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History

One important and often difficult problem in the study of restricted permutations is the enumeration problem:  Given a set $R$ of permutations, enumerate the set $S_n(R)$ consisting of those permutations in $S_n$ which avoid every element of $R$. The earliest solution to an instance of this problem seems to be MacMahon's enumeration of $S_n(123)$. The first explicit solution seems to be Hammersley's enumeration of $S_n(321)$. Knuth shows that for any $\sigma \in S_3$, we have $|S_n(\sigma)| = C_n=\frac{1}{n+1}\binom{2n}{n}$, the $n$th Catalan number. Other authors considered restricted permutations in the 1970s and early 1980s (see, for instance, Rogers, Rotem) but the first systematic study was not undertaken until 1985, when Simion and Schmidt solved the enumeration problem for every subset of $S_3$. Recently, there more than 200 papers on this subject.

The idea of this annual conference is due to Michael Atkinson, who announced in first in July of 2001.

Previous conferences:

2003: Conference on Permutation Patterns PP'03, 10-14 February, 2003, University of Otago Dunedin, New Zealand.

The First International Conference of Permutation Patterns was organized by Michael Atkinson, at the University of Otago, in Dunedin, New Zealand, February 10-14, 2003. Keynote speaker was Herb Wilf, from the University of Pennsylvania. The Electronic Journal of Combinatorics devoted a special issue to the conference.

2004: Conference on Permutation patterns PP'04, 5-9 July, 2004, Malaspina University-College, Nanaimo, British Columbia, Canada.

The Second International Conference of Permutation Patterns is organized by Julian West, at Malaspina College, at Nanaimo, British Columbia, Canada, July 2-9, 2004. Keynote Speaker is Miklos Bona, from the University of Florida. Annals of Combinatorics is devoting a special issue to the conference.

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## Martin Ringo on Principal Components

We have a number of readers who are highly qualified econometricians. I think that initially they find it hard to believe the description of Hockey Team statistical practices. Martin Ringo is one such reader, who has a doctorate in "finite sample properties of a variety of feasible, generalized least squares estimators". He’s sent in the following discussion of principal components reconstruction from a different aspect than any that I’ve considered – whether the temperature PCs from the short calibration period will even accomplish what MBH want. Marty:

Even if we had the exact PC component from the SVD of the long series, it is not enough to reconstruct the temperatures. (That is, even if MBH’s proxies could perfectly estimate the temperature PCs, they still won’t be able to reconstruct what they want to reconstruct.) To reconstruct the temperature, even the averages across all the grids, one needs a better estimate of the post-multiplying matrices associated with the long series SVD than the post-multiply matrices of the short series will give.

Martin Ringo: More on Linear Algebra

I am going to start by reviewing what the proxy variables and all those regressions were about (explained in “MBH Calibration-Estimation Procedure”). There is a relatively short temperature record, roughly 150 years, and Mann et al and most of us would like to have a much longer record. To do this the MBH98 procedure was to use the several linear relationships between the temperature records and proxy variables over that same period to produce a estimate of a set of the principal components of that temperature and then to use that relationship applied to the long period (600 or 1000 years) of the proxy records.

Now for a little linear algebra (although fortunately I am going to avoid all those proxy regression and scaling issues).

MBH98 had 1082 temperature series over 94 years. Call that a 94 by 1082 temperature matrix. The matrix has a most a rank of 94, and thus MBH98 couldn’t use it for principal component analysis (PCA) as such. They expressed on a monthly basis to give a 1128 x 1082 matrix (presumably of full rank 1082). Using the matrix form of MBH98′s equation (in the Methods Section), the temperature matrix could be expressed as:

1) T1 = U1 * S1 * V1′, (for monthly data)

where (for simplicity) I am going to assume that 1) was derived from the compact form of the SVD algorithm with U1 a 1128 x 1082 matrix of principal components, S1 a 1082 x 1082 diagonal matrix, and V1 a 1082 x 1082 matrix. U1 and V1 are, of course, both orthogonal matrices.

MBH98 took the first Neofs PCs of U1 to derive a T^ (“T hat”) estimate of the full temperature matrix.

The thing to note about equation 1) is that it can be used to “reconstruct” T. That is with U1, S1, and V1 we can get T, and if the first Neofs of U1 explain most of the variation, the can approximately reconstruct T from those first Neofs PC. MBH98 took the long series of proxies and relationships from the inverse regression equations to build a U1^ as long as the proxy data. If one goes through the linear algebra in “MBH Calibration-Estimation Procedure,” you will see that the length of the U1^ is independent of the length of T1. Hence the long proxy series allows MBH98 to produce a long estimate of the temperature PCs, a long U^ so to speak.

That is what was done. I looked at the reconstruction a slightly different way. Suppose we don’t have to use U^. Suppose we have the actual U, the actual PCs from the actual, long, temperature series. Then we should be able to do an even better job of reconstructing the full temperature matrix for the long period, say a full 600 or 1200 years, because the actual PCs must be a better estimator than the estimated PCs. Posing the problem this way gives a type of upper bound on what can be accomplished through reconstruction of estimated temperature PCs.

This type problem is just the kind of thing Monte Carlo experiment can — I won’t say solve, but — give insight into. The experiment simulated a set of serially and spatially correlated temperature variables (as deviations from the mean), a matrix T of monthly data. T_full, which is combination of T1 and, say, T2 corresponding to observed and unobserved parts. T1 is decomposed per equation 1) and the annual version of T_full is decomposed similarly:

2) T_full = U_full * S_full * V_full’, (annual)
[where T_full might be 1200 x 1028, U_full the same, S_full 1028 x 1028, and V_full 1028 x 1028]

The first Neofs PCs from U_full are taken: call them W. The U1 from the short period SVD is simply thrown away. In this “world” we know the true PCs. But the S1 and V1 of equation 1) are retained. This is the crucial point: the MBH98 reconstruction has to use the S1 and V1 matrices from the short (observed) period SVD not the S_full and V_full of the full period SVD. We then have two reconstructions

3a) R_short = [W| zeros] * S1 * V1′;
and
3b) R_full = [W| zeros] * S_full * V_full’;
[Note the "[W|zeros]” represents filling out the last 1082-Neofs columns of the matrix with zeros to make the matrix multiplication conform.]

The entire difference in the two estimates are the postmultiplying matrices S and V. with R_short we have a reconstruction just as MBH98 would have done only with a better, in fact perfect, estimate of the temperature PCs, but still limited by the postmultiply matrices being determined from the short (observed) period data. With R_full those matrices are taken from the full period data.

If the MBH98 reconstruction procedure is feasible, it will perform reasonably well in comparison to the R_full reconstruction. That is, the limitation will the fact that Neofs PCs cannot perfectly reconstruction 1082 temperature series.

MBH98 use the Reduction in Error (RE) statistic to “verify” their reconstruction. Alternatives to that are the “Coefficient of Efficiency,” CE, (do not use that term as climatologists use it in economics, sociology, educational research or industrial engineering) and the old standby rho-squared, R2. Let me suggest that Henri Theil’s U statistic — “coefficient of uncertainty [of forecasts],” (Theil, H, Applied Economic Forecasts and Policy, North Holland, 1961), is superior to all three.

4) U = ( sqrt( sum( (Yhat-Y)^2) ) )
/ ( sqrt( sum(Yhat^2))+sqrt( sum(Y^2)) )

John interjects: this might be easier to read

$U = \frac {\sqrt{ \sum {( \hat{Y}-Y)^2}}}{ \sqrt { \sum{ \hat{Y}^2}} + \sqrt{ \sum {Y^2}}}$

It measures the size of the squared errors in proportion to the values being forecasted, both the forecast and the actual values. This (the original) version of the statistic goes from 0, a perfect forecast, to 1, a forecast in which the errors completely overwhelm the forecast, and because of that, I prefer it to the later version with a more tractable distribution but unbounded upper end.

But in any case I calculated all four statistics.

The results: The RE statistics for the R_short reconstructions of the individual temperature series all average well less than 0 for 100 run batches. The grand average is probably something like -20. When the reconstruction is compared on the basis of a spatial average, the RE values improve a bit, to maybe around -5. While it might be fun to hoist Dr. Mann by his own petard, remember the RE is a lousy statistic for Monte Carlos because of its asymmetrical unboundedness. Get 99 RE values of 0.5 and one of -100 (which is can happen), the conclusion on the average is a -0.5, yet the procedures might have been pretty good. The Theil’s U statistic is much more telling, at least to me. There the R_short, i.e. the MBH98 type reconstruction, never performed lower than a 0.70 while the R_full, i.e. what could be done with accurate weightings, never was over 0.2.

A note of caution: It should always be remembered that all Monte Carlo results are dependent on the parameters. For things like the Classical Linear Model (the textbook regression model), changing the parameters doesn’t change the Monte Carlo results (as to the nature or the distribution). But with more complex models, especially when nonlinearity is introduced, use beware. One would like to sample over all the parameter space, but life and computing facilities will run short. And right now I haven’t figured out just what is the appropriate variation of parameters. Also because the time consuming nature of the simulation, I am doing it right now with about half the size of MBH98′s work. I report these preliminary results because they were so dramatic and have remained unchanged across my various parameter changes. Black and white: Neofs — be that 10% of number of series or 75% — principal components constructed from the whole temperature series can reconstruct the full temperature series fairly accurately on an individual basis and very accurately on a spatial average when that reconstruction is based on the postmultiplying matrices that were jointly part of the SVD. But those same principal components cannot even reasonably reconstruct the spatial average when that reconstruction is based on the postmultiplying matrices of a short span of the full period of the temperature reconstruction.

To give a numerical flavor to the differences in the S * V’ matrices from the full versus short series SVDs, below are the two S * V’ matrices for first the full and then the short SVD on a 40 x 4 data set:

Full data S*V’ values:
-0.178 1.432 0.920 -7.440
-0.086 -5.435 3.568 -0.603
-0.849 -2.786 -4.445 -1.065
-4.448 0.398 0.540 0.250

Short data S*V’ values:
0.930 1.379 -0.599 4.414
1.126 -1.833 1.631 0.557
1.244 -0.272 -1.055 -0.320
-0.556 -0.745 -0.548 0.276

And the effect? Below is the 40th observation on the data:
0.161 -0.135 -1.872 1.108

What did the short data SVD reconstruction predict?
0.016 0.031 -0.373 -0.959

Percent differences:
-90.0% -122.7% -80.1% -186.5%

Maybe it isn’t even necessary to go through the Monte Carlo experiments. What MBH98 and the principal components reconstruction advocates appear to be butting their collective heads against is a little linear algebra.

The pseudocode is here and PC reconstruction sample here

1. kim
Posted Feb 15, 2006 at 10:03 AM | Permalink | Reply

B I N G O and bingo was nis nome o.
===========================================

2. jae
Posted Feb 15, 2006 at 10:50 AM | Permalink | Reply

These analyses are way over my head, but they look like pure mathematics. As such, are they contestable?

3. Ian
Posted Feb 15, 2006 at 11:36 AM | Permalink | Reply

Steve,

In your Monte Carlo experiments, what did the spectrum of your noise look like (was it white noise)? I was just wandering if your results would be different if there was a significant low frequency component. Intuitively, it seems like it might.

4. Ian
Posted Feb 15, 2006 at 11:41 AM | Permalink | Reply

I meant ‘simulated signal’ not noise — but I sure you knew what I meant anyways.

5. kim
Posted Feb 15, 2006 at 12:00 PM | Permalink | Reply

jae, I’m ignorant enough of math for the language and the logic to make sense to me.
===================================================================================

6. Ian
Posted Feb 15, 2006 at 12:41 PM | Permalink | Reply

In case they are listening here is my proposal to the hockey stick team for future supporting/refuting papers

1. Drop the use of controversial proxies. They only represent a small percentage so just drop them.
2. Drop the use of principal components. Steve has made it abundantly clear that there are too many issues with its current usage.
3. For each proxy type use the sensor data as known signal and the proxy minus sensor data as noise to create a wiener filter (spectral filter optimized for known signal spectra and noise spectra). Smooth the filter.
3 (alternate). If one doesn’t like this approach (wiener filter) then visually inspect the spectrum and propose a simple low pass filter with appropriate cut off. I am pretty confident that both approaches will lead to similar results (the spectral filtering approach is quite robust).
4. Apply the filter to the data to get the best estimate of the signal.
5. Create a weighted sum of the signals where the weighting is estimated from a best guess geographic area that the proxy applies to.
6. Do the same but only for an ensemble of European proxies to check for the existence of a MWP, where there is the most confidence that one existed.
7. Compare and publish

If you confirmed the existence of MWP in Europe and still get a global hockey stick, then I, and I assume many others, would be convinced. Goodbye global MWP.

Anyone see any flaws with this approach?

If I find the time, I might even take a crack at doing this myself.

7. Mark
Posted Feb 15, 2006 at 12:57 PM | Permalink | Reply

ian, you just described a generalized sidelobe canceller…
mark

8. Ian
Posted Feb 15, 2006 at 12:58 PM | Permalink | Reply

Mark – Ok. And … doesn’t this seem like a good approach to resolve this issue, or what are the flaws with this approach?

9. Ian
Posted Feb 15, 2006 at 1:06 PM | Permalink | Reply

An important point to add/change would be too apply this technique blindly to all proxies, (yes even the bristlecones so scratch step #1), as part of the problem from the beginning is the danger of cherry picking data.

10. fFreddy
Posted Feb 15, 2006 at 1:36 PM | Permalink | Reply

John A , a while ago you mentioned an add-in that would make it possible to represent mathematical formulae more legibly. Is that available yet ?
(Apologies if you’ve been too busy, but it really would make it easier to thrash through these algebra posts.)

11. Posted Feb 15, 2006 at 2:58 PM | Permalink | Reply

Yes, this seems to be about the problem of overfitting the random errors that initally concerned me with the MBH98 use of so many PCs. It would be interesting to see how many Neofs are required to minimize the error of the short program (to borrow a figure skating term). If I understand this correctly 10% is way too many and probably even one might capture all the worthwhile signal.

Posted Feb 15, 2006 at 2:58 PM | Permalink | Reply

Ian, have you ever considered a career in high speed computing?

As the core voltages get lower and lower, and the clocks get faster … well, you can imagine!

Same principles apply and you do clearly get it. Bravo!

13. John Hekman
Posted Feb 15, 2006 at 3:04 PM | Permalink | Reply

Ian
I don’t know why all this filtering would provide better data series with which to test the hypothesis at issue. Assume for the sake of argument that tree ring width and density could be “perfectly” explained if one had good data on sunlight, temp, humidity and soil quality, whereas the Hockey Team has only tested the ring widths and densities against recent temperatures.

Then the regression of ring width on temperature is going to have residuals with a lot of unexplained variance. In reality, this is due to the effects of the omitted variables: sunlight, humidity etc. Now if you do all this filtering and smoothing, how is this going to give you a better estimate of the effect of temperature on tree rings? Why would one have any more confidence in the estimated parameters coming out of this? Thanks in advance.

14. Ian
Posted Feb 15, 2006 at 3:26 PM | Permalink | Reply

John,

I agree this does not help the situation if the proxies themselves are flawed, however if the proxies are flawed, then this will show up as a high level of noise.

The main advantage of this approach is its simplicity and robustness. The current processing technique is basically an exercise in obsfucation. As Steve has shown it is a way to hide unrealistic weighting of bristlecones and other targeted proxies.

Of course the simplicity of the approach is also part of the problem since scientific papers are designed to be complex to impress one’s peers of one’s mathematical prowess (and also to suppress criticism for fear of looking foolish by not understanding some obsfucated process). Writing a paper that anyone can understand is looked down upon. It took Steve quite a while to decipher Mann’s paper in terms of more elementary linear algebra equations. That barrier is part of the point, and may have been a major reason why the paper passed review in the first place.

15. Frank H. Scammell
Posted Feb 15, 2006 at 4:04 PM | Permalink | Reply

Steve, Suppose you were to take all the proxies that DON’T have a kickup in the region of merging with the “instrumental” record (and are not duplicated in other records), and merge them with (for example) the UAH tropospheric record at a single point (the beginning of the UAH record. I expect this will show that the blade of the hockeystick is grossly exxagerated and a MWP and LIA clearly existed.

16. Mark
Posted Feb 15, 2006 at 4:22 PM | Permalink | Reply

Ian, I think there may be some valid direction with such an approach.

FWIW, a generalized sidelobe canceller is sort of the mother of all adaptive filtering schemes (assuming multiple inputs). In directive (beamformed) signal processing applications, the point is to find the common signals, those that are correlated, across an array of sensors. The correlated signals across the array, after blocking the desired signal, represents the sidelobe interference from just about anything, but usually a jammer of some sort.

In our applications (er, the climate application), we’d actually be looking to keep the correlated signals as they would represent the commonality across all the proxy inputs (the proxies are the analogy to my sensor array, I suppose).

At the very least, this seems reasonable…

Mark

17. Mark
Posted Feb 15, 2006 at 4:25 PM | Permalink | Reply

Somehow the first sentence in my last post came out sort of silly sounding… try this:

“Ian, I think there may be validity with such an approach.”

Mark

18. Steve McIntyre
Posted Feb 15, 2006 at 4:35 PM | Permalink | Reply

#15 – Frank, you might look at my splice of Moberg with the satellite record here – http://www.climateaudit.org/?p=179. There’s obviously lots of hair on Moberg as a record, but the difference between the satellite and Jones records is material to the look of the rhetorical hockey stick.

19. Steve McIntyre
Posted Feb 15, 2006 at 4:41 PM | Permalink | Reply

#11 – David, the overfitting is even worse: Mann represents each of the temperature PCs in terms of 22-112 proxy series. The representation is related to multiple regression – I’ll chat about the difference soon. This is a big anvil.

#14. Ian, I don’t necessarily agree that flawed proxies will show up as noise. That will be the case only if the bulk of the proxies actually are “proxies”. Look at the correlations reported in our Reply to Von Storch, where the validity of the proxies is contested.

20. Ian
Posted Feb 15, 2006 at 4:59 PM | Permalink | Reply

Mark,

I have had some experience with beamforming array data, but was not familiar with that term. From my experience with beamforming, it is robust and works even with very low SNR. It’s only downside is minor sidelobe artifacts, which people attempt to attenuate through various window functions.

I have seen many people try fancier techniques, such as maximum entropy, SVD, etc, however these techniques only seem to work under high SNR, and as far as I can tell the only advantage of these techniques (besides sprucing up a paper with unneeded complexity) is they can provide sharper peaks and eliminate sidelobes. With low SNR, these techniques fail horribly providing strong spurious peaks and no peak where they should be one. I think most people will agree that the proxy data has a low SNR.

21. Ian
Posted Feb 15, 2006 at 5:04 PM | Permalink | Reply

Steve,

Even if the flawed proxies do not show up as noise at the state will be given reasonable weightings, far far lower than their current weightings (by a factor of a 100 for bristlecones, if I remember correctly the effective weighting from one of your other posts).

But you’re right, just eliminate them, they only represent a small fraction of the overall number of proxies, however doing so would open one up to claims of cherry picking. At least eliminating 5% of proxies is a lot easier to defend then selecting 5%, which is what is going on now.

22. Ian
Posted Feb 15, 2006 at 5:06 PM | Permalink | Reply

Sorry that should say:

Even if the flawed proxies do not show up as noise at least they will be given reasonable weightings

The dangers of using voice recognition system.

23. Mark
Posted Feb 15, 2006 at 5:21 PM | Permalink | Reply

The generalized sidelobe canceller is nothing more than the top-level description of a beamformer. The underlying principle is simply to find the correlated elements across a signal set. Assuming your primary signal (main-lobe from a directional antenna) is pointing at your desired signal, and each canceller element is omni-directional, the GSC actually steers nulls (spatially) towards sidelobe interferers.

That there are lobes in the directional portion of the desired signal is irrelevant to this discussion. Where I noted similarity is the concept of finding the correlation across an array of proxies. In this sort of system, the correlated bits are actually the desired signals (as opposed to the noise we want to reject in a radar application, for example), sort of a reverse of the canceller problem. I.e. we are in a sense pulling the correlated signals out of the noise.

I typically used a form of a mofified Gram-Schmidt orthogonalization to implement the GSC and generate the weights – one per element or proxy in this case (do a search at the USPTO for “Gerlach, Karl”
and you’ll see the actual algorithm I’m running). The cascaded structure I use has computational benefits that aren’t important when the process is not real-time. However, I’m wondering if a similar approach can be used, without a main-lobe analogy, to find a correlation across a proxy set and “pull out” the signals from the background noise.

Hopefully we haven’t derailed to far off topic, Steve.

Mark

Posted Feb 15, 2006 at 8:03 PM | Permalink | Reply

Mark, no comment yet from Steve M, but here are my two pence. The more we can get people such as yourself, with experience in things like ECM, or folks with experience in noise management in high speed digital circuits, and the like, the more we can bring to bear the best the world has to offer in order to not only challenge the current “climatological” orthodoxy, but also, to come up with entirely new methodologies for analyzing the – pardon the pun – real climate!

25. Posted Feb 16, 2006 at 6:38 AM | Permalink | Reply

John

re your interjection in the original article by Mark – I think you missed out a square on the second Y hat.

26. Posted Feb 16, 2006 at 6:42 AM | Permalink | Reply

Oops sorry – my apologies – I meant in the original article by Martin Ringo – getting old and dyslexic.

John responds: Quite right! Equation updated. At least someone is awake on this blog….

27. Posted Feb 16, 2006 at 8:22 AM | Permalink | Reply

Re: #19 It would be interesting to see the error predicting the known long series with the short clibration series, plotted against number of Neofs. That would tell us the number that maximizes accuracy. I would do it myself if the script was in R.

28. Posted Feb 16, 2006 at 9:11 AM | Permalink | Reply

The more I look at these graphs I think Martin is right. This method couldn’t even predict the mean. You can see this by imagining a process where the proxies are random series with a given mean, from which a small number are chosen based on correlation with temperatures at the end. Then the reconstruction of those selected series is simply the mean – which is an arbitrary parameter! Looking at the reconstructions, including Moberg, Esper and others, you could see (reading backwards in time) the height of the MWP as a reversion of a random series to a predetermined mean, after a purturbation due to the biased selection for a downward impulse matching temperatures.

29. Mark
Posted Feb 16, 2006 at 10:25 AM | Permalink | Reply

Steve S. Agreed. At least, I think that “related” fields provide a different perspective. There’s a common math underlying any form of signal processing (that’s all this is), but a different viewpoint for each area of interest.

I’m still thinking about ways to extract what we want, btw. What I do for radar (ECM as noted) correlates noisy elements against a known signal or primary element, which is something not necessarily present in climate science (if we had a 1000 year true temperature record we could, but we don’t). perhaps we could “test” the known flawed proxies against the noisy ones to show the correlation between them?

Mark

30. Martin Ringo
Posted Feb 19, 2006 at 12:57 AM | Permalink | Reply

Re #27, David,
I do not believe that the will a great deal of sensitivity in “most cases” for changes in the number of principal components used. The reason is that in most cases (my that I mean most of the parameter space for a given model of the Data Generating Process, “DGP,”) the relevant statistics are close to 1 or 0. I have been playing around (euphemism for I’m still debugging parts of the model) and there are specifications that will bring the “full” and “short” reconstructions closer together. I will run the sensitivity to the number of PCs that in the near future.

Re #28, because I am a simple minded old guy (or maybe because I live most of my working life in a business world), I like to see the numbers and the math. So I try to build my simulations in spreadsheets first with a much simplified version. The “Can one reconstruct from principal components with short sample weighting” issue — that’s a mouthful — becomes very clear in the spreadsheet: no REs, no rho-squareds, not even Theil’s Us, just some simple graphs and the interocular test. The results hit you between the eyes: with weightings (the post-multiplying matrix) from the full sample (which can’t be done in reality), one can reconstruct; with weightings from the short sample, one can’t.

Note, this does not rule out reconstruction from one or a few proxies to one temperature series. But it cast considerable doubt on the reconstruction from many proxies to a many temperature or to a temperature aggregate series. Making this claim truly applicable resolves on how well one can model the range of reasonable specifications of temperature. If temperature were strictly periodic with, say, white noise and with constant spatial correlation, then short sampling would do pretty well. It is easy to go to a fully randomly periodic variation, but how does one move on the continuum between the two? Or how much should the spatial correlations be allowed to change over time? All these things make a big difference in the results. The MBH98 type of reconstruction is a complicated way of saying that the relevant temperature patterns of the past are sufficiently strong in the current instrument record that even if we had the full record, we would change how much to weight PC1, PC2, etc. to reconstruct temperature series A and so forth for series B, C … That condition is a “maintained hypothesis” (untested and assumed a priori to be true, and I think unstated) of the MBH98 model. They would say my type simulations only apply if the simulations could produce short sample weightings that reconstructed temperature approximately correctly. They would probably phrase it in the form that the work is irrelevant unless it simulated strong forcings that affected all series approximately the same way. {Ever think about why someone would have to run a climate model to test if the MBH98 transformation mined for hockey stick? Running a tree growth model (e.g. TreeGrOSS, PTAEDA, STIM, TREGRO — go take a look at it to see what the denrochronology guys are missing by not teaming up with the commercial forestry guys) or maybe something that David figured out in his kind of modeling might make sense as the underlying DGP, but not a climate model. It merely serves to restrict the DGP of the proxies, not estimate it from empirical data.}

Finally note, while we can never test the maintained hypothesis on temperature, we can do a “proxy” test, in both senses of the word. Can the proxy variable series be reconstructed from the short sample? If the proxy variable sets can’t be reconstructed by principal component analysis, then there is more than reasonable doubt as to efficacy of doing so on the temperatures. Any interest?

31. Posted Feb 21, 2006 at 9:33 AM | Permalink | Reply

Re #30. There are three possible outcomes of plotting error against number of PCs. (1) errors decrease with increasing PCs (Mann’s maintained hypothesis) (2) errors are unchanged with increasing PC’s (Martin’s hypotheisi) and (3) errors increase with increasing PC’s (David’s hypothesis). This seems like a nice experiment and I will be interested in seeing the result!

Re #30 re #28. I am the same way. I have generally found z tests to underestimate errors, and like to construct simulations to understand what is going on. As to means, I have a theory that the temperature of the MWP from the reconstructions is related to the normalizing temperature for the series. If I get a chance I will collate and plot the zero anomoly temperature against the MWP temperature for a number of series and see if there is a positive slope! Once again, not proof, but suggestive of these proxies are simply expressing hidden ‘maintained hyportheses’.


text:
# Which bond should I pick if both have the same YTM, tenor, credit risk? One has a higher coupon rate than the other.

#### Ixeua

##### New Member
Suppose I am presented 2 bonds

 YTM Coupon A 5% 6% B 5% 4%

Given that both bonds have the same YTM, tenor, credit risk would either bond be better than another?

A couple of things come to mind. In particular that the Macaulay duration (how fast an investor gets their money back) of bond A is shorter than that of bond B. Would there be any scenarios when B would be a better pick?

#### lushukai

##### Well-Known Member
Subscriber
Hi @Ixeua ,

I guess another necessary assumption would be that both bonds would be the same price? In that case, the principal of bond A is less than bond B (that you receive back at the end of the bond tenor).

My experience tells me that when interest rates are decreasing, bond B would be a better choice as the duration of bond B is greater than bond A, which causes the price of bond B to increase greater than bond A. This makes it a better hedge in a bond-equity portfolio.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
HI @lushukai I don't think they can be the same price: For A, y<c such that p>100. For B, y>c such that p<100. I'm not sure that price per se is an issue: I think the (most common) unstated assumption, for comparison purposes, is that you have the same (eg) $1,000 or$10,000 to invest in either. So, you just end up buying different face amounts given the same value.

The textbook answer is that Bond A offers higher re-investment risk: slightly more coupon income to reinvest along the way (which is related to its lower duration, as there is a trade-off between re-investment and rate/duration risk).

The other thought I had is that Bond A is a premium-priced bond (i.e., y<c --> p>100) and, pulling to par, experiences a small capital loss. So there is a slight tax difference, I think. But it's not a really well-worded question, actually. I hope that's interesting!

#### Ixeua

##### New Member
Hi Thanks! Could you go into detail into the hedging part? A lot of books I read don't include strategies on bond hedging so I am interested on how bond hedging strategies work.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
@Ixeua what is the question's source (where did you see/find it)?

#### Ixeua

##### New Member
Hi @Ixeua ,

I guess another necessary assumption would be that both bonds would be the same price? In that case, the principal of bond A is less than bond B (that you receive back at the end of the bond tenor).

My experience tells me that when interest rates are decreasing, bond B would be a better choice as the duration of bond B is greater than bond A, which causes the price of bond B to increase greater than bond A. This makes it a better hedge in a bond-equity portfolio.

Hi @David Harper CFA FRM. The hedging bit was in reply to @lushukai 's answer here.

If you are asking about the original question, that was purely theoretical. Not from any particular source.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Thanks @Ixeua appreciated, just wondered if the original question had a source. I realize your f/up was in reply to Lu Shu's answer. I think he just means that a hedge might prefer the hedge position to be more sensitive (i.e., higher duration) to interest rates. I don't have time to go into that huge topic. Our Tuckman text covers it quite in depth with many examples. Thanks,

#### Ixeua

##### New Member
Thanks @Ixeua appreciated, just wondered if the original question had a source. I realize your f/up was in reply to Lu Shu's answer. I think he just means that a hedge might prefer the hedge position to be more sensitive (i.e., higher duration) to interest rates. I don't have time to go into that huge topic. Our Tuckman text covers it quite in depth with many examples. Thanks,
I am new here. Could you direct me to that? I would really appreciate it.

Staff member
Subscriber

#### lushukai

##### Well-Known Member
Subscriber
HI @lushukai I don't think they can be the same price: For A, y<c such that p>100. For B, y>c such that p<100. I'm not sure that price per se is an issue: I think the (most common) unstated assumption, for comparison purposes, is that you have the same (eg) $1,000 or$10,000 to invest in either. So, you just end up buying different face amounts given the same value.

The textbook answer is that Bond A offers higher re-investment risk: slightly more coupon income to reinvest along the way (which is related to its lower duration, as there is a trade-off between re-investment and rate/duration risk).

The other thought I had is that Bond A is a premium-priced bond (i.e., y<c --> p>100) and, pulling to par, experiences a small capital loss. So there is a slight tax difference, I think. But it's not a really well-worded question, actually. I hope that's interesting!
Thank you David! I should definitely know this (especially on the bond premium/discount portion + pulling to par) ... it is part of my full-time job as well


text:
For $P$ an arbitrary point in $\triangle ABC$, show that $\sum_{cyc}c(\sin \angle CAP+\sin\angle CBP)\leq a+b+c$

In the interior of $$\triangle ABC$$ we take the arbitrary point $$P$$. Prove that the following inequality holds: $$\small c(\sin\angle CAP + \sin\angle CBP) + a(\sin\angle ABP +\sin\angle ACP) + b(\sin\angle BCP + \sin\angle BAP)\\ \le a + b + c$$

I tried to draw the distances from $$P$$ to the sides of the triangle and write the sines from the right angled triangles, but I couldn't get anything out of it. I saw that a case of equality is when the triangle is equilateral and $$P$$ is its center.

Can you help me?

• Interesting inequality. Where does it come from?
– ΑΘΩ
Mar 14, 2019 at 13:14
• It is from the romanian publishment “Gazeta Matematica”. Mar 14, 2019 at 16:17
• If you would be as kind as to tell us what edition of the Gazette that might be, we might be able to look things up ;)
– ΑΘΩ
Mar 14, 2019 at 16:37
• It is from the edition 10/2018, the number of the problem being S:L18.249 :) Mar 14, 2019 at 16:53

To simplify derivation introduce the notation: $$BC=a,\quad AC=b,\quad AB=c,\\ \measuredangle{CAB}=\alpha,\quad \measuredangle{ABC}=\beta,\quad \measuredangle{BCA}=\gamma;\\ \measuredangle{ABP}=\alpha_B,\quad \measuredangle{ACP}=\alpha_C,\\ \measuredangle{BAP}=\beta_A,\quad \measuredangle{BCP}=\beta_C,\\\ \measuredangle{CAP}=\gamma_A,\quad \measuredangle{CBP}=\gamma_B.$$ Then the initial inequality can be written as: $$\small \frac{a(\sin\alpha_B+\sin\alpha_C) +b(\sin\beta_A+\sin\beta_C)+c(\sin\gamma_A+\sin\gamma_B)}{a+b+c}\le1.\tag{*}$$ which can be proved as follows: \begin{align}\small &\scriptsize\frac{a(\sin\alpha_B+\sin\alpha_C) +b(\sin\beta_A+\sin\beta_C)+c(\sin\gamma_A+\sin\gamma_B)}{a+b+c}\\ &\scriptsize=\frac{\sin\alpha(\sin\alpha_B+\sin\alpha_C) +\sin\beta(\sin\beta_A+\sin\beta_C)+\sin\gamma(\sin\gamma_A+\sin\gamma_B)} {\sin\alpha+\sin\beta+\sin\gamma}\tag1\\ &\scriptsize=\frac{(\sin\beta\sin\beta_A+\sin\gamma\sin\gamma_A) +(\sin\alpha\sin\alpha_B+\sin\gamma\sin\gamma_B) +(\sin\alpha\sin\alpha_C+\sin\beta\sin\beta_C)} {\sin\alpha+\sin\beta+\sin\gamma}\tag2\\ &\scriptsize\le\frac{\sin\alpha+\sin\beta+\sin\gamma}{\sin\alpha+\sin\beta+\sin\gamma}=1\tag3. \end{align}

Explanation:

(1) Substitution: $$a=2R\sin\alpha, \quad b=2R\sin\beta, \quad c=2R\sin\gamma.$$

(2) Rearrangement.

(3) Transformation of sine products to cosine differences and then from cosine sums to cosine products: \begin{align} &\scriptsize\sin\beta\sin\beta_A+\sin\gamma\sin\gamma_A\\ &\scriptsize=\frac{\cos(\beta-\beta_A)-\cos(\beta+\beta_A)}2 +\frac{\cos(\gamma-\gamma_A)-\cos(\gamma+\gamma_A)}2\\ &\scriptsize=\frac{\cos(\beta-\beta_A)+\cos(\gamma-\gamma_A)}2 -\frac{\cos(\beta+\beta_A)+\cos(\gamma+\gamma_A)}2\\ &\scriptsize= \underbrace{\cos\frac{\beta+\gamma-\beta_A-\gamma_A}2}_{\ge0} \underbrace{\cos\frac{\beta-\gamma-\beta_A+\gamma_A}2}_{\le1} - \underbrace{\cos\frac{\beta+\gamma+\beta_A+\gamma_A}2}_{=0}\cos\frac{\beta-\gamma+\beta_A-\gamma_A}2\tag4\\ &\scriptsize\le \cos\frac{\beta+\gamma-\alpha}2=\cos\left(\frac\pi2-\alpha\right)=\sin\alpha, \end{align} where we used $$\beta_A+\gamma_A=\alpha$$.

There may arise an interesting question when does the inequality $$(\text{*})$$ degenerate to equality?

It is easily seen from $$(4)$$ that this happens if and only if the following equations hold: $$\begin {cases} \beta-\beta_A=\gamma-\gamma_A\\ \gamma-\gamma_B=\alpha-\alpha_B\\ \alpha-\alpha_C=\beta-\beta_C \end {cases},$$ which together with $$\begin {cases} \beta_A+\gamma_A=\alpha\\ \gamma_B+\alpha_B=\beta\\ \alpha_C+\beta_C=\gamma \end {cases}$$ implies: $$\begin {cases} \beta_C=\gamma_B=\frac\pi2-\alpha\\ \gamma_A=\alpha_C=\frac\pi2-\beta\\ \alpha_B=\beta_A=\frac\pi2-\gamma \end {cases}.$$

Thus the equality holds if and only if the triangle $$ABC$$ is acute and $$P$$ is its circumcenter.

Note:

With the convention that the angle measures at a triangle vertex are signed and the direction towards the adjacent side is positive, the inequality (*) becomes valid for the whole plane. Correspondingly it degenerates to equality at the circumcenter of the triangle, regardless of its shape.

• Thank you very much! Mar 15, 2019 at 14:58
• @MichaelGoldberg You're welcome!
– user
Mar 15, 2019 at 15:09
• Great presentation! Mar 16, 2019 at 10:28
• @Allawonder Thank you!
– user
Mar 16, 2019 at 10:38
• A lovely solution! Just one remark though: the case of equality entails indeed the equations displayed in the final curly brace, however the conclusion on $P$ is not entirely correct. From $\sphericalangle{ABP}=\sphericalangle{BAP}$ one infers $|PA|=|PB|$ and the analogues, to the conclusion that $P$ is the circumcenter. The requirement that the triangle be acute is indeed valid.
– ΑΘΩ
Mar 21, 2019 at 11:06


text:
5:22 AM
3

Consider an object which has been given a speed $v$ on a rough horizontal surface. As time passes, the object covers a distance $l$ until it stops because of friction. Now, Initial kinetic energy = $\frac{1}2mv^2$ And final kinetic energy is zero. Therefore, work done by friction on the object i...

13 hours later…
6:10 PM
2

Lets say we have a frictionless and massless pulley with a massless string over it. There are two masses attached to it $m_1$ and $m_2$ where $m_2>m_1$. So $m_2$ would an experience an acceleration down while $m_1$ experiences an acceleration up and massless string would also move accordingly. Ho...


text:
» » » Choose the option that rhymes with the given word.Boys

# Choose the option that rhymes with the given word.Boys

### Question

Choose the option that rhymes with the given word.

Boys

A)
stays
B)
moist
C)
noise
D)
elbows


text:
### Responsive design phases

In the paper on responsive design, Groves and Heeringa define "design phases." They argue that each phase has a capacity. Once that capacity has been reached, i.e. the current design has exhausted its possibilities, then a design change may be needed.

A difficulty in practice is knowing when this capacity has been reached. There are two related issues:
1. Is there a statistical rule that can be applied to define the end of the phase?
2. Can we identify when the threshold has been met immediately after it occurs, or is there a time lag?

I don't know that anyone has done much to specify these sorts of rules. I would think these are generalizations of stopping rules. A stopping rule says when to stop the last phase, but the same logic could be applied to stopping each phase. I had a paper on stopping rules for surveys a few years back. And there is another from Rao, Glickman, and Glynn. I don't know that anyone has tried this sort of extension. But I think it is an interesting idea.

### "Responsive Design" and "Adaptive Design"

My dissertation was entitled "Adaptive Survey Design to Reduce Nonresponse Bias." I had been working for several years on "responsive designs" before that. As I was preparing my dissertation, I really saw "adaptive" design as a subset of responsive design.

Since then, I've seen both terms used in different places. As both terms are relatively new, there is likely to be confusion about the meanings. I thought I might offer my understanding of the terms, for what it's worth.

The term "responsive design" was developed by Groves and Heeringa (2006). They coined the term, so I think their definition is the one that should be used. They defined "responsive design" in the following way:

1. Preidentify a set of design features that affect cost and error tradeoffs.
2. Identify indicators for these costs and errors. Monitor these during data collection.
3. Alter the design features based on pre-identified decision rules based on the indi…

### An Experimental Adaptive Contact Strategy

I'm running an experiment on contact methods in a telephone survey. I'm going to present the results of the experiment at the FCSM conference in November. Here's the basic idea.

Multi-level models are fit daily with the household being a grouping factor. The models provide household-specific estimates of the probability of contact for each of four call windows. The predictor variables in this model are the geographic context variables available for an RDD sample.

Let $\mathbf{X_{ij}}$ denote a $k_j \times 1$ vector of demographic variables for the $i^{th}$ person and $j^{th}$ call. The data records are calls. There may be zero, one, or multiple calls to household in each window. The outcome variable is an indicator for whether contact was achieved on the call. This contact indicator is denoted $R_{ijl}$ for the $i^{th}$ person on the $j^{th}$ call to the $l^{th}$ window. Then for each of the four call windows denoted $l$, a separate model is fit where each household is assum…

### Is there such a thing as "mode"?

Ok. The title is a provocative question. But it's one that I've been thinking about recently. A few years ago, I was working on a lit review for a mixed-mode experiment that we had done. I found that the results were inconsistent on an important aspect of mixed-mode studies -- the sequence of modes.

As I was puzzled about this, I went back and tried to write down more information about the design of each of the experiments that I was reviewing. I started to notice a pattern. Many mixed-mode surveys offered "more" of the first mode. For example, in a web-mail study, there might be 3 mailings with the mail survey and one mailed request for a web survey. This led me to think of "dosage" as an important attribute of mixed-mode surveys.

I'm starting to think there is much more to it than that. The context matters  a lot -- the dosage of the mode, what it may require to complete that mode, the survey population, etc. All of these things matter.

Still, we ofte…


text:
# Take a break to make a snake!

Our classic snake has developed an inbalance of growth hormones. To make matters worse, his tail is frozen in place! Given directional input as specified in Figure 1, write a program to determine where he will grow.

Figure 1. Directional input.

# Program specifications

• Read the input character by character on STDIN.
• After reading a character, output the snake to STDOUT. Please include a blank line in between each time you print a snake.
• The snake consists of <>v^ and a head. The head of the snake may be any round character of your choosing, such as o, 0, O, or ☺.
• Any combination of wasd is valid for input.
• Your program should not assume the input is within a certain length.
• The snake can pile on top of itself, overwriting <>v^. See examples for snake growing mechanics.
• Trailing whitespace is okay, but your snake must look correct.

# Scoring

This is . Your score is the number of characters in your program. Lowest score wins!

# Example snakes:

Input: ddddssaassdddddww

Output:

>>>>v
v
v<<  ☺
v    ^
>>>>>^

Input: dddsssaaawww

Output:

☺>>v
^  v
^  v
^<<<

Input: dddsssdddddasddddwww

Output:

>>>v
v       ☺
v       ^
>>>>v<  ^
>>>>^

Input: dddwwdddssssaaaaaaaaawww

Output:

      >>>v
☺     ^  v
^  >>>^  v
^        v
^<<<<<<<<<

Input: ddddssssaawwddddd

Output:

>>>>v
v
>>>>>☺
^ v
^<<

Input: dddddssaawwwwddddd

Output:

   >>>>>☺
^
>>>^>v
^ v
^<<

Input:

ddddaaaasssssdddddddddddwwwwwaaaasssssdddddddddddwwwwwwwwwwsssssaaaasssssdddddddwwwwwddddssaaaasssddddaaaassssssssssdddwwwwwwwddddswaaaassssddaasssaaaaaaaaaawwwwddddssssaaaaaaaaaaawwwwddddsssssssssaaaa

Output:

                  v
v
v
v
v
v<<<<  v<<<<  v<<<<  >>>>v
v      v   ^  v   ^  ^   v
v      v   ^  v   ^  v<<<<
v      v   ^  v   ^  v
v      v   ^  v   ^  v
>>>>>>>>>>>>>>>>>>>>>v<<<<
v
v
v  v<<<<
v  v   ^
v  v
>>>>v  >>>>v  v  v
^   v  ^   v  v  v<<
^   v  ^   v  v  v
^   v  ^   v  v  v
^<<<v<<<<<<<<<<<<<
v
v
v
v
O<<<<

• Very nice. But... WTH is a "code gofl"? – John Dvorak Aug 30 '14 at 14:14
• I'd get rid of the bonus as it is way too easy to create your own snake and input string. – Beta Decay Aug 30 '14 at 14:14
• @JanDvorak rotfl my bad. – hmatt1 Aug 30 '14 at 14:16
• @BetaDecay the plane scales. In the code golf example it started at the top left of the c, but since the d is taller it moves down. – hmatt1 Aug 30 '14 at 14:55
• @chilemagic Thanks! If both are acceptable you should probably clarify this in the question. – Ingo Bürk Aug 30 '14 at 23:36

# Ruby, 207 characters

b=[];x=y=0;gets.chars{|c|b[y]||=[];b[y][x]={?\n=>->{?0},?w=>->{y>0?y-=1:b=[[]]+b;?^},?a=>->{x>0?x-=1:b.map!{|r|[' ']+r};b[y][1]=?<},?s=>->{y+=1;?v},?d=>->{x+=1;?>}}[c][]};puts b.map{|r|r.map{|c|c||' '}.join}

Ungolfed:

b=[]  #board
x=y=0 #position
gets.each_char{|c|
b[y] ||= []
b[y][x] = {
"\n" => lambda{0},
"w"  => lambda{if y>0 then y-=1 else b=[[]]+b; "^"},
"a"  => lambda{if x>0 then x-=1 else b.map!{|r|[' ']+r}; b[y][1]="<"},
"s"  => lambda{y+=1; "v"},
"d"  => lambda{x+=1; ">"}
}[c].call}
puts b.map{|r|r.map{|c|c||' '}.join}

(the lambda for a writes back because the row the assignment above writes to is no longer on the board)

# ECMAScript 6 Javascript (399 401431)

Has to be run in a browser supporting ECMAScript 6 due to the arrow functions.

Here are fiddles which have been altered to run in any (common) browser by not using arrow functions. They also print to a textarea instead:

## Golfed Version

i=prompt(),v=[],c=0,x=[0],y=[0],s='unshift',k='slice',t='sort',h=[0,-1,0,1,0]
while(c<i.length){m='wasd'.indexOf(i[c++]);v[s]('^<v>'[m]);x[s](x[0]+h[m]);y[s](y[0]+h[m+1])}f=(a,b)=>a-b
q=x[k]()[t](f)[0],e=x[k]()[t]((a,b)=>b-a)[0],w=y[k]()[t](f)[0],o=[]

## Animated GIF:

One of the OP's examples:

The example from Stretch Maniac:

## Ungolfed

Here is a (slightly) ungolfed version from sometime before I started really golfing it down:

var input = prompt(),
values = [],
c = 0,
x = [0],
y = [0],
s = 'unshift';
while (c < input.length) {
var mapped = 'wasd'.indexOf(input[c++]);
values[s]('^<v>'[mapped]);
x[s](x[0]+[0, -1, 0, 1][mapped]);
y[s](y[0]+[-1, 0, 1, 0][mapped]);
}

var minX = x.slice().sort(function (a,b){return a-b})[0];
var maxX = x.slice().sort(function (a,b){return b-a})[0];
var minY = y.slice().sort(function (a,b){return a-b})[0];

var output = [];
while((i=y.pop())!=null) {
i-=minY;
j=x.pop()-minX;
t=(output[i]||Array(maxX+1-minX).join(" ")).split("");
t.splice(j,1,values.pop()||"@");
output[i]=t.join("");
}

console.log(output.join("\n"));

• The gifs are very cool. Are they made automatically, by a script you've written? – AndoDaan Aug 31 '14 at 20:25
• Thanks! With the power of Google I just looked up how to make a gift from a screen recording in Ubuntu. It just uses a screen recorder and convert. Pretty easy :) – Ingo Bürk Aug 31 '14 at 20:29
• (gif, not gift) – Ingo Bürk Aug 31 '14 at 20:52

# sed, 71

s/w/\^\x1B[D\x1B[A/g
s/a/<\x1B[2D/g
s/s/v\x1B[B\x1B[D/g
s/d/>/g
s/$/@/  # Golfscript, 165 126 ' '*"\33[":e{e'D'}:-{[e'C'+'<'--]]}:a{[-+'>']]}:d{[e'B'+'^'-e'A']]}:w{[e'A'+'v'-e'B']]}:s{][\[}:+7{;}*''\~[e'H'e'J']\'@'e'20H'  Same approach as my previous answer, but correctly positioning the cursor before and afterwards. I'm pretty proud of the approach to cursor positioning -- basically, it first runs the snake in reverse, without printing out characters. • Can you add an example call? echo "dddddssaawwwwddddd" | sed -e 's/w/\^\x1B[D\x1B[A/g' -e 's/a/<\x1B[2D/g' -e 's/s/v\x1B[S\x1B[D/g' -e 's/d/>/g' -e 's/$/@/' doesn't give the correct output for me. – Ingo Bürk Aug 31 '14 at 9:53
• Your prompt after executing is probably overwriting part of the snake. Paste the snake directly into stdin instead of piping, or add a few \ns after the @ so your prompt goes elsewhere. – Sneftel Aug 31 '14 at 9:57
• This might fail if the snake goes up or left of the board. – tomsmeding Aug 31 '14 at 12:31
• @tomsmeding Yeah, I might expand it to deal with that. Language aside, though, I really think ANSI control sequences are the way to go for shorter code. – Sneftel Aug 31 '14 at 21:47

# Java - 646

Might as well be the first one!

I bet you all can beat this.

un(sort of)golfed

import java.util.*;
public class Snake{
public static void main(String[]a) {
int x,y,minX,minY,maxX,maxY;
x=y=minX=maxX=minY=maxY=0;
List<Integer>xs,ys=new ArrayList<Integer>();
xs=new ArrayList<Integer>();
List<Character>p=new ArrayList<Character>();
for(int b=0;b<a[0].length();b++){
int newX=x,newY=y;
switch(a[0].charAt(b)){
}
x=newX;y=newY;
if(x<minX){minX=x;}
if(x>maxX){maxX=x;}
if(y<minY){minY=y;}
if(y>maxY){maxY=y;}
}
char[][]c=new char[maxY-minY+1][maxX-minX+1];
for(int i=0;i<xs.size();i++)c[ys.get(i)-minY][xs.get(i)-minX]=p.get(i);
c[y-minY][x-minX]='@';
for(char[]k:c){for(char l:k){System.out.print(l);}System.out.println();}
}
}

Smaller -

import java.util.*;class S{public static void main(String[]a){int x,y,o,z,s,u;x=y=o=s=z=u=0;List<Integer>j,t=new ArrayList<Integer>();j=new ArrayList<Integer>();List<Character>p=new ArrayList<Character>();for(int b=0;b<a[0].length();b++){int e=x,r=y;switch(a[0].charAt(b)){case'a':e--;p.add('<');break;case's':r++;p.add('v');break;case'd':e++;p.add('>');break;case'w':r--;p.add('^');break;}j.add(x);t.add(y);x=e;y=r;if(x<o)o=x;if(x>s)s=x;if(y<z)z=y;if(y>u)u=y;}char[][]c=new char[u-z+1][s-o+1];for(int i=0;i<j.size();i++)c[t.get(i)-z][j.get(i)-o]=p.get(i);c[y-z][x-o]='@';for(char[]k:c){for(char l:k){System.out.print(l);}System.out.println();}}}

input - dddsssdddwwwwaaaaaaaassssssssssddddddddddddddddd

v<<<<<<<<
v >>>v  ^
v    v  ^
v    v  ^
v    >>>^
v
v
v
v
v
>>>>>>>>>>>>>>>>>@

input - dddsssdddddasddddwww

>>>v
v       @
v       ^
>>>>v<  ^
>>>>^

my personal favorite - dwdwdwddaasassdddddwdwdwddsdswawaasassdddddddwdwdwddsdswawaasassddddwwwwwwwssssssdsdddwwwwddaassddaassddddsssdddwdwdwddaasasassddddwwwwssssssssasasaaawdwwdwddwwdddddddwdwdwddsdswawaasassddddddddddwwdwwwwaasssassdsdddddddwdwdwwwwasasssssssssssdwwwwwwwddd

                    v
v
v
v   v<<
v<<   v<<     v<<v   v
v<    v< ^<   v< ^v   v<<                        v<<     v<
>v    >v   ^  >v   >v  v                          v ^    v<^
>^>>>>>^>>>>>>>^>>>>^>>>>>>>v    v<v               v ^    v ^
v   v< v       v<<    v< ^    v ^
v  v<  v      v< ^<   v >^    v>^
>>>v   v     >v   ^   >v^     v>>>@
>>>>>>>>>>^>>>>>>>>>>>>>>>>v^
^v                      v^
>>^v                      v^
>^  v                      v^
^  v<                      v^
>^ v<                       v^
^<<<                        >^

# C# 607

namespace System{using B=Text.StringBuilder;class P{static void Main(){var f=new Collections.Generic.List<B>(){new B("O")};int w=1,r=0,c=0;for(Action R=()=>f[r].Append(' ',w-f[r].Length+1);1>0;){var key=Console.ReadKey(1>0).KeyChar;if(key=='w'){f[r][c]='^';if(--r<0)f.Insert(r=0,new B());R();f[r][c]='O';}if(key=='a'){f[r][c]='<';if(--c<0){foreach(var s in f)s.Insert(c=0,' ');w++;}R();f[r][c]='O';}if(key=='s'){f[r][c]='v';if(++r>f.Count-1)f.Add(new B());R();f[r][c]='O';}if(key=='d'){f[r][c]='>';if(++c>w++)foreach(var s in f)s.Append(' ');R();f[r][c]='O';}Console.WriteLine(string.Join("\n",f)+"\n");}}}}

"Ungolfed" with whitespace (this will not be kept in sync with the golfed version):

namespace System
{
using B = Text.StringBuilder;
class P
{
static void Main()
{
var f = new Collections.Generic.List<B>() { new B("O") };
int w = 1, r = 0, c = 0;
Action R = () => f[r].Append(' ', w - f[r].Length + 1);
while (true)
{
if (key == 'w')
{
f[r][c] = '^';
if (--r < 0) { f.Insert(0, new B()); r = 0; }
R();
f[r][c] = 'O';
}
if (key == 'a')
{
f[r][c] = '<';
if (--c < 0)
{
foreach (var s in f)
s.Insert(0, ' ');
w++;
c = 0;
}
R();
f[r][c] = 'O';
}
if (key == 's')
{
f[r][c] = 'v';
if (++r > f.Count - 1) f.Add(new B());
R();
f[r][c] = 'O';
}
if (key == 'd')
{
f[r][c] = '>';
if (++c > w++)
{
foreach (var s in f)
s.Append(' ');
}
R();
f[r][c] = 'O';
}

Console.WriteLine(string.Join("\n", f) + "\n");
}
}
}
}

# Python 3: 259 Bytes

x=y=0
b,p,r={},(0,-1,0,1),range
while 1:
d='wasd'.index(input());b[(x,y)]='^<v>'[d];x+=p[d];y-=p[~d];b[(x,y)]='☺';l,m=([k[i]for k in b]for i in(0,1))
for j in r(min(m),max(m)+1):print(''.join(b[(i,j)]if(i,j)in b else' 'for i in r(min(l),max(l)+1)))
print()

I decided to store the snake in a dict, with coordinates for the keys. Then find and iterate over the output range, substituting blank spaces.

x = y = 0
board = {}
while 1:
d = 'wasd'.index(input())
board[(x, y)] = '^<v>'[d] # body
x += (0, -1, 0, 1)[d]
y -= list(reversed((0, -1, 0, 1)))[d]

xs, ys= ([coord[dim] for coord in board] for dim in(0, 1))
for j in range(min(ys), max(ys)+1):
print(''.join(board[(i,j)] if (i,j) in board else ' '
for i in range(min(xs), max(xs)+1)))
print()

PS. My first Golf :) Let me know if my answer is inappropriate

• Using O instead of ☺ saves you 2 bytes. ☺ is a red herring. – Erik the Outgolfer Jul 31 '16 at 18:20
• @EʀɪᴋᴛʜᴇGᴏʟғᴇʀ ☺ wasn't actually counted as 3 bytes. – Martin Ender Jul 31 '16 at 18:30
• @MartinEnder It must be, the default encoding is UTF-8. I had a feeling it wasn't counted. I have a feeling it was accidental and must be immediately fixed by Gilly. – Erik the Outgolfer Jul 31 '16 at 18:39
• @EʀɪᴋᴛʜᴇGᴏʟғᴇʀ Well technically, the answerer is free to use any encoding supported by their interpreter (and I'm pretty sure some ASCII-compatible code page contains that character) but that's beside the point. I'm saying the byte count is already the same as that of using O, so I wouldn't worry about it. It's clear from the code that it would still work for any other character but using ☺ lets you conveniently run the test cases from the challenge without changes. – Martin Ender Jul 31 '16 at 18:43
• @MartinEnder Python uses UTF-8, proved by its ability to support unicode strings. To enable this functionality, the first or second line must be #coding=utf-8. Note that #coding=utf-16 does not work. Hence ☺ must be counted as 3. – Erik the Outgolfer Jul 31 '16 at 18:45

## Python 2.7 - 274 bytes

x,y,m,d,r=0,0,{},(0,-1,0,1),range
for c in raw_input():b='wasd'.index(c);m[(x,y)]='^<v>'[b];x+=d[b];y-=d[~b];m[(x,y)]='@';l,n=([k[h] for k in m] for h in (0, 1))
for j in range(min(n),max(n)+1):print(''.join(m[(i,j)] if (i,j) in m else ' 'for i in range(min(l),max(l)+1)))

## Ungolfed version

x,y,matrix,delta = 0,0,{},(0, -1, 0, 1)
for c in raw_input('Command: '):
d = 'wasd'.index(c)
matrix[(x, y)] = '^<v>'[d]
x += delta[d]
y -= list(reversed(delta))[d]
matrix[(x, y)] = '@'
xs, ys = ([xy[i] for xy in matrix] for i in (0, 1))
for j in range(min(ys), max(ys)+1):
print(''.join(matrix[(i, j)] if (i, j) in matrix else ' '
for i in range(min(xs), max(xs)+1)))

• Welcome to Programming Puzzles & Code Golf! According to the rules outlined in our help center, all solutions to challenges should be a serious contender for the winning criteria in use. For example, an entry to a code golf contest needs to be golfed. – Dennis Mar 19 '16 at 17:09
• I'm added the byte count for you, but there's quite a lot of unnecessary spaces in there which you might want to remove. – Martin Ender Mar 19 '16 at 18:38
• Thanks guys, I made the necessary changes to my first entry. Any extra advice will be greatly appreciated. – Adriaan Erasmus Mar 20 '16 at 7:20
• – Erik the Outgolfer Jul 31 '16 at 19:04
• @EʀɪᴋᴛʜᴇGᴏʟғᴇʀ That just prints SyntaxError: invalid syntax. – Dennis Jul 31 '16 at 19:34

# 05AB1E, 353430 28 bytes

.•₃º•"<>v^"©‡0ªÐUĀ>sŽO^®XkèΛ

Uses 0 as head of the snake.

-4 bytes thanks to @Grimy.

Try it online (no test suite for all test cases at once, because there is no way to reset the Canvas, so outputs would overlap..).

Explanation:

.•₃º•          # Push compressed string "adsw"
"<>v^"    # Push string "<>v^"
©   # Save it in variable r (without popping)
‡  # Transliterate the (implicit) input-string,
# replacing all "adsw" with "<>v^" respectively
#  i.e. "ddddssaassdddddww" → ">>>>vv<<vv>>>>>^^"
0ª             # Convert the string to a list of characters, and append a 0 (for the head)
#  → [">",">",">",">","v","v","<","<","v","v",">",">",">",">",">","^","^","0"]
Ð            # Triplicate this list of characters
U           # Pop and store one of the three lists in variable X
Ā           # Trutify each character ("0" remains 0; everything else becomes 1)
>          # And then increase each integer by 1
#  → [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,1]
s            # Swap the two lists on the stack
®Xk      # Get the index of each character of variable X in variable r ("<>v^")
#  i.e. [">",">",">",">","v","v","<","<","v","v",">",">",">",">",">","^","^","0"]
#   → [1,1,1,1,2,2,0,0,2,2,1,1,1,1,1,3,3,-1]
ŽO^   è     # And use those to index into the compressed number 6240
#  → [2,2,2,2,4,4,6,6,4,4,2,2,2,2,2,0,0,0]
Λ              # Use the Canvas builtin with these three lists

See this 05AB1E tip of mine (sections How to compress strings not part of the dictionary? and How to compress large integers?) to understand why .•₃º• is "adsw" and ŽO^ is 6240.

As for a brief explanation of the Canvas builtin Λ and its three arguments:

First argument: length(s): the sizes of the lines we want to draw. Since we have to keep in mind overlapping, we use size 2 for every character, and an additional 1 for the head of the snake.
Second argument: string(s): the characters we want to display. Which are the characters in this case, appended with the head-character of the snake.
Third argument: direction(s): the directions these character-lines of the given length should be drawn in. In general we have the directions [0,7] which map to these directions:

7   0   1
↖ ↑ ↗
6 ← X → 2
↙ ↓ ↘
5   4   3

Which is why we have the integer 6240 for the directions $$\[←,→,↓,↑]\$$ respectively.

See this 05AB1E tip of mine for a more detailed explanation about the Canvas builtin Λ.

• I might have missed something, but I don't think γ is needed at all. This seem to work just fine. – Grimmy Jun 4 at 10:37
• @Grimy Thanks, it indeed does; nice alternative approach with the list of 2s and 1 for the head! And I've been able to save 2 more bytes based off your program. – Kevin Cruijssen Jun 4 at 12:02
• Thanks! I am currently at 23, though that's a significantly different approach, so I might make it its own answer if that's fine by you. – Grimmy Jun 4 at 12:22
• @Grimy That indeed is quite a different approach than mine, so feel free to post it. I was indeed kinda expecting some unicode conversion and modulo might make this shorter than the transliterate, but I'm quite honestly pretty bad with those kind of magic integer/string conversions. :) – Kevin Cruijssen Jun 4 at 12:24

# Perl - 394

Not the shortest, but it beats Javascript, C# and Java at least.

use List::Util qw(min max);sub c{()=$_[0]=~/$_[1]/g}%l=(a,['<',-1,0],d,['>',1,0],w,['^',0,-1],s=>['v',0,1]);($s,$x,$y,$w,$h)=($ARGV[0],0,0,max(c($s,a),c($s,d)),max(c($s,w),c($s,'s')));@s=split'',$s;map$x=min($x,$i+=$l{$_}[1]),@s;$i=0;map$y=min($y,$i+=$l{$_}[2]),@s;$x=abs$x;$y=abs$y;map{$m[$y][$x]=$l{$_}[0];$x+=$l{$_}[1];$y+=$l{$_}[2]}@s;$m[$y][$x]='o';map{map{print$_||' '}@$_;print"\n"}@m

Some tricks:

• Warnings and strict not turned on to allow barewords and not declaring variables before using them
• Thin commas instead of fat commas to save a few characters
• Not setting initial values for variables when not necessary
• Leaving out semi-colons when possible
• Defining arrays and hashes not as references to avoid having to use ->
• Allowing width, height to be larger than necessary to avoid having to calculate them accurately (which would take extra code)

Things that hurt:

• No built-in way to count number of characters in a string (might have been longer anyway)
• No built-in min/max functions, thus need to waste 27 characters to import library that does it (less than defining our own)

# C - 273 bytes - with Interactive Input!

#define F for(i=w*w
*g,*G,x,i,j,w,W,u;main(w){putch(1);F;j=-~getch();g=G){if(!(x%w&&~-~x%w&&x/w&&x/w^~-w)){W=w+6;G=calloc(W*W,4);F-1;u=i%w+i/w*W-~W*3,i==x?x=u:8,i;)G[u]=g[i--];free(g);w=W;}G[x]="<^X>v"[j%=7];G[x+=1-G[x]%3+W*(!!j-j/2)]=1;F;i;)putch(i--%W?G[i]?G[i]:32:10);}}

The field is printed each time a character is entered and grows if the snake's head nears the edge. I don't know how portable it is--someone on the Internet said getch() doesn't work on non-Windows platforms. Hard to say whether ASCII 1 will look like a smiley face either.

The golfed version is quite annoying as there is no way to gracefully exit the program. Control-C doesn't work for me. On the other hand, the ungolfed version terminates if a character other than 'w', 'a', 's', or 'd' is entered.

So-called "ungolfed":

#define SMILEYFACE 1
int main()
{
int o;
int w = 1;
int *g = 0, *g2;
int c, n;
int x = 0;
for( putch(SMILEYFACE);c = getch(); ) {
if(c!='w'&&c!='a'&&c!='s'&&c!='d')
return 1;
if(!(x%w) | !(~-~x%w) | !(x/w)  | !(x/w-~-w) ) {
int wnew = w + 4;
int off = 2;
g2 = calloc(wnew*wnew,sizeof(int));
for(n = w*w; --n; )
g2[ n%w+off + (n/w+off)*wnew ] = g[n];
free(g);
g = g2;
x = (x/w+off)*wnew + x%w + off;
w = wnew;
}
int i = -~c%7;
g[x] = "<^X>v"[i];
int dx = 1-g[x]%3 + w * (!!i-i/2);
x += dx;
g[x] = SMILEYFACE;
for(o = w*w; o; )
putch(o--%w?g[o]?g[o]:32:10);

}
return 0;
}

# 05AB1E, 23 bytes

Ç7%DÉ+D"^>>v"ºsè0ªDĀ>rΛ

Try it online!

Explanation:

                      # implicit input (eg: "wasd")
Ç                     # codepoints (eg: [119, 97, 115, 100])
7%                   # modulo 7 (eg: [0, 6, 3, 2])
DÉ+                # plus itself modulo 2 (eg: [0, 6, 4, 2])
# This is the list of directions that will be passed to 05AB1E's canvas function, Λ.
# 0 means up, 6 left, 4 right, 2 down.

"^>>v"º              # "^>>v", horizontally mirrored (namely "^>>vv<<^")
D       sè            # index into this with a copy of the list of directions
0ª          # append "0"
# This is the list of strings that will be drawn.

D                     # duplicate the list of strings
Ā                    # truthify (maps letters to 1, 0 stays 0)
>                   # increment each
# This is the list of lengths to draw.

r                     # reverse the stack because Λ takes arguments in the opposite order
Λ                    # draw!


text:
# zbMATH — the first resource for mathematics

Logical and semantic purity. (English) Zbl 1159.03004
Preyer, Gerhard (ed.) et al., Philosophy of mathematics. Set theory, measuring theories, and nominalism. Frankfurt: Ontos Verlag (ISBN 978-3-86838-009-5/hbk). LOGOS. Studien zur Logik, Sprachphilosophie und Metaphysik 13, 40-52 (2008).
Starting with Hilbert’s formulation of the concern for the purity of the method (“one strives to use in the proof of a theorem as far as possible only those auxiliary means that are required by the content of the theorem”), the author distinguishes between two kinds of pure proofs of theorems: (i) logically pure proofs, which are proofs carried out from a minimal subset of axioms of a given axiom system, and (ii) semantically pure proofs, which “draw only on what must be understood and accepted in order to understand that theorem”.
He then shows that: (1) “Some results require more concepts and/or propositions to be proved than to be understood”, and (2) “Some results require more concepts and/or propositions to be understood than to be proved”. To establish (1), he uses the example of the casus irreducibilis for cubic polynomials (which requires complex numbers for its solution, but not for its understanding) and that of Gödel sentences in Peano Arithmetic (which can be understood as arithmetical statements, but not proved as such statements), whereas for (2) he uses the theorem stating that there are infinitely many primes, which can pe proved in fragments of arithmetic, but requires a more generous axiom system to be understood.
For the entire collection see [Zbl 1149.03003].

##### MSC:
 03A05 Philosophical and critical aspects of logic and foundations 00A30 Philosophy of mathematics 03B30 Foundations of classical theories (including reverse mathematics) 03F07 Structure of proofs


text:
kidzsearch.com > wiki   Explore:images videos games

# Triton (moon)

Triton
The orbit of Triton (red) is different from most moons' orbit (green) in the movement's direction, and the orbit is tilted
Retrograde orbit: the satellite (red) orbits in the direction opposite to the rotation of its primary (blue/black)
The Kuiper belt (green), in the Solar System's outskirts, is where Triton is thought to have originated.

Triton, or Neptune I, is the largest moon of the planet Neptune.[1][2] It is the seventh largest moon in the Solar System. Triton is slightly smaller than Earth's Moon. Triton has a complicated geological history and it is thought to have a comparatively young surface compared to the age of the Solar System itself.

It was discovered by the British astronomer William Lassell on 10 October 1846, just 17 days after Neptune itself was discovered by the German astronomer Johann Gottfried Galle and Heinrich Louis d'Arrest.[1]

Triton is believed to be a captured Kuiper Belt object,[1] and is the coldest known body in the Solar System.[1] The surface temperature of Triton was recorded by Voyager 2 as -235 °C (-391 °F).[1] Triton has its own magnetic field and has a faint trace of an atmosphere.

## Orbit

Drag and tidal interaction cause Triton to fall into a lower orbit. This process is immensely slow. 3.6 billion years from now, Triton will pass within Neptune's Roche limit.[3] It will either collide with Neptune's atmosphere or break up, forming a ring system similar to that found around Saturn.[3]

The reason why Triton is thought to be a captured object is its orbit, which is unique in the Solar System. It is both retrograde (see diagram) and highly tilted. Although there are other satellites with retrograde orbits, they are much smaller than Titan, and much more distant from their host planets (their "primaries"). Triton, like our Moon, shows only one face to the planet: it rotates in synchrony to Neptune.

## Surface

All we know about the surface of Titan comes from one fly-by of Voyager 2 in 1989. There are few impact craters. This suggests the surface is quite young: estimates are six to 50 million years.[4]

Despite its very cold temperature, the surface of Triton is geologically active. There are events which look like geysers, volcanos and earthquakes. All the materials involved are quite different from those on Earth. Most of the gases are frozen most of the time. Gases like nitrogen are frozen until something occurs to melt them, then it turns back into a gas. A temperature increase of just 4° K is enough to make this happen.

## References

1. Chyba C.F; Jankowski D.G. & Nicholson P.D. 1989. Tidal evolution in the Neptune-Triton system. Astronomy and Astrophysics 219 (1–2): L23–L26. [1]
2. Schenk, Paul M. & Zahnle, Kevin 2007. On the negligible surface age of Triton. Icarus 192 (1): 135–49. [2]
• Sir Patrick Moore (2003). Atlas of the Universe. Great Britain: Philips'.
      .


text:
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 Diskr. Mat.: Year: Volume: Issue: Page: Find

 Random polynomials over a finite fieldG. I. Ivchenko, Yu. I. Medvedev 3 Random permutations with cycle lengths in a given finite setA. N. Timashev 25 The Kloss convergence principle for products of random variables with values in a compact group and distributions determined by a Markov chainI. A. Kruglov 38 On enumeration of labelled connected graphs by the number of cutpointsV. A. Voblyi 52 On random 2-adjacent 0/1-polyhedraV. A. Bondarenko, A. G. Brodskiy 64 The intersection number of complete $r$-partite graphsN. S. Bol'shakova 70 Skew Laurent series rings and the maximum condition on right annihilatorsA. A. Tuganbaev 80 On Mazurov triples of the sporadic group $B$ and Hamiltonian cycles of the Cayley graphA. I. Makosiy, A. V. Timofeenko 87 Classification of indecomposable Abelian $(v,5)$-groupsA. A. Frolov 94 On complexity of linear operators on the class of circuits of depth 2D. Yu. Cherukhin 109 On complexity of realisation of a class of almost symmetric functions by formulas of depth 3S. E. Cherukhina 120 On complexity of the anti-unification problemE. V. Kostylev, V. A. Zakharov 131 A block algorithm of Lanczos type for solving sparse systems of linear equationsM. A. Cherepnev 145 Some characteristics of dependencies in discrete random sequencesM. I. Tikhomirova 151


text:
# Let X be a first countable space and â‚¬ a limit point of a subset A of X: Prove that

###### Question:

Let X be a first countable space and â‚¬ a limit point of a subset A of X: Prove that there is a sequence of points of A| {~} which converges to â‚¬.

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text:
# Why is $9 \times 11{\dots}12 = 100{\dots}08$?

While I was working on Luhn algorithm implementation, I discovered something unusual.

$$9 \times 2 = 18$$ $$9 \times 12 = 108$$ $$9 \times 112 = 1008$$ $$9 \times 1112 = 10008$$

Hope you can observe the pattern here.

What to prove this? What is it's significance?

• It's so good to see when people get interested in these beautiful things! and they start looking for explanation :) – Daniel Jan 21 '17 at 14:07
• Try doing the multiplications with pencil and paper rather than relying on a calculator ... – hmakholm left over Monica Jan 21 '17 at 14:45
• Hint $\ 9 = 10\!-\!1\$ divides $\ 10^N\!+\!8 = 10^N\!-\!1 + (10-1)\ \$ See Casting Out Nines. – Bill Dubuque Jan 21 '17 at 15:26

The repunit, $R_k = \overbrace{111\ldots 111}^{k \text{ ones}}$ , can be written as $R_k = \dfrac{10^k-1}{9}$

Your nice pattern corresponds to $9\times (R_k+1) = (10^k-1)+9 = 10^k+8$

Here's an idea.

This pattern is easier to understand: $$9 \times 11 = 99$$

$$9 \times 111 = 999$$

$$9 \times 1111 = 9999$$

Then see what happens when you add 9 to both sides.

One logic behind.

$9 \times (0 + 2) = 9 \times 0 + 9 \times 2 = 18$

$9 \times (10 + 2) = 9 \times 10 + 9 \times 2 = 108$

$9 \times (110 + 2) = 9 \times 110 + 9 \times 2 = 1008$

So on.

$$9 \times 11\cdots12 = 9 \times (11\cdots 11+1) = 99\cdots99 + 9.$$

Write $1...12$ as $$\overline{1...(\times\ n)...1} +1=\frac{10^n-1}{9}+1$$

Hence $$9\times (\frac{10^n-1}{9}+1)=10^n-1+9=10^n+8$$


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# Regular ring

In commutative algebra, a regular ring is a commutative noetherian ring, such that the localization at every prime ideal is a regular local ring: that is, every such localization has the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension.

The origin of the term regular ring lies in the fact that an affine variety is nonsingular (that is every point is regular) if and only if its ring of regular functions is regular.

For regular rings, Krull dimension agrees with global homological dimension.

Jean-Pierre Serre defined a regular ring as a commutative noetherian ring of finite global homological dimension. His definition is stronger than the definition above.

Examples of regular rings include fields (of dimension zero) and Dedekind domains. If A is regular then so is A[X], with dimension one greater than that of A.

In particular if k is a field, the polynomial ring ${\displaystyle k[X_{1},\ldots ,X_{n}]}$ is regular. This is Hilbert's syzygy theorem.

Any localization of a regular ring is regular as well.

A regular ring is reduced[1] but need not be an integral domain. For example, the product of two regular integral domains is regular, but not an integral domain.[2]


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# CSS - Have each item in a horzontal list have an equal share of the available padding

Consider a simple CSS layout, where containing wrapper div defines a fixed width of 900 or so pixels, everything inside it therefore expands to 100% width.

In here, I have a navigation div, containing 1 UL and 6 list items, left-floated so they appear in a line horizontally.

Each list item should growing variably to fit its text contents exactly, but the spacing between each item should be shared so that the whole menu fits into the 100% space, such as:

------------------------------------------------------
-N-        -N-          -N-               -N-      -N-
- -        - -          - -               - -      - -
- -ITEM1111- -ITEM222222- -ITEM33333333333- -ITEM44- -
- -        - -          - -               - -      - -
------------------------------------------------------
<-------------------- 100% -------------------------->

I hope that is illustrative! 'N' is constant but grows to fit the 100% width accordingly (i.e. for accessibility - someone increases the font size).

Happy to take alternative suggestions although i'm aiming for no javascript or images, just css purity.

Thanks!

-
Does the LI padding matter at all? If not then you might consider removing the reference to padding in your question title. –  cfx Aug 1 '13 at 21:31

Here is my simple solution for this based on CSS3 Flexible Box Layout:

Styles

ul {
width: 100%;
}

ul li {
list-style: none;
text-align: center;
border: 1px solid Gray;
}

display: -webkit-box;
display: -moz-box;
display: box;
}

-webkit-box-flex: 1;
-moz-box-flex: 1;
box-flex: 1;

-webkit-flex-basis: auto;
-moz-flex-basis: auto;
flex-basis: auto;
}

HTML

short

looooooooooooooooooooooooooooooong

short

short

Result

-
+1 for flexi-sexi but the problem stated that the list items should be as wide as they need. the space left from 900px should then be divided equally and used as the padding between each item..the padding is therefore the same width and the ul width = 900 –  jenson-button-event Oct 29 '12 at 22:35
In my example the second item is bigger, than others, and as you can see on the image, it takes as much space, as it needs, decreasing paddings respectively. I've specified width: 400px; here just for example. You can change it to any value and it will work. –  s.ermakovich Oct 30 '12 at 7:53
@jenson-button-event I've updated answer to fit the question precisely. –  s.ermakovich Oct 30 '12 at 8:09

If IE6/IE7 compatibility is not critical to you, this layout is trivial using display: table

http://jsfiddle.net/5SFG5/

-
Not sure what you mean by "the padding issue" gordian. If you read the original question, it speaks in terms of width and spacing, not padding. –  Gaurav Jul 31 '13 at 14:46
Your LIs do not have any padding, which the OP posed as a major point in the question. I don't see a way for them to all share in the available padding in your fiddle. –  cfx Jul 31 '13 at 18:30
gordion, read the question. The word padding is only mentioned in the title, and the user wants the LIs spaced evenly. –  Gaurav Jul 31 '13 at 21:40
Difficult to discern what the OP means by "padding". –  cfx Aug 1 '13 at 0:54

Problem is that when an element is floated, its width is reduced to the actual width of the content. You can toy with this using Javascript though (I know you requested a no-js version, but I can't think of another option). In jQuery it should be fairly easy. Given this structure:

ITEM1

ITEM2

ITEM3

you could do something like this:

var width = 0, maxwidth = 900;
$('#nav li').each(function(){ width +=$(this).width();
});
var count = 2 * $('#nav').size(); var margin = (maxwidth - width) / count;$('#nav li').css({
marginLeft: margin + 'px',
marginRight: margin + 'px'
});

Sorry for the messy code, I'm in a bit of a hurry :D. Anyways, what you do is, basically: check the total width occupied by the li elements, check how much is unoccupied and fill it with li margins.

-

I'm not sure how N can be both a constant and a variable width.

Here's a start..

CSS

#nav { width: 100%; display: inline; }
#nav ul li {
display: block;
float: left;
margin: 2%;
width: 10%;
}

#nav ul li a {margin: 0 auto; display: block;}

html

Item 1

Item 1

Item 1

Item 1

Item 1

Item 1

-
N is constant across all columns. N=100%-(width-of-all-columns)/number-of-columns –  jenson-button-event Dec 18 '10 at 14:35


text:
# Quarkonia meet Dark Matter

15-18 June 2021
Kavli IPMU, Kashiwa, Japan
Asia/Tokyo timezone

## A model of electroweakly interacting non-abelian vector dark matter

15 Jun 2021, 23:20
35m
Remote access (Kavli IPMU, Kashiwa, Japan)

Kashiwa, Japan

Motoko Fujiwara

### Description

In this talk, we propose a new electroweakly interacting spin-1 dark matter (DM) model. We consider the non-Abelian extension of electroweak symmetry. Namely, we extend the SU(2)$_L$ group in the Standard Model (SM) into the direct products of three SU(2) groups. We also impose the exchange symmetry between two of these SU(2) groups to realize the spin-1 stable spectrum. In this setup, the DM pair efficiently annihilate into SM particles through the electroweak interaction. Therefore, we can obtain the DM energy density correctly via the freeze-out mechanism. We also find not only electroweak processes but also Higgs exchange processes give the relevant contribution to determine the DM energy density. We conclude a next-generation DM searches will be an excellent probe of this spin-1 DM.

### Presentation Materials

 Motoko_FUJIWARA_poster.png Motoko_Fujiwara-short.pdf