.75}
\usepackage{tabularx}

\begin{document}

\begin{exercise}
\begin{tabularx}{\textwidth}{r @{\hskip .4em} X}
a) & This is a test. \\
b) & This is a long test. \\
c) & This is a very long test.
\end{tabularx}
\end{exercise}

\end{document}

What I want is the number of the exercise to be aligned with the top of the tabularx (question a)) instead of being aligned with the middle of the tabularx.

Use t (top alignment) for the optional argument of tabularx:

\documentclass[12pt]{report}
\usepackage{amsmath, amssymb, amsthm}

\usepackage{tikz}
\newcommand*\squared[1]{\tikz[baseline=(char.base)]{\node[shape=rectangle, draw, inner color = white, drop shadow = {opaque, black}, inner sep=3pt, text justified] (char) {#1};}}

\newtheoremstyle{exercise}
{\topsep}
{\topsep}
{\hangindent=2em}
{}
{}
{}
{.8em}
{\squared{\thmnumber{#2}}}

\theoremstyle{exercise}
\newtheorem{exercise}{}

\renewcommand{\arraystretch}{1.75}
\usepackage{tabularx}

\begin{document}

\begin{exercise}
\begin{tabularx}{\textwidth}[t]{r @{\hskip .4em} X}
a) & This is a test. \\
b) & This is a long test. \\
c) & This is a very long test.
\end{tabularx}
\end{exercise}

\end{document}

Not related to the question, but to prevent an overfull \hbox you'll need to reduce the width for the tabularx.

• You mean I should change the argument \textwidth when I use \tabularx ? Then how could I still have the width of the text without risking an overfull \hbox ? – Amateur Jul 27 '14 at 0:37
• @Amateur Yes, precisely that. Since you declared a 2em hangindet for the exercise environment, you'll need \begin{tabularx}{\dimexpr\linewidth-2em\relax}[t]{r @{\hskip .4em} X} to prevent the bad box in this case. – Gonzalo Medina Jul 27 '14 at 0:39


text:
# Dearbitraging a weak smile on SVI with Damghani's method

Yesterday, I wrote about some calendar spread arbitrages with SVI. Today I am looking at the famous example of butterfly spread arbitrage from Axel Vogt.

$$(a, b, m, \rho, \sigma) = (−0.0410, 0.1331, 0.3586, 0.3060, 0.4153)$$
The parameters obey the weak no-arbitrage constraint of Gatheral, and yet produce a negative density, or equivalently, a negative denominator in the local variance Dupire formula. Those parameters are mentioned in Jim Gatheral and Antoine Jacquier paper on arbitrage free SVI volatility surfaces and also in Damghani’s paper dearbitraging a weak smile.

Gatheral proposes to adjust just two of the parameters (corresponding to call wing and minimum variance) and run an optimizer with a large penalty on arbitrage to find the best-fit SVI parameters. The arbitrage can be easily detected by just computing the local variance denominator, which is not much more costly than computing the implied variance (the analytical derivatives come almost for free):

$$g(y)=1 - \frac{y}{w}\frac{\partial w}{\partial y} + \frac{1}{4}\left(-\frac{1}{4}-\frac{1}{w}+\frac{y^2}{w^2}\right)\left(\frac{\partial w}{\partial y}\right)^2 + \frac{1}{2}\frac{\partial^{2} w}{\partial y^2}$$
where $$w$$ is the total implied variance in log-moneyness $$y$$ coming from the SVI formula.

It is also possible include this penalty directly in the Nelder-Mead minimization of Zeliade’s quasi explicit calibration. In this case, all the parameters will be optimized. It is also not much slower than the more classic minimization without penalty. While it results a much lower RMSE, the solution might not be as visually pleasing as Gatheral’s one.

I was curious to see what Damghani’s technique would produce on this same example. Instead of doing the minimization with the full denominator evaluation, Damghani uses a simpler no-arbitrage necessary constraint on the first derivative:

$$|\frac{\partial w}{\partial y}|<4 e$$
and iteratively search for $$e \in [0,1]$$ so that no arbitrage remains in $$w$$. The standard weak no-arbitrage constraint stands for $$e=1$$. Dhamgani thus proposes to enforce a stronger “weak” constraint during the minimization, in order to de-arbitrage. Note, the equations in the paper have a typo (everywhere), the $$b$$ parameter should be a factor in front of the parenthesis.

Now, it seems twisted to not enforce directly the real no arbitrage constraint in the minimization, which is actually not much more complex to evaluate, and to instead rely on some weak condition, that we make stronger by steps. There would be no loop over the minimization needed. Does this produce any good result?

The graph is clear, the technique produces crap. This is the local variance denominator:

Maybe I misunderstood something big in this paper, but so far it was just a waste of time.


text:
# The Rockfall Hazard Rating System

This system of rating the risk of rockfalls was developed for the Oregon Department of Transportation. The goal behind a comprehensive system for evaluating the risk is to allow the DOT to address the risk in a systematic and proactive way instead of simply responding to incidents retroactively. The report has directions for how any agency can implement their rating systems for rockfall hazards. Although the system addresses just the risk of rockfalls, the general principles behind it can also be extrapolated to any specific risk that a transportation system needs to classify.


text:
1. ## I need help understanding this please.

Use the product rule for exponents to simplify the expression. Write the result using exponents.a8b4 · a8b9

2. ## Re: I need help understanding this please.

Basically they want you to do $x^a x^b = x^{a+b}$

For ex, $2^2 * 2^3 = 2^{2+3}$

3. ## Re: I need help understanding this please.

$a^8b^4 \cdot a^8b^9 = a^{8+8}b^{4+9}$


text:
# Topic detection for a sentence or an article using Machine learning

Besides LDA (Latent Dirichlet allocation), are there other ways or methods to detect a topic or category from a sentence?

For example, all the categories or tags from news websites can be used to train a classifier to predict the topic for a new sentence or article or a paragraph.

Are there any known public datasets with keywords to topics relationship which can be used a training set in a classic classification problem?

A simple training data example of a news website:

Article 1: category x
Article 2: category x
Article 3: category x
Article 4: category y
Article 5: category y
Article 6: category y

It's a very simple example but it's enough to paint the picture. Now use this data to predict the category of a new article.

A bit of explanation: By topic I mean if a text is talking about politics, entertainment, business, finance, lifestyle etc and ideally, a classification into sub-categories of such types. A similar categorization is used in the news website where they place each article in a specific category.

It sounds like you are talking about a text classification problem. There are lots of data sets for this.

1. 20 newsgroups data http://qwone.com/~jason/20Newsgroups/

3. DBPedia ontology dataset (can't find a link right now)

There is also something called supervised LDA that might interest you.

• 1 and 2 are relevant but too small for my use case. Sep 26 '17 at 8:35
• You could do hashtag prediction on Twitter Sep 26 '17 at 15:55

I really liked Gensim package for Python. There they use a collection of articles from wikipedia as training, you can find it here (8GB).

As transformations they use:

1. Term Frequency * Inverse Document Frequency, Tf-Idf
2. Latent Semantic Indexing, LSI (or sometimes LSA)
3. Random Projections, RP

and then LDA and its transformation HDP. You can find a good tutorial here.