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# Bonferroni Adjustment/Correction and Confidence Intervals

#### Jake-Finn

##### New Member
Hello,

I have two questions here.

RESOLVED (for now): You perform the bonferroni for multiple tests and/or multiple comparisons.
FIRST: I have performed 4 mixed between-within subjects ANOVAs, that look at the differences between time (2 time points) and group (2 groups). I have been told in some places that you divide you alpha level by the amount of tests you're running, but in the below quote from Pallant 2011 is says "comparisons" rather that "tests", and it has me wondering whether, with 4 mixed between-within subjects ANOVAs looking at time points and group differences, do I need to divide the alpha level by 4 or by 8?

SECOND: If I'm reporting confidence intervals with the rest of my output, is there anything I need to do with them, any adjustments, etc, any calculations for the confidence intervals, relating to the Bonferroni adjustment. I know that for the Bonferroni adjustment I am examining my p values according to the adjustment (in this case I think it's: 0.0125/0.013), but is there anything different I need to do when reporting the confidence intervals?

I have searched the forum, looked in my textbooks, but there I can't find a clear (to me) answer. Thank you for your help.

"The other alternative is to apply what is known as a Bonferroni adjustment to the alpha level that you will use to judge statistical significance. To achieve this, you can divide your alpha level (usually .05) by the number of comparisons that you intend to make, and then use this new value as the required alpha level. For example, if you intend to make three comparisons the new alpha level would be .05 divided by 3, which equals .017."

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#### hlsmith

##### Less is more. Stay pure. Stay poor.
You typically correct for the number of comparisons. So if you did post ANOVA ttests for the 3 group means, you would either divide your alpha by 3 for the cut0ff or multiply your p-values by 3. If you are also reporting values with CI, then you need to modify them too. So you would report the 95% CI, but due to correction they would actually be 96.666...% CI

#### Jake-Finn

##### New Member
I'm doing the Bonferroni adjustment/correction due to performing multiple ANOVAs, as I was not able to perform a MANOVA. Following this advice from Pallant (just re-found this; so, I think I understand that you do it for multiple tests or comparisons):

"One way to control for the Type 1 error across multiple tests is to use a Bonferroni adjustment. To do this, you divide your normal alpha value (typically .05) by the number of tests that you intend to perform. If there are three dependent variables, you would divide .05 by 3 (which equals .017 after rounding) and you would use this new value as your cut-off. Differences between your groups would need a probability value of less than .017 before you could consider them statistically significant."

I have performed 4 mixed between-within subjects ANOVAs, aka split-plot ANOVA design (SPANOVA). For each ANOVA I have the same 2 groups, and 2 time points.

So, I performed a Bonferroni adjustment due to performing multiple tests, following Pallant's (2011) advice, I'm happy that dividing the alpha by 4 is what I should be doing for the 4 ANOVAs (the, first question is, for me, for now, resolved).

QUESTIONS: How did you get to that 96.666...% CI number?
In my case, if I'm using the Bonferroni adjustment to account for the multiple ANOVAs, rather than comparisons, should I alter anything to do with the confidence intervals?
And, how do I do that?
What is the calculation?
Do I just write in a different number for the CI percentage, or do I adjust the upper and lower confidence intervals themselves?

Also, there's no mention of post-hoc tests in the guide that I'm following, but I'm starting that as a separate question/thread, for the purposes of being concise: http://www.talkstats.com/showthread...NOVA-aka-split-plot-ANOVA?p=174939#post174939

#### Jake-Finn

##### New Member
Thank you.

Then if I were to use the Bonferroni adjustment, following Pallant's advice, and adjusted my alpha level, diving it by 4, then I'd do (5%/4)+95%?

Is there a text source for that calculation? (I have to cite/reference decisions like this).

And, do I just write in a different number for the CI percentage, or do I adjust the upper and lower confidence intervals themselves?

#### Mean Joe

##### TS Contributor
So you would report the 95% CI, but due to correction they would actually be 96.666...% CI
This is an interesting idea that I've never seen before.

(5%/3)+95%
...but this can't be right. If you only do 1 comparison, then 5%/1 + 95% = 100%.

#### hlsmith

##### Less is more. Stay pure. Stay poor.
Yeah, perhaps not clear enough. You would then look up 96.6666 up on the standard normal table and multiply it by the SE and +/- it from the estimate.

Does that seem right.

#### Mean Joe

##### TS Contributor
**** lost connection while trying to post.

It's interesting to me, because I've always heard of Bonferroni to adjust the alpha level, but nothing done to the ORs/CIs.

Let's say you do 1,000 tests. In one test, you get a p=.0001 with OR=1.50 [1.25, 1.80].
Doing a Bonferroni correction to alpha, you would say you do not have a statistically significant result. Yet clearly your OR is greater than 1, and 1 is outside of your CI.
I always thought it would be strange to see that in a paper, the OR=1.50 [1.25, 1.80] is not signficant.

Looking at Wikipedia now, I see the confidence level for the CI would be 1 - (alpha)/(number of tests). ie confidence level is increasing with more tests.
Which doesn't seem right to me, shouldn't you have less confidence? You know you've done so many tests, that the probability of rejecting a true null result has increased.

Returning, we've done 1,000 tests, so our alpha=.00005. We've made 1,000 CIs, and we expect 999.95 of them to contain the true OR.
So we're even more confident (almost certainly confident), that the OR is clearly greater than 1.
But at the same time, we conclude that result is not statistically significant. Because. P values.

#### Dason

Looking at Wikipedia now, I see the confidence level for the CI would be 1 - (alpha)/(number of tests). ie confidence level is increasing with more tests.
Which doesn't seem right to me, shouldn't you have less confidence? You know you've done so many tests, that the probability of rejecting a true null result has increased.
There are several things going on here. If we were to just consider that specific interval will then be a $$(1 - \alpha/n)%[/code] confidence interval. But in the grand scheme of things if we want to be 95% confident that all of the intervals contain the parameters of interest then we are only 95% confident in that after modifying the intervals to be wider.$$

#### hlsmith

##### Less is more. Stay pure. Stay poor.
It is just adjusting your alpha or level of significance, which gets used with p-value and CI. So in odds ratios or comparing multiple groups to the reference, you would correct the pvalues and CI.

#### Jake-Finn

##### New Member
I am still unsure on what to do, and what to report.

I am doing 4 ANOVAs, with a Bonferroni adjustment resulting in an adjusted alpha of 0.0125/0.013 (0.05 / by 4 = 0.0125).

What should I do re: reporting CIs? Should I report the confidence interval percentage differently (97/98/99% etc) differently (and if so, how do I work out what to report), and/or do I need to do anything to the upper and lower confidence interval numbers themselves?

Thanks for your help so far.

You can't over-simplify. I have no degree in maths or statistics.

#### Dragan

##### Super Moderator
Very simply, use the Bootstrap i.e. construct 95% bootstrap confidence intervals for each of your research questions, which ostensibly appear to orthogonal to each other and also obviate the usual assumptions associated with the parametric ANOVA.

Subsequently, look at the confidence interavals and look to see if they overlap...which is a conservative approach.

Well, how convenient

#### Jake-Finn

##### New Member
Thank you for your assistance, but could you explain that in simpler terms please? And/or provide any resources that do?

Very simply, use the Bootstrap i.e. construct 95% bootstrap confidence intervals for each of your research questions,
I have never been taught anything about bootstrapping, what it is, how or why to do it, and there appears to be a lot of information out there about different types/applications of it.

which ostensibly appear to orthogonal to each other and also obviate the usual assumptions associated with the parametric ANOVA.
I have no idea what you mean here. Can you please clarify?

Subsequently, look at the confidence interavals and look to see if they overlap...which is a conservative approach.
And, consequently, because I don't understand that previous information, I have no context to understand this simpler sentence.

format: plain


text:
Lemma 43.16.1. Let $X$ be a nonsingular variety and $W, V \subset X$ closed subvarieties which intersect properly. Let $Z$ be an irreducible component of $V \cap W$ with generic point $\xi$. Assume that $\mathcal{O}_{W, \xi }$ and $\mathcal{O}_{V, \xi }$ are Cohen-Macaulay. Then

$e(X, V \cdot W, Z) = \text{length}_{\mathcal{O}_{X, \xi }}(\mathcal{O}_{V \cap W, \xi })$

where $V \cap W$ is the scheme theoretic intersection. In particular, if both $V$ and $W$ are Cohen-Macaulay, then $V \cdot W = [V \cap W]_{\dim (V) + \dim (W) - \dim (X)}$.

Proof. Set $A = \mathcal{O}_{X, \xi }$, $B = \mathcal{O}_{V, \xi }$, and $C = \mathcal{O}_{W, \xi }$. By Auslander-Buchsbaum (Algebra, Proposition 10.111.1) we can find a finite free resolution $F_\bullet \to B$ of length

$\text{depth}(A) - \text{depth}(B) = \dim (A) - \dim (B) = \dim (C)$

First equality as $A$ and $B$ are Cohen-Macaulay and the second as $V$ and $W$ intersect properly. Then $F_\bullet \otimes _ A C$ is a complex of finite free modules representing $B \otimes _ A^\mathbf {L} C$ hence has cohomology modules with support in $\{ \mathfrak m_ A\}$. By the Acyclicity lemma (Algebra, Lemma 10.102.8) which applies as $C$ is Cohen-Macaulay we conclude that $F_\bullet \otimes _ A C$ has nonzero cohomology only in degree $0$. This finishes the proof. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).


text:
## A new approach to Bayesian hypothesis testing

Posted in Books, Statistics with tags , , , , , on September 8, 2016 by xi'an

“The main purpose of this paper is to develop a new Bayesian hypothesis testing approach for the point null hypothesis testing (…) based on the Bayesian deviance and constructed in a decision theoretical framework. It can be regarded as the Bayesian version of the likelihood ratio test.”

This paper got published in Journal of Econometrics two years ago but I only read it a few days ago when Kerrie Mengersen pointed it out to me. Here is an interesting criticism of Bayes factors.

“In the meantime, unfortunately, Bayes factors also suffers from several theoretical and practical difficulties. First, when improper prior distributions are used, Bayes factors contains undefined constants and takes arbitrary values (…) Second, when a proper but vague prior distribution with a large spread is used to represent prior ignorance, Bayes factors tends to favour the null hypothesis. The problem may persist even when the sample size is large (…) Third, the calculation of Bayes factors generally requires the evaluation of marginal likelihoods. In many models, the marginal likelihoods may be difficult to compute.”

I completely agree with these points, which are part of a longer list in our testing by mixture estimation paper. The authors also rightly blame the rigidity of the 0-1 loss function behind the derivation of the Bayes factor. An alternative decision-theoretic based on the Kullback-Leibler distance has been proposed by José Bernardo and Raúl Rueda, in a 2002 paper, evaluating the average divergence between the null and the full under the full, with the slight drawback that any nuisance parameter has the same prior under both hypotheses. (Which makes me think of the Savage-Dickey paradox, since everything here seems to take place under the alternative.) And the larger drawback of requiring a lower bound for rejecting the null. (Although it could be calibrated under the null prior predictive.)

This paper suggests using instead the difference of the Bayesian deviances, which is the expected log ratio integrated against the posterior. (With the possible embarrassment of the quantity having no prior expectation since the ratio depends on the data. But after all the evidence or marginal likelihood faces the same “criticism”.) So it is a sort of Bayes factor on the logarithms, with a strong similarity with Bernardo & Rueda’s solution since they are equal in expectation under the marginal. As in Dawid et al.’s recent paper, the logarithm removes the issue with the normalising constant and with the Lindley-Jeffreys paradox. The approach then needs to be calibrated in order to define a decision bound about the null. The asymptotic distribution of the criterion is  χ²(p)−p, where p is the dimension of the parameter to be tested, but this sounds like falling back on frequentist tests. And the deadly .05% bounds. I would rather favour a calibration of the criterion using prior or posterior predictives under both models…

## the art of war

Posted in Books, Kids, pictures with tags , , , on December 30, 2015 by xi'an

My son offered me this illustrated version of The Art of War for Xmas. (Illustrated with traditional Chinese and Japanese drawings.) And I found this citation within the fourth chapter:

Measurement owes its existence to Earth;
Estimation of quantity to Measurement;
Calculation to Estimation of quantity;
Balancing of chances to Calculation;
and Victory to Balancing of chances.

Which could somewhat be seen as an early version of Bayesian decision theory….

## Bayesian propaganda?

Posted in Books, Kids, pictures, Statistics, University life with tags , , , , , , , , , on April 20, 2015 by xi'an

“The question is about frequentist approach. Bayesian is admissable [sic] only by wrong definition as it starts with the assumption that the prior is the correct pre-information. James-Stein beats OLS without assumptions. If there is an admissable [sic] frequentist estimator then it will correspond to a true objective prior.”

I had a wee bit of a (minor, very minor!) communication problem on X validated, about a question on the existence of admissible estimators of the linear regression coefficient in multiple dimensions, under squared error loss. When I first replied that all Bayes estimators with finite risk were de facto admissible, I got the above reply, which clearly misses the point, and as I had edited the OP question to include more tags, the edited version was reverted with a comment about Bayesian propaganda! This is rather funny, if not hilarious, as (a) Bayes estimators are indeed admissible in the classical or frequentist sense—I actually fail to see a definition of admissibility in the Bayesian sense—and (b) the complete class theorems of Wald, Stein, and others (like Jack Kiefer, Larry Brown, and Jim Berger) come from the frequentist quest for best estimator(s). To make my point clearer, I also reproduced in my answer the Stein’s necessary and sufficient condition for admissibility from my book but it did not help, as the theorem was “too complex for [the OP] to understand”, which shows in fine the point of reading textbooks!

## Dennis Lindley (1923-2013)

Posted in Books, Statistics, University life with tags , , , , , , on December 16, 2013 by xi'an

Dennis Lindley most sadly passed away yesterday at the hospital near his home in Somerset. He was one of the founding fathers of our field (of Bayesian statistics), who contributed to formalise Bayesian statistics in a coherent theory. And to make it one with rational decision-making, a perspective missing in Jeffreys’ vision. (His papers figured prominently in the tutorials we gave yesterday for the opening of O’Bayes 250.) At the age of 90, his interest in the topic had not waned away: as his interview with Tony O’Hagan last Spring showed, his passionate arguing for the rationale of the Bayesian approach was still there and alive! The review he wrote of The Black Swan a few years ago also demonstrated he had preserved his ability to see through bogus arguments. (See his scathing “One hardly advances the respect with which statisticians are held in society by making such declarations” in his ripping discussion of Aitkin’s 1991 Posterior Bayes factors.) He also started this interesting discussion last year about the five standard deviations “needed” for the Higgs boson…  My personal email contacts with Dennis over the re-reading of Jeffreys’ book  were a fantastic experience as he kindly contributed by expanding on how the book was received at the time and correcting some of my misunderstanding. It is a pity I can no longer send him the (soon to come?) final version of my Jeffreys-Lindley paradox paper as I intended to do. The email thomasbayes@gmail.com will no longer answer our queries… I figure there will be many testimonies and shared memories of his contributions and life at the Bayes-250 conference tomorrow. Farewell, Dennis, and I hope you now explore the paths of a more coherent world than ours!

## Đôi nét về GS. Xi’an

Posted in Books, Travel, University life with tags , , , , on May 28, 2013 by xi'an

Here is a short bio of me written in Vietnamese in conjunction with the course I will give at CMS (Centre for Mathematical Sciences), Ho Chi Min City, next week:

Christian P. Robert là giáo sư tại Khoa Toán ứng dụng của ĐH Paris Dauphine từ năm 2000. GS Robert đã từng giảng dạy ở các ĐH Perdue, Cornell (Mỹ) và ĐH Canterbury (New-Zealand). Ông đã làm biên tập cho tạp chí Journal of the Royal Statistical Society Series B từ năm 2006 đến năm 2009 và là phó biên tập cho tạp chí Annals of Statistics. Năm 2008, ông làm Chủ tịch của Hiệp hội Thống kê Quốc tế về Thống kê Bayes (ISBA). Lĩnh vực nghiên cứu của GS Robert bao gồm Thống kê Bayes mà tập trung chính vào Lý thuyết quyết định (Decision theory) và Mô hình lựa chọn (Model selection), Lý thuyết về Xích Markov trong mô phỏng và Thống kê tính toán.

## testing via credible sets

Posted in Statistics, University life with tags , , , , , , , , , , , on October 8, 2012 by xi'an

Måns Thulin released today an arXiv document on some decision-theoretic justifications for [running] Bayesian hypothesis testing through credible sets. His main point is that using the unnatural prior setting mass on a point-null hypothesis can be avoided by rejecting the null when the point-null value of the parameter does not belong to the credible interval and that this decision procedure can be validated through the use of special loss functions. While I stress to my students that point-null hypotheses are very unnatural and should be avoided at all cost, and also that constructing a confidence interval is not the same as designing a test—the former assess the precision in the estimation, while the later opposes two different and even incompatible models—, let us consider Måns’ arguments for their own sake.

The idea of the paper is that there exist loss functions for testing point-null hypotheses that lead to HPD, symmetric and one-sided intervals as acceptance regions, depending on the loss func. This was already found in Pereira & Stern (1999). The issue with these loss functions is that they involve the corresponding credible sets in their definition, hence are somehow tautological. For instance, when considering the HPD set and T(x) as the largest HPD set not containing the point-null value of the parameter, the corresponding loss function is

$L(\theta,\varphi,x) = \begin{cases}a\mathbb{I}_{T(x)^c}(\theta) &\text{when }\varphi=0\\ b+c\mathbb{I}_{T(x)}(\theta) &\text{when }\varphi=1\end{cases}$

parameterised by a,b,c. And depending on the HPD region.

Måns then introduces new loss functions that do not depend on x and still lead to either the symmetric or the one-sided credible intervals.as acceptance regions. However, one test actually has two different alternatives (Theorem 2), which makes it essentially a composition of two one-sided tests, while the other test returns the result to a one-sided test (Theorem 3), so even at this face-value level, I do not find the result that convincing. (For the one-sided test, George Casella and Roger Berger (1986) established links between Bayesian posterior probabilities and frequentist p-values.) Both Theorem 3 and the last result of the paper (Theorem 4) use a generic and set-free observation-free loss function (related to eqn. (5.2.1) in my book!, as quoted by the paper) but (and this is a big but) they only hold for prior distributions setting (prior) mass on both the null and the alternative. Otherwise, the solution is to always reject the hypothesis with the zero probability… This is actually an interesting argument on the why-are-credible-sets-unsuitable-for-testing debate, as it cannot bypass the introduction of a prior mass on Θ0!

Overall, I furthermore consider that a decision-theoretic approach to testing should encompass future steps rather than focussing on the reply to the (admittedly dumb) question is θ zero? Therefore, it must have both plan A and plan B at the ready, which means preparing (and using!) prior distributions under both hypotheses. Even on point-null hypotheses.

Now, after I wrote the above, I came upon a Stack Exchange page initiated by Måns last July. This is presumably not the first time a paper stems from Stack Exchange, but this is a fairly interesting outcome: thanks to the debate on his question, Måns managed to get a coherent manuscript written. Great! (In a sense, this reminded me of the polymath experiments of Terry Tao, Timothy Gower and others. Meaning that maybe most contributors could have become coauthors to the paper!)

## a paradox in decision-theoretic interval estimation (solved)

Posted in pictures, Statistics, Travel, University life with tags , , , , , , , , , , , on October 4, 2012 by xi'an

In 1993, we wrote a paper [with George Casella and Gene/Juinn Hwang] on the paradoxical consequences of using the loss function

$\text{length}(C) - k \mathbb{I}_C(\theta)$

(published in Statistica Sinica, 3, 141-155) since it led to the following property: for the standard normal mean estimation problem, the regular confidence interval is dominated by the modified confidence interval equal to the empty set when is too large… This was first pointed out by Jim Berger and the most natural culprit is the artificial loss function where the first part is unbounded while the second part is bounded by k. Recently, Paul Kabaila—whom I met in both Adelaide, where he quite appropriately commented about the abnormal talk at the conference!,  and Melbourne, where we met with his students after my seminar at the University of Melbourne—published a paper (first on arXiv then in Statistics and Probability Letters) where he demonstrates that the mere modification of the above loss into

$\dfrac{\text{length}(C)}{\sigma} - k \mathbb{I}_C(\theta)$

solves the paradox:! For Jeffreys’ non-informative prior, the Bayes (optimal) estimate is the regular confidence interval. besides doing the trick, this nice resolution explains the earlier paradox as being linked to a lack of invariance in the (earlier) loss function. This is somehow satisfactory since Jeffreys’ prior also is the invariant prior in this case.


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# Lab Notebook

#### Notes

10 Dec 2013

Consider the model

$d X_t = \alpha \left(\theta - X_t\right)dt + \sigma dB_t$

By Ito Isometry, we have:

$\langle X \rangle_t = E_t(X) = \theta \left(1 - e^{-\alpha t} \right) + X_0 e^{-\alpha t}$

and

$\langle X^2 \rangle_t - \langle X \rangle_t^2 = V_t(X) = \frac{\sigma^2}{2 \alpha }\left(1 - e^{-2 \alpha t} \right)$

If we assume discrete, uniform sampling of spacing $$\tau$$, then we have,

$P(\theta | X; \alpha, \sigma) = \sqrt{\frac{1}{2\pi V_{\tau} }}^{T-1} \exp\left(-\frac{\sum_t^{T-1} \left(X_{t+1} - E\right)^2 }{2V}\right)$

or more explicitly (taking $$\tau_i = 1$$ without loss of generality),

$P(\theta | X; \alpha, \sigma) = \sqrt{\frac{\tfrac{\alpha}{\sigma}}{\pi (1- e^{-\alpha t}) }}^{T-1} \exp\left( - \frac{\sum_t^{T-1} \left(X_{t+1} - ( \theta (1 - e^{-\alpha}) + X_0 e^{-\alpha}) \right)^2 }{\tfrac{\sigma^2}{\alpha}(1-e^{-2\alpha})}\right)$

To integrate out $$\theta$$, $$P(X | \alpha, \sigma) = \int P(X | \theta, \alpha, \sigma ) P(\theta) d\theta$$, we’ll make this look like a Gaussian in $$\theta$$ by completing the square. First, let us introduce a more compact notation to manipulate terms independent of $$\theta$$:

$A_t := X_{t+1} - X_0e^{-\alpha}$ $B := 1-e^{-\alpha}$

Our sum in the exponent can then be written more succinctly:

$\sum_t^{T-1} (A_t - \theta B)^2$

Squaring out inside the sum and distributing the summation operator (by linearity) and extracting or summing terms constant in $$t$$, we have:

$\sum A_t^2 - 2 \theta B \sum A_t + \theta^2 B^2 (T-1)$

For which we complete the square in $$\theta$$,

$B^2 (T-1) \left( \left(\theta - \frac{\sum_t A_t}{B(T-1)}\right)^2 + \frac{\sum A_t ^2 - (\sum A_t)^2}{B^2 (T-1)} \right)$

Which lets us write $$P(\theta | X; \alpha, \sigma)$$ as a normal distribution in $$\theta$$ times some constant terms:

$P(\theta | X; \alpha, \sigma) = \exp \left(\frac{-\sum A_t^2 + (\sum A_t)^2 }{V_{\tau} }\right) \sqrt{\frac{1}{2\pi V_{\tau} }}^{T-1} \int d\theta\exp\left( -\frac{ \left(\theta - \frac{\sum_t A_t}{B(T-1)} \right)^2 }{\tfrac{2V_{\tau}}{B^2 (T-1)}}\right)$

In which we recognize the integral as Gaussian with mean $=$ and variance $$\nu = V_{\tau}/(B^2(T-1))$$, and thus can replace the integral with $$\sqrt{2 \pi \nu} = \sqrt{\frac{2 \pi V_{\tau} }{B^2(T-1)}}$$,

$P(\theta | X; \alpha, \sigma) = \exp \left(\frac{-\sum A_t^2 + (\sum A_t)^2 }{V_{\tau} }\right) \sqrt{\frac{1}{2\pi V_{\tau} }}^{T-1} \sqrt{\frac{2 \pi V_{\tau} }{B^2(T-1)}}$

Collecting common terms in $$V_t$$,

$P(\theta | X; \alpha, \sigma) = \exp \left(\frac{f(A_t)}{V_{\tau}}\right) \sqrt{\frac{1}{2\pi}}^{T-1} \sqrt{\frac{2 \pi }{B^2(T-1)}} V_{\tau}^{\tfrac{1}{2}} V_{\tau}^{\tfrac{1-T}{2}}$

$= \exp \left(\frac{f(A_t)}{V_{\tau}}\right) \sqrt{\frac{(2 \pi)^{2-T}}{B ^ 2(T-1)}} V_{\tau}^{1 - \tfrac{T}{2}}$

where

$f(A_t) = - \sum A_t^2 + \left( \sum A_t \right)^2$

Then substituting in $$V_t = \tfrac{\sigma^2}{2\alpha} \left(1 - e^{-2\alpha}\right)$$

$= \exp \left(\frac{\tfrac{2\alpha f(A)}{1 - e^{-2\alpha}} }{\sigma^2}\right) \sqrt{\frac{(2 \pi)^{2-T}}{B ^ 2(T-1)}} \left(\tfrac{1}{2\alpha} \left(1 - e^{-2\alpha}\right)\right)^{1 - \tfrac{T}{2}}\sigma^{2(1 - T/2)}$

Which can be written to look like an Inverse Gamma integral in $$\sigma^2$$

$= \mathcal{N} e^{\tfrac{-\beta}{\sigma^2}} (\sigma^2)^{-\gamma - 1}$

where

$\gamma = T/2 -2,$

$\beta = \frac{2 \alpha \left( \left( \sum A_t \right)^2- \sum A_t^2 \right) }{1 - e^{-2\alpha}}$ and

$\mathcal{N} = \sqrt{\frac{(2 \pi)^{2-T}}{B ^ 2(T-1)}} \left(\tfrac{1}{2\alpha} \left(1 - e^{-2\alpha}\right)\right)^{1 - \tfrac{T}{2}}$

Which we can integrate as Inverse Gamma (again assuming uniform prior, now in $$\sigma^2$$),

$= \mathcal{N} \int e^{\tfrac{-\beta}{\sigma^2}} (\sigma^2)^{-\gamma - 1} d\sigma^2 = \mathcal{N} \frac{\beta^{\gamma}}{\Gamma(\gamma)}$

with $$\beta$$, $$\gamma$$, and $$\mathcal{N}$$ as defined above.

26 Nov 2013

## RNeXML

EML provides a nice illustration of the kind of metadata we would want to typically accompany character trait data; at minimum, at the attribute level.

Some of this is already there – we know if states are continuous or discrete, what possible states they can take on, and can add labels to state nodes to define what the symbols mean (however, perhaps an additional meta annotation node would be preferred). We don’t have a way to specify the units of a continuous trait, or longer definitions of the traits themselves, etc.

Obviously this is a hard problem, particularly in building any semantic reasoning around these traits. I gather that the phenoscape project is tackling this, but I’m not really up to speed on that. Long-term it would be nice to allow users to read and write trait metadata from R even without a working knowledge of the ontologies.

One idea I’m pondering is if we might be able to generate an EML metadata file the NeXML. The most trivial and straight forward approach is simply to use EML vocabularly itself in the metadata.

### Software to integrate into NeXML (long-term)

• Visualization: automatically generate the appropriate HTML+javascript page to render tree with jsPhyloSVG/phylotouch (example)
• Support for phenoscape onotlogy tools, e.g. see data as generated by phenex
• PhyloWS Generate a REST web service based on NeXML library(?)

### Issues

• Trying to understand if using nexml for comparative trait data will generate any confusion. (#44)

19 Nov 2013

### RNeXML

• Feedback from Rutger, need to add about attributes so that RDFa abstraction references the right level of the DOM (issue #35).
• Looking for strategy for distilling RDF from RDFa in R, see my question on SO. Hopefully don’t have to wrap some C library myself…

### nonparametric-bayes

Writing writing.

• Update pandoc templates to use yaml metadata for author, affiliation, abstract, etc. Avoids having to manually edit the elsarticle.latex template with this metadata. Added fork for my templates, e.g. see my elsarticle.latex. Example metadata in manuscript.

• fixing xtable caption (as argument)

• Extended discussion. Adjustments to figures. See commit log /diffs for details.

Mace (2013) , e.g.

a new kind of ecology is needed that is predicated on scaling up efforts, data sharing and collaboration

hear hear.

• PNAS with a somewhat confused take on error rates, suggesting a revised threshold p-value…

• AmNat Asilomar schedule (pdf) is up.

17 Nov 2013

(From issue #20)

a question of how the user queries that metadata. Currently we have a metadata function that simply extracts all the metadata at the specified level (nexml, otus, trees, tree, etc) and returns a named character string in which the name corresponds to the rel or property and the value corresponds to the content or href, e.g.:

birds <- read.nexml("birdOrders.xml")
meta <- get_metadata(birds)

prints the named string with the top-level (default-level) metadata elements as so:

> meta
##                                             dc:date
##                                        "2013-11-17"
## "https://creativecommons.org/publicdomain/zero/1.0/"

Which we can subset by name, e.g. meta["dc:date"]. This is probably simplest to most R users; though exactly what the namespace prefix means may be unclear if they haven’t worked with namespaces before. (The user can always print a summary of the namespaces and prefixes in the nexml file using birds@namespaces).

This approach is simple, albeit a bit limited.

### XPath queries

For instance, the R user has a much more natural and powerful way to handle these issues of prefixes and namespaces using either the XML or rrdf libraries. For instance, if we extract meta nodes into RDF-XML, we could handle the queries like so:

xpathSApply(meta, "//dc:title", xmlValue)

which uses the namespace prefix defined in the nexml; or

xpathSApply(meta, "//x:title", xmlValue, namespaces=c(x = "https://purl.org/dc/elements/1.1/"))

defining the custom prefix x to the URI

### Sparql queries

Pretty exciting that qe can make arbitrary SPARQL queries of the metadata as well.

library(rrdf)
sparql.rdf(ex, "SELECT ?title WHERE { ?x <https://purl.org/dc/elements/1.1/title> ?title })

Obviously the XPath or SPARQL queries are more expressive / powerful than drawing out the metadata from the S4 structure directly. On the other hand, because both of these approaches use just the distilled metadata, the original connection between metadata elements and the structure of the XML tree is lost unless stated explicitly. An in-between solution is to use XPath on the nexml XML instead, though I think we cannot make use of the namespaces in that case, since they appear in attribute values rather than structure.

Anyway, it’s nice to have these options in R, particularly for more complex queries where we might want to make some use of the ontology as well. On the other hand, simple presentation of basic metadata is probably necessary for most users.

Would be nice to illustrate with a query that required some logical deduction from the ontology.

#### So, you're active on Research Gate?

14 Nov 2013

I have occassionally been getting this question:

So, you’re active on ResearchGate?

Sounds like being accused of some scandal, doesn’t it?

I’m not generally active on it - my impression is that the open science community is mostly skeptical about ResearchGate and any other “Social Network” for scientists, largely on the grounds that “we already use the same social networks everyone else uses.” Some object on more philosophical grounds (profit, Mendeley, etc), but heck I publish in Elsevier/Springer/Wiley so I won’t preach. That’s perhaps a US/elite institution centric view though; it seems more popular with a more international audience where things like basic access to pdfs may be more of an issue. I present no data to back any if that up.

Personally, it hasn’t added any value for me, in contrast to the value I get from interacting with other researchers on Github, G+ or Twitter. Still, as RG recently got $35 million from Bill Gates, they might actually build something useful. Certainly traditional publishers have left plenty of room for innovation in the space of sharing data, networking, etc. So I have a profile there to wait and see, next to a disclaimer that says “please see my website for updated information.” However, I was actually impressed by ResearchGate this morning. While I thought I had successfully blocked most of their email notifications, one this morning had successfully found the full text of a recent paper of mine (albeit a few months after it had appeared). Instead of asking me to upload something, RG was able to obtain the full text from the publisher (Springer). On so doing, it also asks me if I would like to “follow” several of the researchers I cited who are also on ResearchGate. Why is that impressive? Mendeley, for all it’s much more natural fit into most researcher’s workflows, never automatically discovers papers I publish. If I want them in my Mendeley profile, I have to add them manually. Manually maintaining profiles across different networks is so entirely a waste of time and the antithesis of a linked data web where I have already made this information machine readable that I find it the most annoying feature by far in any of these sites. Here, ResearchGate is actually doing the intelligent thing, whether by connecting my RG identity to my ORCID ID, or something more heuristic. (Google Scholar automatically adds things to my profile, but in a far less selective algorithm that can be easily gamed, see 10.1002/asi.23056). By obtaining the full-text directly from the publisher, they show the considerable advantage of a well-funded network. Presumably this indicates that access was negotiated directly with the publisher, who agrees and even facilitates me sharing the full-text of my otherwise paywalled article on my RG profile. That’s a non-trivial contribution towards open access: Contrast this to Mendeley’s more murky policy which encourages me to provide full text access through my user profile but places the legal responsibility directly on me to confirm that this permissible, or an organization like ORCID which despite (because of?) it’s more non-profit and utilitarian values does not have permission to distribute my paywalled pdfs on my profile. (Sure, my papers on arXiv already, but that isn’t the point). Likewise, using the citation data against the RG data on which researchers have profiles shows a vaguely intelligent use of data other platforms mostly ignore (providing useful suggestions using some understanding of the academic process rather than mindless application of some friend-of-a-friend network algorithm). (The fact that RG pings unfortunate souls who might have signed up once but have no desire to see my “Activity” on RG is one of it’s potentially effective marketing but more pernicious decisions. Use does not necessarily imply trust). #### Notes 14 Nov 2013 ## ropensci • ropensci stategic planning: wrote personal vision statement ### RNeXML • Comments on issues #12 and #15, thinking about character matrices • comments on #20, thinking about metadata parsing. (If only everyone knew xpath…) Commit details on Github #### Do we need a culture of Data Science in Academia? 13 Nov 2013 Just my draft copy of a Guest blog post I wrote for Dynamic Ecology. On Tuesday the Whitehouse Office of Science and Technology Policy announced the creation of a$37.8 million dollar initiative to promote a “Data Science Culture” in academic institutions, funded by the Gordon and Betty Moore Foundation, Alfred P. Sloan Foundation, and hosted in centers at the universities UC Berkeley, University of Washington, and New York University. Sadly, these announcements give little description of just what such a center would do, beyond repeating the usual the hype of “Big Data.”

Fernando Perez, a research scientist at UC Berkeley closely involved with the process, paints a rather more provocative picture in his own perspective on what this initiative might mean by a “Data Science Culture.” Rather than motivating the need for such a Center merely by expressing terabytes in scientific notation, Perez focuses on something not mentioned in the press releases. In his view, the objective of such a center stems from the observation that:

the incentive mechanisms of academic research are at sharp odds with the rising need for highly collaborative interdisciplinary research, where computation and data are first-class citizens

His list of problems to be tackled by this Data Science Initiative includes some particularly catching references to issues that have raised themselves on Dynamic Ecology before:

• people grab methods like shirts from a rack, to see if they work with the pants they are wearing that day
• methodologists tend to only offer proof-of-concept, synthetic examples, staying largely shielded from real-world concerns

Well that’s a different tune than the usual big data hype1. While it is easy to find anecdotes that support each of these charges, it is more difficult to assess just how rare or pervasive they really are. Though these are not new complaints among ecologists, the solutions (or at least antidotes) proposed in a Data Science Culture given a rather different emphasis. At first glance, the Data Science Culture sounds like the more familiar call for an interdisciplinary culture, emphasizing that the world would be a better place if only domain scientists learned more mathematics, statistics and computer science. It is not.

##### the problem, part 1: statistical machismo?

As to whether ecologists choose methods to match their pants, we have at least some data beyond anecdote. A survey earlier this year by Joppa et al. (2013) Science) has indeed shown that most ecologists select methods software guided primarily by concerns of fashion (in other words, whatever everybody else uses). The recent expansion of readily available statistical software has greatly increased the number of shirts on the rack. Titles in Ecology reflect the trend of rising complexity in ecological models, such as Living Dangerously with big fancy models and Are exercises like this a good use of anybody’s time?). Because software enables researchers to make use of methods without the statistical knowledge of how to implement them from the ground up, many echo the position so memorably articulated by Jim Clark that we “handing guns to children.” This belittling position usually leads to a call for improved education and training in mathematical and statistical underpinnings (see each of the 9 articles in another Ecology Forum on this topic), or the occassional wistful longing for a simpler time.

##### the solution, part 1: data publication?

What is most interesting to me in Perez’s perspective on the Data Science Institute in an emphasis on changing incentives more than changing educational practices. Perez characterizes the fundamental objective of the initiative as a cultural shift in which

“The creation of usable, robust computational tools, and the work of data acquisition and analysis must be treated as equal partners to methodological advances or domain-specific results”

While this does not tackle the problem of misuse or misinterpretation of statistical methodology head-on, I believe it is a rather thought-provoking approach to mitigate the consequences of mistakes or limiting assumptions. By atomizing the traditional publication into such component parts: data, text, and software implementation, it becomes easier to recognize each for it’s own contributions. A brilliantly executed experimental manipulation need not live or die on some minor flaw in a routine statistical analysis when the data is a product in its own right. Programmatic access to raw data and computational libraries of statistical tools could make it easy to repeat or alter the methods chosen by the original authors, allowing the consequences of these mistakes to be both understood and corrected. In the current system in which access to the raw data is rare, statistical mistakes can be difficult to detect and even harder to remedy. This in turn places a high premium on the selection of appropriate statistical methods, while putting little selective pressure on the details of the data management or implementation of those methods. Allowing the data to stand by itself places a higher premium on careful collection and annotation of data (e.g. the adoption of metadata standards). To the extent that misapplication of statistical and modeling approaches could place a substantial error rate on the literature (Economist, Ioannidis 2005), independent data publication might be an intriguing antidote.

##### the problem, part 2: junk software

As Perez is careful to point out, those implementing and publishing methods aren’t helping either. Unreliable, inextensible and opaque computational implementations act both as barriers to adoption and validation. Trouble with scientific software has been well recognized by the literature (e.g. Merali (2010), Nature, Inces et al. (2012), Nature), the news (Times Higher Education) and funding agencies (National Science Foundation). While it is difficult to assess the frequency of software bugs that may really alter the results (though see Inces et al.), designs that will make software challenging or impossible to maintain, scale to larger tasks or extend as methods evolve are more readily apparent. Cultural challenges around software run as deep as they do around data. When Mozilla’s Science Lab undertook a review of code associated with scientific publications, they took some criticism from other advocates of publishing code. I encountered this first hand in replies from authors, editors and reviewers on my own blog post suggesting we raise the bar on the review of methodological implementations. Despite disagreement about where that bar should be, I think we all felt the community could benefit from clearer guidance or consensus on how to review papers in which the software implementation plays an essential part and contribution.

##### the solution, part 2: software publication?

As in the case of data, educational practices are the route usually suggested to address better programming practices, and no doubt these are important. Once again though, it is interesting to think how a higher incentive on such research products might also improve their quality, or at least facilitate distilling the good from the bad from the ugly, more easily. Yet in this case, I think there is a potential downside as well.

###### Or not?

While widespread recognition of its importance will no doubt help bring us faster software, fewer bugs and more user-friendly interfaces, it may do more harm than good. Promotion of software as a product can lead to empire-building, for which ESRI’s ArcGIS might be a poster child. The scientific concepts become increasingly opaque, while training in a conceptually rich academic field gives way to more mindless training in the user interface of a single giant software tool. I believe that good scientific software should be modular – small code bases that can be easily understood, inter-operable, and perform a single task well (the Unix model). This lets us build more robust computational infrastructure tailored to the problem at hand, just as individual Lego bricks may be assembled and reassembled. Unfortunately, I do not see how recognition for software products would promote small modules over vast software platforms, or interoperability with other software instead of an exclusive walled garden.

#### So, change incentives how?

If this provides some argument as to why one might want to change incentives around data and software publication, I have said nothing to suggest how. After all, as ecologists we’re trained to reflect on the impact a policy would have, not advocate for what should be done about it. If the decision-makers agree about the effects of the given incentives, then choosing what to reward should be easier.

1. Probably for reasons discussed recently on Dynamic Ecology about politicians and dirty laundry.

#### Mbi Day Five Notes

08 Nov 2013

Panel discussion

• Hugh’s question on the usefulness of dynamic vs static models: do we have dynamical systems envy?
• Chris: are temporal dynamics historical artefact, and space the new frontier?
• Hugh: though decision theory is fundamentally temporal. really question of sequential decision vs single decision
• Hugh, on what would be his priority if he had time for new question: Solve the 2 player, 2 step SDP competition closed form.
• Paul: the narrow definitions of “math biology” with 1980s flavor.
• @mathbiopaul: Formulating the hard problems arising in application in an appropriate abstraction that mathematicians will attack.
• Leah raises issue of publishing software and reproducibility
• Julia mentions Environmental modeling and software journal

### pdg-control

Trying to understand pattern of increasing ENPV with increasing stochasticity. Despite having the same optimal policy inferred under increasing stochasticity (i.e. still in Reed’s self-sustaining criterion, below $$\sigma_g$$ of 0.2 or so) the average over simulated replicates is higher. We don’t seem to obtain the theoretical ENPV, but something less, in either case. See code noise_effects.md.

### ropensci

Schema.org defines a vocabulary for datasets (microdata/rdfa)

Rutger gives a one-liner solution for tolweb to nexml using bio-phylo perl library:

perl -MBio::Phylo::IO=parse -e 'print parse->to_xml' format tolweb as_project 1 url 'https://tolweb.org/onlinecontributors/app?service=external&page=xml/TreeStructureService&node_id=52643'

Hmm, there’s a journal of Ecological Informatics.

## References

• Rebecca S. Epanchin-Niell, Robert G. Haight, Ludek Berec, John M. Kean, Andrew M. Liebhold, Helen Regan, (2012) Optimal Surveillance And Eradication of Invasive Species in Heterogeneous Landscapes. Ecology Letters 15 803-812 10.1111/j.1461-0248.2012.01800.x
• Rebecca S. Epanchin-Niell, James E. Wilen, (2012) Optimal Spatial Control of Biological Invasions. Journal of Environmental Economics And Management 63 260-270 10.1016/j.jeem.2011.10.003
• J. Esper, U. Buntgen, D. C Frank, D. Nievergelt, A. Liebhold, (2007) 1200 Years of Regular Outbreaks in Alpine Insects. Proceedings of The Royal Society B: Biological Sciences 274 671-679 10.1098/rspb.2006.0191
• unknown Fagan, unknown Meir, unknown Prendergast, unknown Folarin, unknown Karieva, (2001) Characterizing Population Vulnerability For 758 Species. Ecology Letters 4 132-138 10.1046/j.1461-0248.2001.00206.x
• A. R. Hall, A. D. Miller, H. C. Leggett, S. H. Roxburgh, A. Buckling, K. Shea, (2012) Diversity-Disturbance Relationships: Frequency And Intensity Interact. Biology Letters 8 768-771 10.1098/rsbl.2012.0282
• Elizabeth Eli Holmes, John L. Sabo, Steven Vincent Viscido, William Fredric Fagan, (2007) A Statistical Approach to Quasi-Extinction Forecasting. Ecology Letters 10 1182-1198 10.1111/j.1461-0248.2007.01105.x
• Brian Leung, Nuria Roura-Pascual, Sven Bacher, Jaakko Heikkilä, Lluis Brotons, Mark A. Burgman, Katharina Dehnen-Schmutz, Franz Essl, Philip E. Hulme, David M. Richardson, Daniel Sol, Montserrat Vilà, Marcel Rejmanek, (2012) Teasing Apart Alien Species Risk Assessments: A Framework For Best Practices. Ecology Letters 15 1475-1493 10.1111/ele.12003
• A. D. Miller, S. H. Roxburgh, K. Shea, (2011) How Frequency And Intensity Shape Diversity-Disturbance Relationships. Proceedings of The National Academy of Sciences 108 5643-5648 10.1073/pnas.1018594108
• Adam David Miller, Stephen H. Roxburgh, Katriona Shea, (2011) Timing of Disturbance Alters Competitive Outcomes And Mechanisms of Coexistence in an Annual Plant Model. Theoretical Ecology 5 419-432 10.1007/s12080-011-0133-1

07 Nov 2013

## Morning session

• Chadès, I., Carwardine, J., Martin, T.G., Nicol, S., Sabbadin, R. & Buffet, O. (2012) MOMDPs: a solution for modelling adaptive management problems. The Twenty-Sixth AAAI Conference on Artificial Intelligence (AAAI-12), pp. 267-273. Toronto, Canada.

• 10.1098/rspb.2013.0325 Migratory connectivity magnifies the consequences of habitat loss from sea-level rise for shorebird populations

#### Jake LaRiviera

presents the challenges of the uncertainty table. Additional challenges in making an apples-to-apples comparison of the benefit of decreasing noise of different systems (e.g. in pricing information?)

#### Me

Some good questions following talk, primarily on BNP part.

• Where does the risk-adverse vs risk-prone behavior come from? Adjusting curvature of the uncertainty appropriately.
• Any lessons after stock collapsed, e.g. Rebuild a stock rather than mantain it? (Perhaps, but may face hysteresis in a way the intial collapse does not).
• Brute-force passive learning?

## Afternoon discussion

1. Is an active learning approach more or less valuable in a changing environment
2. Embracing surprise: how do we actually mathematically do this.
3. Limitations due to constraints on frequency of updating. (e.g. we don’t get to change harvest, we get to set a TAC once every ten years).
4. Uncertainty affecting net present value vs affecting model behavior


text:
# What is the slope of the line passing through the following points: (0, -2), (-1, 5) ?

Apr 3, 2016

$- 7$

#### Explanation:

use the formula

$\text{slope} = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}$

Here ${x}_{1} = 0$, ${x}_{2} = - 1$, ${y}_{1} = - 2$, and ${y}_{2} = 5$

So after arranging the values according to the formula so the answer would be $- 7$


text:
# Interaction Between Different Skew-Partitions in a Graph

For , let be a skew partition of , where there are no edges between and , and is complete to . For each , choose one of , say , and let be the union of all these . Call a chunk. If there is an odd hole or antihole in , then it belongs to the chunk (for some choice of the 's), so to check Bergeness of , it is enough to check Bergeness of all the chunks. But even if is linear in the size of , the number of chunks can be exponential. Maybe there is a way around this. With decomposition theorems that come up from excluded minors, using separations instead of skew partitions, the same exponential blowup happens, but it can be avoided by using separations that are pairwise noncrossing - then you only get linearly many pieces. Is there an analogous nice way for skew partitions to fit together, so that we only get linearly many (or polynomially many) chunks?

Contributed by Paul Seymour

Back to the main index for Perfect Graphs.


text:
Definition of entropy
Entropy of a source is defined as follows:
$$H(X) = \sum\limits_{i=1}^N P_{i}\cdot\log_{2}\left(\frac{1}{P_{i}}\right)$$ $P_i$ is the probability of the symbol $x_i$
Files on a PC are organized in bytes. One byte consists of 8 bit. So there are N=256 differen bytes possible. The source has an alphabet of 256 different symbols. The symbols can be described as integer numbers of the range from 0 to 255. The following slide shows different distributions of the symbols for different files.
The maximum entropy of a source is: $$H_{max}(X)=H_0=\log_2(N)$$
$N$ is the number of different symbols of the source.
The redundancy ist defined as follows: $$R(X) = H_{0} - H(X) = \log_2(N) - H(X)$$


text:
Article Contents
Article Contents

# Minimal dynamics for tree maps

• We prove that, given a tree pattern $\mathcal{P}$, the set of periods of a minimal representative $f: T\rightarrow T$ of $\mathcal{P}$ is contained in the set of periods of any other representative. This statement is an immediate corollary of the following stronger result: there is a period-preserving injection from the set of periodic points of $f$ into that of any other representative of $\mathcal{P}$. We prove this result by extending the main theorem of [6] to negative cycles.
Mathematics Subject Classification: Primary: 37E25.

 Citation:


text:
proving trigonometric identities calculator with steps

Mathway. prove cot ( x) + tan ( x) = sec ( x) csc ( x) trigonometric-identity-proving-calculator. trigonometric-identity-proving-calculator, Please try again using a different payment method. Pizazz worksheets, simplifying trigonometric equations, Lowest Common Denominator Calculator, beginners graphing in algebra, ti 89 laplace transform, algebra … In a previous post, we talked about trig simplification. High School Math Solutions – Trigonometry Calculator, Trig Identities. Proving Trigonometric Identities Calculator. cosec = 1/sec = cot/cosec. Trig identities are very similar to this concept. Verifying Trigonometric Identities Verify the Identity Start on the left side. Proving a trigonometric identity refers to showing that the identity is always true, no matter what value of x x x or θ \theta θ is used. 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Solved example of proving trigonometric identities, Combine fractions with different denominator using the formula: $\displaystyle\frac{a}{b}+\frac{c}{d}=\frac{a\cdot d + b\cdot c}{b\cdot d}$, When multiplying two powers that have the same base ($\cos\left(x\right)$), you can add the exponents, Apply the trigonometric identity: $1-\cos\left(x\right)^2$$=\sin\left(x\right)^2, Multiplying polynomials \cos\left(x\right) and 1+\sin\left(x\right), Factor the polynomial \cos\left(x\right)+\cos\left(x\right)\sin\left(x\right) by it's GCF: \cos\left(x\right), Simplify the fraction \frac{\sin\left(x\right)\left(\sin\left(x\right)+1\right)}{\cos\left(x\right)\left(1+\sin\left(x\right)\right)} by \sin\left(x\right)+1, Apply the trigonometric identity: \frac{\sin\left(x\right)}{\cos\left(x\right)}$$=\tan\left(x\right)$, Since both sides of the equality are equal, we have proven the identity. Download free on Google Play. 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The formulas for double angle identities are as follows: … prove csc ( θ) + cot ( θ) tan ( θ) + sin ( θ) = cot ( θ) csc ( θ) $prove\:\cot\left (x\right)+\tan\left (x\right)=\sec\left (x\right)\csc\left (x\right)$. If you’ve ever taken a ferris wheel ride then you know about periodic motion, you go up and down over and over... prove\:\tan^2(x)-\sin^2(x)=\tan^2(x)\sin^2(x), prove\:\cot(2x)=\frac{1-\tan^2(x)}{2\tan(x)}, prove\:\frac{\sin(3x)+\sin(7x)}{\cos(3x)-\cos(7x)}=\cot(2x), prove\:\frac{\csc(\theta)+\cot(\theta)}{\tan(\theta)+\sin(\theta)}=\cot(\theta)\csc(\theta). en. $\frac{1}{\cos\left(x\right)}-\frac{\cos\left(x\right)}{1+\sin\left(x\right)}=\tan\left(x\right)$, $\frac{1}{\cos\left(x\right)}+\frac{-\cos\left(x\right)}{1+\sin\left(x\right)}=\tan\left(x\right)$, $\frac{1+\sin\left(x\right)-\cos\left(x\right)\cos\left(x\right)}{\cos\left(x\right)\left(1+\sin\left(x\right)\right)}=\tan\left(x\right)$, $\frac{1+\sin\left(x\right)-\cos\left(x\right)^2}{\cos\left(x\right)\left(1+\sin\left(x\right)\right)}=\tan\left(x\right)$, $\frac{\sin\left(x\right)^2+\sin\left(x\right)}{\cos\left(x\right)\left(1+\sin\left(x\right)\right)}=\tan\left(x\right)$, $\frac{\sin\left(x\right)^2+\sin\left(x\right)}{\cos\left(x\right)+\cos\left(x\right)\sin\left(x\right)}=\tan\left(x\right)$, $\frac{\sin\left(x\right)\left(\sin\left(x\right)+1\right)}{\cos\left(x\right)+\cos\left(x\right)\sin\left(x\right)}=\tan\left(x\right)$, $\frac{\sin\left(x\right)\left(\sin\left(x\right)+1\right)}{\cos\left(x\right)\left(1+\sin\left(x\right)\right)}=\tan\left(x\right)$, $\frac{\sin\left(x\right)}{\cos\left(x\right)}=\tan\left(x\right)$, $\cot\left(x\right)\cdot\sec\left(x\right)=\csc\left(x\right)$, $\frac{\csc\left(x\right)}{\cot\left(x\right)}=\sec\left(x\right)$, $\tan\left(x\right)\cdot \cos\left(x\right)\cdot \csc\left(x\right)=1$, $\sin\left(x\right)^2+\cos\left(x\right)^2=1$, $\csc\left(x\right)\cdot\tan\left(x\right)=\sec\left(x\right)$, $\tan\left(x\right)+\cot\left(x\right)=\sec\left(x\right)\csc\left(x\right)$, $\frac{1-\sin\left(x\right)}{\cos\left(x\right)}=\frac{\cos\left(x\right)}{1+\sin\left(x\right)}$. Proving Trigonometric Identities The 7 step method works both sides and meets in the middle, like a V. Some teachers will ask you to prove the identity directly (from one … get Go. Trigonometry… Spinning The Unit Circle (Evaluating Trig Functions ).

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# Tag Info

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The operator $a$ has a polar decomposition $a=u|a|$, where $u$ is a partial isometry with initial space $\ker a^\perp$ and final space given by the closure of the range of $a$. Then $a=u|a|$ and $a^*=|a|u^*$. We can rewrite the given inequalities as $$\Vert |a|^2 - p \Vert < 1/4, \qquad \Vert u|a|^2 u^* -q \Vert < 1/4.$$ Looking at the second ...

0

Here's a more elaborate example to give you a bounded group that is not $C_{0}$. To do this, I'll define a function $F : \mathbb{R}\rightarrow\mathbb{R}$ such that $F(x+y)=F(x)+F(y)$. Then $E(x)=e^{iF(x)}$ satisfies $E(x+y)=e^{iF(x+y)}=e^{iF(x)}e^{iF(y)}=E(x)E(y)$. Automatically $F(0)=0$ because $F(0)=F(0+0)=F(0)+F(0)$, which gives $E(0)=1$. I'll describe ...

1

Let $P_{k}$ be the orthogonal projection onto the closure of the range of $T_{k}$. Then $P_{k}P_{k'}=P_{k'}P_{k}=0$ for $k\ne k'$ because $(T_{k}x,T_{k'}y)=(T_{k'}^{\star}T_{k}x,y)=0$ for all $x,y \in H$. Similarly, if $Q_{k}$ is the orthogonal projection onto the closure of the range of $T_{k}^{\star}$, then $Q_{k}Q_{k'}=0$ for $k\ne k'$. Furthermore, $$... 1 First observation: For all u,v\in\mathcal H and j\ne k$$ \langle T_ju,T_k v\rangle=\langle T_j^*u,T_k^* v\rangle=0. $$Thus the linear subspaces T_j^*\mathcal H, j\in\mathbb Z, are perpendicular to each other, and so are their closures. Set$$ Y=\bigoplus_{j\in\mathbb Z}\overline{T_j^*\mathcal H}. $$This infinite direct sum contains elements of ... 3 Yes, let X=L^{2}[0,1], and let (Tx)(t)=tx(t), i.e., multiplication by t. This is the prototypical example in many regards. You don't have any eigenfunctions because Tx=\lambda x would require (t-\lambda)x(t)=0 for a.e. t\in[0,1] which means that x=0 a.e.. However, you have approximate eigenfunctions. For example, if \lambda\in(0,1) and ... 0 Sorry, but I don't have the rights to add a comment. Can you explain the line$$ C_\pm(w)(A\pm i\mu I)=(A\pm i\mu I) +w B $$I cannot see that this is true. 0 Let H = K = \ell_2. Define a function \Phi by \Phi( x,y ) = \tfrac{1}{2}\|x\|\|y\|I_{H\otimes K}. If there were an extenstion of \Phi to a bounded linear operator \ell_2(\mathbb{N}\times \mathbb{N})\to B(H\otimes K) (still dented by \Phi), it would map the weakly null sequence (e_n\otimes e_n)_{n=1}^\infty in the domain to a weakly null ... 0 An idea: maybe you can show that if x_1M \subset M then (x_1M \cap H^2) \subset (M\cap H^2) (+ remaining a CLOSED subspace, which I have no idea how to show). Once there, you could apply Beurling's Thm and you know that M \cap H^2 is of the form \phi H^2. By density of H^2 as a subspace of H^1, you can probably show that the space M is also of ... 1 Compact operators on Banach spaces can in general be upper-triangularized---generalizing the Jordan canonical form. This means that the space is Schauder-spanned by the generalized eigenvectors. So what you would need to show is that, given any non-zero \lambda \in \sigma(L), the finite dimensional subspace \mbox{ker}(L - \lambda) is spanned by ... 1 You are given that$$ (-2\pi i)Tf = \int_{-\pi}^{\pi}K(x-y)f(y)\,dy \\ =\int_{-\pi}^{\pi}(\pi\,\mbox{sgn}(x-y)-(x-y))f(y)\,dy\\ =\int_{-\pi}^{x}(\pi\,-(x-y))f(y)\,dy\\ +\int_{x}^{\pi}(-\pi-(x-y))f(y)\,dy\\ =(\pi-x)\int_{-\pi}^{x}f(y)\,dy+\int_{-\pi}^{x}yf(y)\,dy \\ -(\pi+x)\int_{x}^{\pi}f(y)\,dy+\int_{x}^{\pi}yf(y)\,dy. $$... 1 I think you are thinking about it backwards. Every Banach space is a vector space. Every vector space has a basis. If your eigenfunctions are: (1) linearly independent under the inner product (2) have the same dimension as the space. They must be an equivalent basis. However, neither (1) nor (2) are necessary conditions of eigenfunctions as far as I know. ... 1 See if you can extend your operator to some L^2 space containing your functions. Maybe your operator is even self adjoint and still compact. Use the result for compact self adjoint operators on Hilbert spaces to find a basis of eigenvectors in that L^2 space. Show that in fact your eigenfunctions are already in your original space. Show that any ... 1 The following is a solution to part (c). One direction was proved in part (a). Therefore, I will only prove the other direction: assuming \sum_{n = 1}^\infty |\lambda_n|^2 < \infty, I will show that T is a Hilbert-Schmidt operator (where \lambda_n are the eigenvalues corresponding to some orthonormal basis of eigenvectors, such that T(f) = \sum_{n ... 2 Let H be a separable Hilbert space as desired. For simplicity assuming we are working over the real field. Then being an integral operator, we have$$ K: H\rightarrow H, (Kf)(x)=\int K(x,y)f(y)dy $$Since K is a symmetric compact operator, it is normal and can be diagonalized. Let us write its eigenvectors as \phi_{j} with eigenvalue \lambda_{j}. ... 1 There is no any relation between stability and connectivity. Take two non intersecting invariant sets ( it might be any 2 trajectories of the system, which in regular autonomous case do not intersect). The union of these trajectories is also an invariant set, which is not connected. I assume you question is more relevant when the invariant set is minimal. ... 0 I don't think stability and connectedness of an invariant set is related. Consider the linear system$$ \dot{x} = A x, ~~ x(0) = x_0 $$Obviously, all trajectories (invariant sets) are connected with the mapping x(t) = e^{At} x_0 regardless of the stability of the system. However, I don't know if all of the invariant sets are connected for an arbitrary ... 0 In short, yes. One thing that you need to clarify is, \epsilon is not a vector while the argument of \Psi is. I'll assume that \epsilon is multiplied by some constant vector v. Consider \Psi(\vec{r}+\epsilon\vec{v}) as a function of both \vec{r} and \epsilon. Name it \psi for clarity.$$ ...

2

Let $(e_i)_{i\in I}$ be an an orthonormal basis, and $x =\sum_{i\in I} y_i e_i$ with norm $1$, i.e. $\sum_{i\in I} |y_i|^2 = 1$, then $$\|Tx\| \leq \sum_{i\in I}|y_i|\|Te_i\| \leq \sqrt{\sum_{i\in I} |y_i|^2}\sqrt{\sum_{i\in I}\|Te_i\|^2} = \|T\|_{HS}$$

0

Slightly different approach: Assume to the contrary that $T$ is compact. Clearly $\|T\|=1$, in fact $\|T^n\|=1,$ so spectral radius is equal to 1. Since the ambient space in infinite dimensional then there exists a sequence of eigenvalues that converges to zero . in particular we can find eigenvalue $\lambda$ with $|\lambda|<1$ and $g\in C[0,1]$ such ...

0

I assume we're talking about normed linear spaces here, and (for convenience) real scalars. 1) If $Y$ is not complete, there is a Cauchy sequence $y_n$ in $Y$ that has no limit there. Take linear operators $T_n$ on $\mathbb R$ such that $T_n 1 = y_n$. Then these form a Cauchy sequence in $B(\mathbb R, Y)$ with no limit. 2) I'm not sure I understand your ...

1

Suppose that $Ax$ is not a scalar multiple of $x$ for some $x \in X$. Then $x\ne 0$, and there is a linear functional $\phi$ defined on the linear span of $\{x,Ax\}$ such that $\phi(x)=1$ and $\phi(Ax)=0$. Such a linear functional is continuous on the two-dimensional subspace spanned by $Ax,x$ and, so, by the Hahn-Banach Theorem, extends to a continuous ...

2

I think we can just play around with some special compact operators, unless I'm missing some subtlety. If $x$ and $y$ are linearly independent, then we can choose a basis for $X$ which contains $x$ and $y$. Let $K_{x, y}$ be the linear operator which swaps $x$ and $y$ and which sends all of our other basis vectors to zero. Then $K_{x, y}$ is compact, ...

4

It is not necessarily true that $\pi(A)$ will have non-trivial projections. But $\pi(A)''$ will, because it is a von Neumann algebra. Now, the assertion you are looking at is basically trivial. You want to prove, for $A$ abelian, "if $\dim H\geq2$ then $\pi$ is not irreducible". This is the same as its contrapositive, which is "If $\pi$ is irreducible, ...

0

The new operator $T$ is going to be also compact, as the product of a bounded and a compact and a bounded operator is also compact, and hence its spectrum, which could now contain even complex numbers, could only have as a limit point the $0$. The fact that $P$ is also self-adjoint does not change anything.

1

Elements in the strong closure are only obviously limits of nets. The Corollary says that they are limits of sequences if the Hilbert space is separable.

3

The strong operator topology on $\mathscr{B}(H)$ is not metrisable if the underlying Hilbert space $H$ is infinite-dimensional. Therefore it need not be sequential a priori. What the Kaplansky density theorem really tells you is that elements in the SOT-closure of a C*-algebra are limits of (norm) bounded nets. Moreover, if the Hilbert space is separable, ...

2

As stated in the comments, you should specify what class of $\phi$ you are considering or what restrictions you impose. Anyway, proceeding a bit vaguely and assuming a "nice class" (for example, at least the inverse of $\phi(D)$ and $\phi(0)$ should be defined) you can transform your equation to (multiplying by $\phi(D)$ on the left and $\phi(0)$ on the ...

0

If $\alpha_{nm}=\delta_{nm}$, the operator $A$ is the identity, which is not compact. So, there is a problem with your attempt... here: $A=\lim A_n$ This is true in the sense of pointwise convergence (strong operator topology). But to conclude compactness, you need norm convergence: $\|A-A_n\|\to 0$. This is not necessarily the case, as the example ...

0

A basis $\{ e_{1},e_{2},\cdots,e_{N} \}$ of an $N$-dimensional vector space $X$ can be thought of as a way to transform a vector $x \in X$ into a coefficient function $\hat{x} : \{ 1,2,3,\cdots,N \}\rightarrow X$, where the inverse map is $(\hat{x})^{\vee} = \hat{x}(1)e_{1}+\cdots+\hat{x}(N)e_{N}=x$. A linear map $L$ on $X$ is diagonalized by the basis ...

1

Restating the requirements, with $D = \dfrac{d}{dx}$ for convenience, we want $$\sum_k a_k(x) D^k(D(f)) = \sum_k b_k(x) D^k(x f) \tag 1$$ for any function $f$. Now by Leibniz's rule, $D(x f) = f + x D(f)$, and by induction $D^k(x f) = k D^{k-1}(f) + x D^k(f)$, so that (1) becomes \eqalign{ \sum_k a_k(x) D^{k+1}(f) &= \sum_k b_k(x) (k D^{k-1}(f) + x ... 1 For existence, one can often find an example. The book has given an example. Here is how it works: A(\frac{d}{dx}(f))=x^2f' and B(x(f))=(x\frac{d}{dx}-1)(xf)=x(\frac{d}{dx}xf) - xf=x(f+xf')-xf=x^2f'+xf-xf =x^2f' And so both are equal. If I have misunderstood you, please say. 1 In general, it is fully acceptable for a bounded self-adjoint operator to not have any eigenvalues. Consider for instance the operater M\in L^2([0,1]) defined on a representative by (Mf)(x)=xf(x). $$It is not hard to see that the spectrum of this operator is [0,1] (because it is a multiplication operator corresponding to a real continuous function ... 3 Rule of thumb: compactness should be expected when Tf is at least a little bit nicer than a generic element of the target space. Integration against a continuous kernel does this because Tf inherits modulus of continuity from the kernel, which is an improvement compared to having totally unknown modulus of continuity. But for compactness from C^0 to ... 1 The spectral theorem for normal operators on a Hilbert space gives you the result, but you don't get a unique selfadjoint T. Any unitary U can be written as$$ U = \int_{|\lambda|=1}\lambda dE(\lambda), $$where E is the unique spectral measure associated with U. Choose a branch of \log and define$$ ...

0

Be careful that $T$ is not assumed to be bijective, so things like $T^{-1}x$ do not make sense. Maybe the following hint (in the spirit of the one you indicated for forward orbits) will be enough for you. As above, let $(y_n)_{n\geq 1}$ be dense in $X$, and let $x_0\in X$. Find $x_1$ close to $y_1$ and an integer $n_1$ such that $T^{n_1}x_1$ is close to ...

1

Diagonalization is synonymous with finding a basis of eigenvectors. If you have a diagonal matrix, the elements of the standard basis are all clearly eigenvectors, and conversely, if you write the matrix in a basis of eigenvectors it will be diagonal. The exponentials form a basis of eigenvectors,so in this sense they diagonalize convolution. Here follows a ...

0

I obtained the following upper bound. Let $D_z$ denote a diagonal matrix whose diagonal entries are given by vector $z$. Furthermore, write the SVD of $X$ as $X=UD_\sigma V^\mathrm{T}=\sum _{i=1}^r\sigma_i u_iv_i^\mathrm{T}$ where $r$ denotes the rank of $X$. Then we have \begin{align*} F\odot X &=\sum_{i=1}^r F\odot \left(\sigma_i u_i ...

2

First observe that the proof given in Bounded linear operator maps norm-bounded, closed sets to closed sets. Implies closed range? does not use the fact that $X,Y$ are assumed to be complete. It hence remains to show that we can reduce to the case of an injective operator. To this end, we "factor out the kernel": As $T$ is bounded, the kernel $Z := ... 1 This is just a thought which might be helpful but is a bit long to be a comment. Consider a$N\times N$matrix$A$whose$(1,1)$entry is$|z|^2$. Let$e_1=[1,0,\dots,0]^T$denote first column of$N\times N$Identity matrix. Then note that, $$\lambda_{min}(A) = \min_{x^Tx=1}~x^TAx\leq e_1^TAe_1=|z|^2$$ 2 A common way to prove that some operator$T$is not compact is to exhibit an infinite-dimensional subspace$M$on which$T$has a lower bound: that is, there exists$c>0$such that $$\|Tx\|\ge c\|x\|,\quad \forall\ x\in M \tag{1}$$ If (1) holds, then the image of unit ball under$T$contains a ball of radius$c$in the infinite-dimensional subspace$TM$, ... 1 Jonas Meyer's comment gives an example of a self-adjoint$T$whose spectrum is$[-1,1]$. Then we may take the absolute value function, for instance, or even any continuous function whose zeros accumulate in$(-1,1)$to see that that the space of continuous functions on the spectrum contains functions which are not power series. Since$C(\sigma (T))$is ... 0 I'll assume the underlying space of interest is$L^{2}[0,L]$because any discussion of 'selfadjoint' requires a Hilbert space. And I'm assuming you're starting with a domain$C^{2}[0,L]$. There are variations where you assume$f \in C^{2}(0,L)$with$f''\in L^{2}[0,L]$, and I'll leave those to you. Define$\Delta f = f''$on$\mathcal{D}(\Delta)=\{ f \in ...

3

In the following, I will assume that $\mu$ is $\sigma$-finite for simplicity. One can probably drop that assumption with a bit of extra work. Also, I do not claim that my proof is one of the simpler ones. Let us define $$\left\Vert k\right\Vert _{L^{p,q}}:=\left(\int\left(\int\left|k\left(x,y\right)\right|^{p}d\mu\left(x\right)\right)^{q/p}\, ... 3$$\int_0^1 \underbrace{|k(s,t)-k(s',t)|}_{\leq \sup_{r \in [0,1]} |k(s,r)-k(s',r)|} \underbrace{|f(t)|}_{\leq \sup_{r \in [0,1]} |f(r)|} \, dt \leq \sup_{r \in [0,1]} |k(s,r)-k(s',r)| \cdot \|f\|_{\infty} \cdot \int_0^1 \, dt.$$1 Nota Bene: In my haste and arrogance I misread the question as written, blithely and blindly assuming that the only case of interest was t \to \infty; when my error was graciously pointed out by our colleague PhoemueX (see comments), I deleted my answer until such time as I could correct it. Having done so, I present my rectified answer below; however, I ... 2 You can find P so that PGP^{-1}=T=\begin{pmatrix}\lambda_1 & * & *\\&\ddots & *\\&&\lambda_n\end{pmatrix} where the \lambda_is are the eigenvalues and * means "whatever". Then, \exp(tG)=\exp(tP^{-1}TP)=P^{-1}\exp(tT)P. \exp(tT)=\begin{pmatrix}e^{t\lambda_1} & * & *\\&\ddots & ... 2 If you have a bounded operator A, then the holomorphic functional calculus is always an option, and it is based on Cauchy's integral representation:$$ f(A) = \frac{1}{2\pi i} \oint_{C} f(\lambda)\frac{1}{\lambda I-A}"\,d\lambda = \frac{1}{2\pi i} \oint_{C} f(\lambda)(\lambda I-A)^{-1}\,d\lambda.  The contour $C$ is any simple ...

2

The nearest point projection onto a closed subset $E\subset \mathbb R^n$ is single-valued if and only if $E$ is convex*. In this case, the Lipschitz constant is equal to $1$. If $E$ is not convex, there is at least one point $x\in \mathbb R^n$ for which $\min_{y\in E}\|x-y\|$ is attained at more than one point. We could try to discuss the continuity of ...

1

This is a comment to @ABC 's answer. I seem to have difficulties to understand things correctly, maybe it's simply too late. Here is a list of a: the alternating series of units, b: the partial sums (=using Cesaro-order 0), c: the partial sums using Cesaro-order 1, d: and the squares of $c$, (where it is focused that the ...

7

We have the following characterization of adjoint operators: Suppose that $X$ and $Y$ are normed spaces. If $T:X\rightarrow Y$ is a bounded linear operator, then $T^*$ is weak*-weak* continuous. Conversely, if $S$ is a weak*-weak* continuous linear operator from $Y^*$ to $X^*$, then there is a bounded linear operator $T:X\rightarrow Y$ such that $T^*=S$. ...

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# Anomalies for not-on-site discrete gauge symmetries

If a symmetry group $G$ (let's say finite for simplicity) acts on a lattice theory by acting only on the vertex variables, I will call it ultralocal. Any ultralocal symmetry can be gauged. However, in general there are discrete symmetries that cannot be gauged. For example, Freed and Vafa in http://link.springer.com/article/10.1007%2FBF01212418 discuss how in 1+1d one needs the pullback of a certain class in $H^3(G,U(1))$ to have some trivial periods.

So, is the converse true--that if $G$ can be gauged then there is a formulation of the theory where $G$ acts ultralocally? In other words, is a symmetry with no ultralocal action necessarily anomalous?

And if so, can we see this anomaly as an explicit class in $H^3(G,U(1))$ for 1+1d theories, for example?

It seems to me like the answer is yes. If we have no anomaly and go ahead and gauge $G$, then we can put the result on a lattice where the $G$ gauge field will live on edges. These edge variables will have the flatness condition that the start and end vertex variables differ by the action of the edge variable (an element of $G$). It seems like $G$ shouldn't act anywhere else, since in some sense gauging $G$ is a type of "fat quotient" of the theory by $G$. Thus, if we take this lattice formulation and forget the gauge field, we end up again with the original theory, but now with an ultralocal action of $G$. What remains is how to quantify this anomaly in the cohomology group.

• Hi Ryan, just read your profile and thought maybe you are interested in this too ... – Dilaton Aug 18 '13 at 19:11
• We also had some discussion about the topic here and here. What is really needed to (re)start is to bring a large enough group if interested people together ... – Dilaton Aug 18 '13 at 20:03
• I would say after 3.5 years working on this problem that it's still open :) – Ryan Thorngren Mar 16 '18 at 16:53

Your question is very interesting. I would like to mention something along the line of your question, but perhaps from another viewpoint. Recently there are some better understanding along the thinking between

(1)"whether a theory is free from anomaly (the anomaly matching condition satisfied),"

(2)"whether the symmetry of a theory is on-site symmetry,"

(3)"whether the symmetry of a theory can be gauged,"

(4)"whether the theory can exist alone in its own dimension without an extra bulk dimension,"

(5)"whether the massless modes of the theory can be gapped (opened up a mass gap) without breaking the assigned symmetry."

The insight connects to a topic in condensed matter physics, such as the intrinsic topological order and symmetric protected topological order(such as the topological insulator).

(A) In this paper: Classifying gauge anomalies through SPT orders and classifying gravitational anomalies through topological orders, it is proposed that the anomaly can be classified by a cohomology group of $$\text{Free}[\mathcal{H}^{d+1}(G,U(1))] \oplus \mathscr{H}_\pi^{d+1}(BG,U(1))$$ The ABJ anomalies are classified by $\text{Free}[\mathcal{H}^{d+1}(G,U(1))]$, while $\mathscr{H}_\pi^{d+1}(BG,U(1))$ is beyond ABJ type, such as for discrete gauge anomaly.

In this 1303.1803, it is explained the above notions, to a certain degree (1),(2),(3),(4) are related, or even identical.

(B) In this paper: A Lattice Non-Perturbative Definition of 1+1D Anomaly-Free Chiral Fermions and Bosons, it has been shown the relation between (1),(4) and (5), i.e. the anomaly matching condition = the massless modes of the theory can be fully gapped, for a specific case that the theory has a U(1) symmetry:

$$%{\boxed{ \text{ ABJ's U(1) anomaly matching condition in 1+1D} \Leftrightarrow\\ \text{the boundary fully gapping rules of 1+1D boundary/2+1D bulk }\\ \text{with unbroken U(1) symmetry.} %}}$$

There in 1307.7480, based on this understanding, the chiral fermions on the lattice is proposed by including strong interactions. It avoids Fermion doubling problem due to the theory is not free, but interacting. A similar idea to put a SO(10) chiral gauge theory and its induced standard model on the lattice is proposed in 1305.1045.

Back to your question, you had said that $$\text{Any ultralocal symmetry can be gauged}$$ I suspect this understanding can connect to Dijkgraaf-Witten theory. It seems to me your converse statement: $$\text{if G can be gauged then there is a formulation of the theory where G acts ultralocally.}$$ would also be true. If one use the understanding that my above listed notions, (3) a theory can be gauged $\leftrightarrow$ (1) a theory is free from anomaly $\leftrightarrow$ (2) the symmetry is on-site symmetry. We suppose one can further use the idea of Dijkgraaf-Witten theory, and the correspondence between "the gauge symmetry $G$ variables acted on the links(the gauge symmetry $G$ of a gauge theory)" and "the symmetry $G$ acted on the vertices(the global symmetry $G$ of a Symmetry Protected Topological order)", in principle "$G$ acts on the links" and "$G$ acts on the vertices" are dual to each other, then we may argue your statement is an "if and only if" statement.

• Thanks for your answer, Idear. You point out some interesting references I didn't know about. SPT phases are indeed the context I've been thinking about this problem in and that any ultralocal symmetry can be gauged does connect to DW theory. – Ryan Thorngren Aug 19 '13 at 18:49
• . @Idear: About your recent question about unexplained breakthroughs, I think it was a really good question. It may be on-topic on Math Overflow (which also does accept advanced physics questions; many of your questions may in fact be rather on-topic there.). – Abhimanyu Pallavi Sudhir Aug 25 '13 at 12:43
• Thanks DIMension 10, are you really 14 years-old??? – wonderich Dec 20 '13 at 16:32


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lassnags

0th

Percentile

Snag classification

Snag classification/segmentation using several possible algorithms (see details). The function attributes a number identifying a snag class (snagCls attribute) to each point of the point cloud. The classification/segmentation is done at the point cloud level and currently only one algorithm implemented, which uses LiDAR intensity thresholds and specified neighborhoods to differentiate bole and branch from foliage points (see details).

Usage
lassnags(las, algorithm, attribute = "snagCls")
Arguments
las

An object of class LAS or LAScatalog.

algorithm

function. An algorithm for snag segmentation. lidR has wing2015.

attribute

character. The returned LAS object automatically has a new attribute (a new column). This parameter is the name of this new attribute.

Value

If the input is a LAS object, return a LAS object. If the input is a LAScatalog, returns a LAScatalog.

Working with a LAScatalog

This section appears in each function that supports a LAScatalog as input.

In lidR when the input of a function is a LAScatalog the function uses the LAScatalog processing engine. The user can modify the engine options using the available options. A careful reading of the engine documentation is recommended before processing LAScatalogs. Each lidR function should come with a section that documents the supported engine options.

The LAScatalog engine supports .lax files that significantly improve the computation speed of spatial queries using a spatial index. Users should really take advantage a .lax files, but this is not mandatory.

Supported processing options

Supported processing options for a LAScatalog (in bold). For more details see the LAScatalog engine documentation:

• chunk size: How much data is loaded at once.

• chunk buffer*: Mandatory to get a continuous output without edge effects. The buffer is always removed once processed and will never be returned either in R or in files.

• chunk alignment: Align the processed chunks.

• cores: How many cores are used. More cores means more data is loaded at once.

• progress: Displays a progression estimation.

• output_files*: Mandatory because the output is likely to be too big to be returned in R and needs to be written in las/laz files. Supported templates are {XLEFT}, {XRIGHT}, {YBOTTOM}, {YTOP}, {XCENTER}, {YCENTER} {ID} and, if chunk size is equal to 0 (processing by file), {ORIGINALFILENAME}.

• laz_compression: write las or laz files

• select: The function will write files equivalent to the original ones. Thus select = "*" and cannot be changed.

• filter: Read only points of interest.

• lassnags
Examples
# NOT RUN {
LASfile <- system.file("extdata", "MixedConifer.laz", package="lidR")
las <- readLAS(LASfile, select = "xyzi", filter="-keep_first") # Wing also included -keep_single

# For the Wing2015 method, supply a matrix of snag BranchBolePtRatio conditional
# assessment thresholds (see Wing et al. 2015, Table 2, pg. 172)
bbpr_thresholds <- matrix(c(0.80, 0.80, 0.70,
0.85, 0.85, 0.60,
0.80, 0.80, 0.60,
0.90, 0.90, 0.55),
nrow =3, ncol = 4)

# Run snag classification and assign classes to each point
las <- lassnags(las, wing2015(neigh_radii = c(1.5, 1, 2), BBPRthrsh_mat = bbpr_thresholds))

# Plot it all, tree and snag points...
plot(las, color="snagCls", colorPalette = rainbow(5))

# Filter and plot snag points only
snags <- lasfilter(las, snagCls > 0)
plot(snags, color="snagCls", colorPalette = rainbow(5)[-1])

# Wing et al's (2015) methods ended with performing tree segmentation on the
# classified and filtered point cloud using the watershed method
# }

Documentation reproduced from package lidR, version 2.0.0, License: GPL-3

Community examples

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text:
Summer Deals - hours only!Up to 80% off on all courses and bundles.-Close
Introduction
Inserting and updating NULLs
Conditions in UPDATE and DELETE
Using values from another column
Return rows in INSERT, UPDATE, DELETE
14. Display inserted values
Summary

## Instruction

Okay. You know how to display column values during a DELETE operation. You can do the same thing during an INSERT:

INSERT INTO Student
OUTPUT Inserted.Id, Inserted.LastName
VALUES (13, N'Alan', NULL, N'Stanley');

This returns a table with the names of the columns listed after OUTPUT, assigning them inserted values. You use the Inserted prefix to refer to newly inserted rows.

Id LastName
13 Stanley

Notice that the OUTPUT clause is placed after the table name (Student) and before VALUES. After OUTPUT, we list the names of the columns getting the data, separated by commas (Id, LastName). Notice that these column names must be preceded by the Inserted prefix followed by a dot(Inserted.Id, Inserted.LastName).

As you've noticed, OUTPUT will display only chosen data or all inserted data (as in the DELETE statement we did previously). In the example above, we showed data from only two columns and from the inserted row.

## Exercise

Are you ready to do a task on your own? The teacher would like to insert Alan Stanley's score on his written Spanish exam. Alan's student ID is 13, he scored 23 points, the exam date was 2018-06-23, and the exam score date was 2018-06-30. The ID of the exam is 16.

Show the exam score and the score date in the inserted records.


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# Tag Info

8

Just up front, your code doesn't have any blatant bugs which would prevent its use. There are a few things that you could do to improve it though: One thing that I'd try to avoid is singletons. These are effectively globals and bring with them almost all their problems. In particular, it prevents isolating code for unit tests. The alternative is that you ...

8

It's probably worth accepting a second template argument for an Allocator to be passed through to the std::deque: template<class T, class Allocator = std::allocator<T>> class ThreadSafeDeque { std::deque<T, Allocator> deque; }; Technically, we do need to include <condition_variable>; we're not allowed to assume that <mutex>...

5

Redundant summing // The hand vector contains the maximum number of cards // We can now compute its value That is not completely true, the value could have been built up incrementally each time a card was added to the hand, which would remove some duplicated work: hands that share a common prefix would not each recompute the sum of the values of ...

5

This code is broken. Imagine an Account with $100, two threads, one depositing$1, the other withdrawing $50. Thread one acquired lock1, and begins executing: this.balance = getBalance() + amount; and reads$100 ... Thread two acquired lock2, reads the balance (still $100), subtracts$50, stores the new balance (\$50) and releases lock2. ... (back to ...

4

Is it worth parallelising? When you are parallelizing code you have to ask yourself if it's worth doing that. Parallel code doesn't magically give a speedup, because there are various things that can actually slow you down when using parallelism, for example: Spawning and waiting for threads itself costs some time. Threads might be fighting for resources, ...

4

Invoke is a blocking call that returns only after that call has competed. That means your loop is also including the time it takes to marshal over to the GUI thread and complete. You probably don't want that. I would use BeginInvoke instead, which does not wait for the method to complete on the GUI thread. This is also the difference between ...

4

Here's a better algorithm: It has been benchmarked and reduces the time from 29 hours to 1.1 seconds This is approximately 95,000 times faster. Edit: Faster version, described below, reduces the execution time to 0.68 seconds, which is 153,500 times faster. We only need to consider the number of each type of honor and then combinations of remaining spot ...

4

how can I improve the performance of my program? A standard answer is use a better algorithm. Trying to improve performance by shaving cycles while enumerating a 635 013 559 600 strong set is futile. Consider instead enumerating subsets of valuable cards. There are merely $2^{16} = 65536$ of them; a trillion time acceleration. Given a popcount function, ...

3

Design The design is OK; it seems to use data classes and some classes that perform operations. In true OO fashion, I would expect the player to send the message. For instance, you could do Player.sendMessageTo(int otherPlayerID, String message). You could initialize the player with a MessageHandler that does the actual sending of the message. The player ...

3

Other answers cover a lot but I also would suggest replacing while(deque.empty()) { condition.wait(lock); // unlocks, sleeps and relocks when woken up } with condition.wait(lock, [this]{return !deque.empty();}); because it means the same and is more compact (readability being the same or also better).

3

Your singleton initialization pattern is unsafe for a logger that you may want to use from other static initializers (that have undefined initialization order) because you cannot guarantee that the initialize function has been executed. Typically you would use C++11 magic statics for this, but then you can't give a filename to it. Providing a filename as ...

2

It seems your code will work expected. I am sure that there are other improvements for multithreading but I will just suggest some improvements in terms of readability and usability. I hesitated to share but maybe it could be beneficial for you or others. Firstly, intuitive expectation of most of developers is using operator << to stream out things. ...

1

Yes, your approach is correct. However, you parallelize various queries, which probably go to a remote backing database. As you told us nothing about the runtime of these queries, the nature of the database, the complexity of the underlying statements, I can only ask you to measure. Parallelism induces overhead - in the runtime AND in program complexity. ...

1

Logging after an infinite loop The only way that this: while True: time.sleep(1) message = random.randint(1, 101) logging.info("Producer_1 got message: %s", message) queue_1.put(message) event_1.set() is going to terminate is on an exception. That in itself isn't a bad thing - stop iteration exceptions are a common pattern in Python. ...

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text:
Abstract view

# Some norm inequalities for operators

Published:1999-03-01
Printed: Mar 1999
Let $A_i$, $B_i$ and $X_i$ $(i=1, 2, \dots, n)$ be operators on a separable Hilbert space. It is shown that if $f$ and $g$ are nonnegative continuous functions on $[0,\infty)$ which satisfy the relation $f(t)g(t) =t$ for all $t$ in $[0,\infty)$, then $$\Biglvert \,\Bigl|\sum^n_{i=1} A^*_i X_i B_i \Bigr|^r \,\Bigrvert^2 \leq \Biglvert \Bigl( \sum^n_{i=1} A^*_i f (|X^*_i|)^2 A_i \Bigr)^r \Bigrvert \, \Biglvert \Bigl( \sum^n_{i=1} B^*_i g (|X_i|)^2 B_i \Bigr)^r \Bigrvert$$ for every $r>0$ and for every unitarily invariant norm. This result improves some known Cauchy-Schwarz type inequalities. Norm inequalities related to the arithmetic-geometric mean inequality and the classical Heinz inequalities are also obtained.
 MSC Classifications: 47A30 - Norms (inequalities, more than one norm, etc.) 47B10 - Operators belonging to operator ideals (nuclear, $p$-summing, in the Schatten-von Neumann classes, etc.) [See also 47L20] 47B15 - Hermitian and normal operators (spectral measures, functional calculus, etc.) 47B20 - Subnormal operators, hyponormal operators, etc.


text:
Timezone: »

Poster
Training Recurrent Neural Networks via Forward Propagation Through Time
Anil Kag · Venkatesh Saligrama

Wed Jul 21 09:00 PM -- 11:00 PM (PDT) @ None #None
Back-propagation through time (BPTT) has been widely used for training Recurrent Neural Networks (RNNs). BPTT updates RNN parameters on an instance by back-propagating the error in time over the entire sequence length, and as a result, leads to poor trainability due to the well-known gradient explosion/decay phenomena. While a number of prior works have proposed to mitigate vanishing/explosion effect through careful RNN architecture design, these RNN variants still train with BPTT. We propose a novel forward-propagation algorithm, FPTT, where at each time, for an instance, we update RNN parameters by optimizing an instantaneous risk function. Our proposed risk is a regularization penalty at time $t$ that evolves dynamically based on previously observed losses, and allows for RNN parameter updates to converge to a stationary solution of the empirical RNN objective. We consider both sequence-to-sequence as well as terminal loss problems. Empirically FPTT outperforms BPTT on a number of well-known benchmark tasks, thus enabling architectures like LSTMs to solve long range dependencies problems.


text:
# Macro or \def to create lines in a table

I have a certain table to which I would like to add lines by a command.

One way I can get to this looks like this:

\newcommand\arow[3]{
#1 & #2 & #3 \tabularnewline
}

and the table

\begin{tabularx}{\unitlength}{ X X X }

First & Second & Third \tabularnewline
\arow{I}{AM}{ROW}
\end{tabularx}

which is ok for simple tables, but when the situation gets more involved I would like to be able to separate the initialization of the row from the placing of the table.

By more involved I mean that the command will take possibly a large number of variable, some of which numbers. Meanwhile the table shall reside amidst other code for the page, somewhere. Ideally, I have a neat .tex file where I list all my definition and either save them to a easy name that I then add in the table (so that I also don't lose an idea of the content of the table, which I would if I were to, say, use \input{my_list_of_rows.tex} altought this might be a solution.

I imagined something like saving the resulting row from the command in a more succintly named variable and have a separate sheet for it.

An example would be:

my_rows.tex:
row_complex = \arow{this}{is}{a}{nastyyyy}{$\mathcal{C}$ row}
row_real    = \arow{this}{is}{a}{less nasty}{$\mathcal{R}$ow}

with the table taking

\begin{tabularx}{\unitlength}{ ... }
...
row_complex
row_real
\end{tabularx}

Is it possible and how would one do such a thing? Or, what are other standard way of dealing with this kind of situation?

• I don't understand what you mean by a "more involved" situation; "[separating] the initialization of the row from the placing in the table." Can you elaborate? – Werner May 25 '16 at 23:05
• Can you also complete your code so that it can be compiled? If your code is completed 'in the obvious way', it certainly will not work because the macro name you have chosen is not allowed. (If you are using expl3 syntax to do this, then you are abusing it badly and should please stop doing it in public. What you do in private is up to you. ;) ) – cfr May 25 '16 at 23:17
• Ehm..I do not abuse syntax either in private or in public. I am badly new to making latex doing more complicated than basic command. I have read around stuff like \def and macros might be on the lines of what I am looking for, but honestly I don't even know where the documentation on this features should be (which is the con of how "we" teach ourselves latex). – Three Diag May 25 '16 at 23:25
• @ThreeDiag I thought it unlikely that was the reason, but I wasn't sure so it seemed worth pointing out. Because you you did say your code was OK for simple cases, but unless you are changing cat codes, it can't possibly work because those are not legal macro names. – cfr May 26 '16 at 0:03

Basic option would be my_rows.tex with

\rowdef{row_complex}{\arow{this}{is}{a}{nastyyyy}{$\mathcal{C}$ row}}
\rowdef{row_real}{\arow{this}{is}{a}{less nasty}{$\mathcal{R}$ow}}

\newcommand*\arow[3]{#1&#2&#3\tabularnewline}
\newcommand*\rowdef[1]{\expandafter\def\csname row@#1\endcsname}
\newcommand*\rowuse[1]{\csname row@#1\endcsname}
\input{my_rows}

with the table taking

\begin{tabularx}{\unitlength}{..}
...
\rowuse{row_complex}
\rowuse{row_real}
\end{tabularx}
• By the way, \arow has five arguments… – Manuel May 25 '16 at 23:57