text: Using the packages lmtest and multiwayvcov causes a lot of unnecessary overhead. Is it more efficient to send a fleet of generation ships or one massive one? Generation of restricted increasing integer sequences. Why did George Lucas ban David Prowse (actor of Darth Vader) from appearing at sci-fi conventions? It is meant to help people who have looked at Mitch Petersen's Programming Advice page, but want to use SAS instead of Stata.. Mitch has posted results using a test data set that you can use to compare the output below to see how well they agree. ��q�����y�kz"���1qb:n�(�i��.�V|.p'�]\��'���dp��@�q�A*IYm�.�L!7�s��.���aG�ɾ[�Z2�C1���WHې=�7�����ud�� Weighted least squares. But at least For more discussion on this and some benchmarks of R and Stata robust SEs see Fama-MacBeth and Cluster-Robust (by Firm and Time) Standard Errors in R. See also: Clustered standard errors in R using plm (with fixed effects) stream Ecclesiastical Latin pronunciation of "excelsis": /e/ or /ɛ/? It only takes a minute to sign up. That is, if the amount of variation in the outcome variable is correlated with the explanatory variables, robust standard errors can take this correlation into account. 584 That is, if the amount of variation in the outcome variable is correlated with the explanatory variables, robust standard errors can take this correlation into account. endobj stream endobj Robust standard errors vs clustered standard errors 09 Sep 2015, 09:46. Key words: White standard errors, longitudinal data, clustered standard errors JEL codes: C23, C12 1 We thank Alberto Abadie, Gary Chamberlain, Guido Imbens, Doug Staiger, Hal White, and the referees for helpful comments and/or discussions, Mitchell Peterson for providing the data in footnote 2, and Anna Mikusheva for research assistance. Clustered standard errors belong to these type of standard errors. The formulation is as follows: Clustered errors have two main consequences: they (usually) reduce the precision of ̂, and the standard estimator for the variance of ̂, V [̂] , is (usually) biased downward from the true variance. %PDF-1.3 '$�:����y �rYNb��dHB���(+1bhHىGC. 7 0 obj About robust and clustered standard errors. Finally, I verify what I get with robust standard errors provided by STATA. 6 0 obj 1 Standard Errors, why should you worry about them ... Heteroskedasticity (i.n.i.d) Now Var (b)=E h X0X i 1 X0ee0X h X0X i 1 = h X0X i 1 E h With panel data it's generally wise to cluster on the dimension of the individual effect as both heteroskedasticity and autocorrellation are almost certain to exist in the residuals at the individual level. Generalized least squares estimator. I ^ is still unbiased for Molly Roberts Robust and Clustered Standard Errors March 6, 2013 12 / 35. However, when comparing random effects (xtreg, re cluster()) and pooled OLS with clustered standard errors (reg, cluster()), I have hard time understanding how one should choose between the two. ��������ieJI9I�zGFn%���r���/%NzR�V@ng��Y�ć*�s���f*Ŷ�zmu9�Ngꛊ�BQ�ѡ$���c,˜�E�_hxO�A$�"�,��]�����vh��)A��r۫�,�U� 2. By default vcovHC() estimates a heteroskedasticity consistent (HC) variance covariance matrix for the parameters. The way to accomplish this is by using clustered standard errors. I accidentally added a character, and then forgot to write them in for the rest of the series, Panshin's "savage review" of World of Ptavvs. Robust standard errors can account for heteroskedasticity in the data and prevent incorrect inferences. DeepMind just announced a breakthrough in protein folding, what are the consequences? Who first called natural satellites "moons"? The Huber-White robust standard errors are equal to the square root of the elements on the diagional of the covariance matrix. 5 0 obj How to avoid boats on a mainly oceanic world? HETEROSKEDASTICITY-ROBUST STANDARD ERRORS 159 (T t=1 X˜ itu it) (the second equality arises from the idempotent matrix identity). << /T1.0 9 0 R /T3.0 12 0 R /T2.0 11 0 R >> >> a��\��Ʌ+�����Ĥ��+�~?ب�9 ������% ����. Hence, I was hoping that I can address both issues simultaneously. << /ProcSet [ /PDF /Text ] /ColorSpace << /Cs2 10 0 R /Cs1 8 0 R >> /Font Dear all, I am doing an analysis of the pollution haven effect in the German manufacturing industry. 2 Estimating xed-e ects model The data set Fatality in the package Ecdat cover data for 48 US states over 7 ... Heteroskedasticity-robust standard errors for xed e ects panel data regression. We do not impose any assumptions on the What do I do to get my nine-year old boy off books with pictures and onto books with text content? Clustered standard errors are popular and very easy to compute in some popular packages such as Stata, but how to compute them in R? Clustering errors in Panel Data at the ID level and testing its necessity, How to estimate a fixed effects regression WITH robust standard errors AND instrument variables, Double-clustered standard errors and large panel, R | Robust standard errors in panel regression clustered at level != Group Fixed Effects. This page shows how to run regressions with fixed effect or clustered standard errors, or Fama-Macbeth regressions in SAS. Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. by Dhananjay Ghei.$${\displaystyle {\widehat {\beta }}_{\text{OLS}}=(\mathbb {X} '\mathbb {X} )^{-1}\mathbb {X} '\… Convert negadecimal to decimal (and back). ... Panel Data: 7. Is that a severe problem? Computing cluster -robust standard errors is a fix for the latter issue. You just need to use STATA command, “robust,” to get robust standard errors (e.g., reg y x1 x2 x3 x4, robust). I ^ is still unbiased for Molly Roberts Robust and Clustered Standard Errors March 6, 2013 12 / 35. Heteroskedasticity just means non-constant variance. This function allows you to add an additional parameter, called cluster, to the conventional summary() function. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. As Wooldridge notes, the heteroskedasticity robust standard errors for this specification are not very different from the non-robust forms, and the test statistics for statistical significance of coefficients are generally unchanged. Actually, I have run such a regression and detected heteroskedasticity. In practice, heteroskedasticity-robust and clustered standard errors are usually larger than standard errors from regular OLS — however, this is not always the case. ”Robust” standard errors is a technique to obtain unbiased standard errors of OLS coefficients under heteroscedasticity.In contrary to other statistical software, such as R for instance, it is rather simple to calculate robust standard errors in STATA. One could use information about the within-cluster correlation of errors to What happens when the agent faces a state that never before encountered? All you need to is add the option robust to you regression command. Making statements based on opinion; back them up with references or personal experience. Remember, the Fisher information matrix is E [hi(Yij )]. (solved). We see that the standard errors are much larger than before! Actually, I have run such a regression and detected heteroskedasticity. The use of cluster robust standard errors (CRSE) is common as data are often collected from units, such as cities, states or countries, with multiple observations per unit. Notice in fact that an OLS with individual effects will be identical to a panel FE model only if standard errors are clustered on individuals, the robust option will not be enough. ��&��$Ñ��z(�qh��h�v��<=���t% Asking for help, clarification, or responding to other answers. endstream 13 0 obj I was wondering if, when running a regression on panel data, clustered standard errors are already correcting for heteroskedasticity. We call these standard errors heteroskedasticity-consistent (HC) standard errors. C23, C12 ABSTRACT ... Another estimator of Σ is the clustered (over entities) variance estimator, 11 1 Robust standard errors can account for heteroskedasticity in … Their gener-alized method of moments{based covariance matrix estimator is an extension of White’s y��\ _ �Թb� eb]�R1��k�$�A\ Find the farthest point in hypercube to an exterior point. Since standard model testing methods rely on the assumption that there is no correlation between the independent variables and the variance of the dependent variable, the usual standard errors are not very reliable in the presence of heteroskedasticity. For this reason,we often use White's "heteroskedasticity consistent" estimator for the covariance matrix of b, if the presence of heteroskedastic errors is suspected. plm can be used for obtaining one-way clustered standard errors. Heteroskedasticity-Robust Standard Errors for Fixed Effects Panel Data Regression James H. Stock and Mark W. Watson NBER Technical Working Paper No. Clustered standard errors are an additional method to deal with heteroscedastic data. About robust and clustered standard errors. However, what happens if I correct for heteroscedasticity by means of clustered standard errors, even though there is prove that the initial results are homoscedastic. Second, in general, the standard Liang-Zeger clustering adjustment is conservative unless one Zx�~�,��ג���Ȯ'{#>II���w 2v� �T@0y�oh)�>y���[�d�1��K��7u��n���V��,���� !+���c[0M;k3N��� ��ُu|^�0Ê��@l�Sf Since I used the pooled OLS model I have to cluster the standard errors anyway. Clustered Standard Errors. Σˆ and obtain robust standard errors by step-by-step with matrix. Is that right? Clustered errors have two main consequences: they (usually) reduce the precision of ̂, and the standard estimator for the variance of ̂, V [̂] , is (usually) biased downward from the true variance. An Introduction to Robust and Clustered Standard Errors GLM’s and Non-constant Variance RSEs for GLMs This shouldn’t be too unfamiliar. There are several ways to estimate such a HC matrix, and by default vcovHC() estimates the “HC3” one. rev 2020.12.2.38106, The best answers are voted up and rise to the top, Cross Validated works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. Comment: On p. 307, you write that robust standard errors “can be smaller than conventional standard errors for two reasons: the small sample bias we have discussed and their higher sampling variance.” A third reason is that heteroskedasticity can make the conventional s.e. The ﬁrst of these expressions leads to the “clustered” (over entities) variance estimator Σˆcluster = 1 nT n i=1 T t=1 X˜ it uˆ˜ it T s=1 X˜ is uˆ˜ is (10) endobj Molly Roberts Robust and Clustered Standard Errors March 6, 2013 20 / 1 The ﬁrst of these expressions leads to the “clustered” (over entities) variance estimator Σˆcluster = 1 nT n i=1 T t=1 X˜ it uˆ˜ it T s=1 X˜ is uˆ˜ is (10) This procedure is reliable but entirely empirical. x�WwXS��[�P�Ы�Бf@z�� ҋ#&!Đ� 6dQ���ˊ.���V֊kǮT�uq�77�����{��o��9s��9�wf���r�X,D �"�$,��:!5��C?7_̊��"�h���s͑�5\$�g����s��q� For further detail on when robust standard errors are smaller than OLS standard errors, see Jorn-Steffen Pische’s response on Mostly Harmless Econometrics’ Q&A blog. 3 0 obj An Introduction to Robust and Clustered Standard Errors Linear Regression with Non-constant Variance Review: Errors and Residuals ... heteroskedasticity. %��������� Thanks for contributing an answer to Cross Validated! Robust standard errors account for heteroskedasticity in a model’s unexplained variation. I a first specification, I am using robust standard errors as I have heteroscedasticity. The easiest way to compute clustered standard errors in R is the modified summary() function. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. >> Fortunately, the calculation of robust standard errors can help to mitigate this problem. If the errors are homoscedastic, Heteroskedastic consistent errors are biased. << /Length 6 0 R /Filter /FlateDecode >> In the presence of heteroskedasticity, the errors are not IID. MathJax reference. We illustrate Actually, I have run such a regression and detected heteroskedasticity. I was wondering if, when running a regression on panel data, clustered standard errors are already correcting for heteroskedasticity. Clustered standard errors are popular and very easy to compute in some popular packages such as Stata, but how to compute them in R? Jusha Angrist and Jorn Pischke have a nice discussion around that topic in the book Mostly Harmless Econometrics (Chapter 8), Clustered standard errors and robust standard errors, A Practitioner's Guide to Cluster-Robust Inference, “Question closed” notifications experiment results and graduation, MAINTENANCE WARNING: Possible downtime early morning Dec 2, 4, and 9 UTC…, Why don't my cluster-robust (panel-robust) standard errors match those in Stata? Consequently, if the standard errors of the elements of b are computed in the usual way, they will inconsistent estimators of the true standard deviations of the elements of b. This video explains how to calculate heteroskedasticity-robust standard errors in Stata. Estimating robust standard errors in Stata 4.0 resulted in ... Clustered data . If the answer to both is no, one should not adjust the standard errors for clustering, irrespective of whether such an adjustment would change the standard errors. where the elements of S are the squared residuals from the OLS method. Introduction to Robust and Clustered Standard Errors Miguel Sarzosa Department of Economics University of Maryland Econ626: Empirical Microeconomics, 2012. Running a robust regression in Stata 4.0 results in . To subscribe to this RSS feed, copy and paste this URL into your RSS reader. As Wooldridge notes, the heteroskedasticity robust standard errors for this specification are not very different from the non-robust forms, and the test statistics for statistical significance of coefficients are generally unchanged. x}��n1��y Answering you question: Cluster Robust is also Heteroskedastic Consistent. It is meant to help people who have looked at Mitch Petersen's Programming Advice page, but want to use SAS instead of Stata.. Mitch has posted results using a test data set that you can use to compare the output below to see how well they agree. With panel data it's generally wise to cluster on the dimension of the individual effect as both heteroskedasticity and autocorrellation are almost certain to exist in the residuals at the individual level. How can a company reduce my number of shares? 3. 323 June 2006 JEL No. They allow for heteroskedasticity and autocorrelated errors within an entity but not correlation across entities. u�s����E|����p Or it is also known as the sandwich estimator of variance (because of how the calculation formula looks like). I was wondering if, when running a regression on panel data, clustered standard errors are already correcting for heteroskedasticity. mechanism is clustered. ��}��}��X�A��KJ���:��%�Sh�T#ѷ�й��q�0M]>���`/���u̲�u1���/K�{e/��Y.=/�YsR�J�rcQ�R�M��An,��ϻu�e�� Since I used the pooled OLS model I have to cluster the standard errors anyway. Clustering of Errors Cluster-Robust Standard Errors More Dimensions A Seemingly Unrelated Topic Estimates and their VCV Note that the heteroskedasticity-robust and cluster-robust estimators for standard errors have no impact whatsoever on point estimates. How to get ANOVA table with robust standard errors? I am confused to why the OLS estimated coeffecients (column 1) is the same as those from clustering the standard errors on both time and firm (column 9). I thought, that by clustering on these two dimensions, I would be able to remove serial correlation and heteroskedasticity and as such, the coeffecients would be different from those of OLS? HETEROSKEDASTICITY-ROBUST STANDARD ERRORS 159 (T t=1 X˜ itu it) (the second equality arises from the idempotent matrix identity). << /Length 14 0 R /N 1 /Alternate /DeviceGray /Filter /FlateDecode >> x(ٳ)�H������?K����"D��U �L� How to draw a seven point star with one path in Adobe Illustrator. How to explain the LCM algorithm to an 11 year old? Robust and clustered standard errors. Why is frequency not measured in db in bode's plot? You should use clustered standard errors if there are several different co-variance structures in your data. To learn more, see our tips on writing great answers. An Introduction to Robust and Clustered Standard Errors Linear Regression with Non-constant Variance Review: Errors and Residuals ... heteroskedasticity. Even in the second case, Abadie et al. I think so, yes, but you might want to provide more detail on how you're handling the clustering. Do I get the heteroskedasticity-robust standard errors from my OLS or WLS regression? I use an IV approach with time, country, and industry fixed effects. JakubMućk SGHWarsawSchoolofEconomics Jakub MućkAdvanced Applied Econometrics Heteroskedasticity and serial correlation 1 / 45 << /Type /Page /Parent 4 0 R /Resources 7 0 R /Contents 5 0 R /MediaBox [0 0 362.835 272.126] So, similar to heteroskedasticity-robust standard errors, you want to allow more flexibility in your variance-covariance (VCV) matrix (Recall that the diagonal elements of the VCV matrix are the squared standard errors of your estimated coefficients). By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. Use MathJax to format equations. Computing cluster -robust standard errors is a fix for the latter issue. This page shows how to run regressions with fixed effect or clustered standard errors, or Fama-Macbeth regressions in SAS. We illustrate Consequentially, it is inappropriate to use the average squared residuals. Standard errors based on this procedure are called (heteroskedasticity) robust standard errors or White-Huber standard errors. upward-biased. Many blog articles have demonstrated clustered standard errors, in R, either by writing a function or manually adjusting the degrees of freedom or both (example, example, example and example).These methods give close approximations to the standard Stata results, but they do not do the small sample correction as the Stata does. Where did the concept of a (fantasy-style) "dungeon" originate? option, that allows the computation of so-called Rogers or clustered standard errors.2 Another approach to obtain heteroskedasticity- and autocorrelation (up to some lag)-consistent standard errors was developed by Newey and West (1987). Heteroskedasticity and serial correlation. I would recommend that you read the A Practitioner's Guide to Cluster-Robust Inference which is a nice piece from Colin Cameron on several aspects of clustered/heteroskedastic robust errors. Since I used the pooled OLS model I have to cluster the standard errors anyway. endobj Robust standard errors account for heteroskedasticity in a model’s unexplained variation. Of course, you do not need to use matrix to obtain robust standard errors. The following post describes how to use this function to compute clustered standard errors in R: You can refer to Zeileis (2004) for more details. note that both the usual robust (Eicker-Huber-White or EHW) standard errors, and the clustered standard errors (which they call Liang-Zeger or LZ standard errors) can both be correct, it is just that they are correct for different estimands. site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa.


text:
# How to manipulate this functions to an identity involving the Riemann zeta function

The identity I want to prove is the following (from Stein's book, an introduction to Fourier analysis):

$$\pi^{-s/2} \Gamma(s/2) \zeta (s)=\frac{1}{2} \int_{0}^{\infty}t^{\frac{s}{2}-1}(v(t)-1)dt$$

for $s>1$, where $v(t)$ is the theta function and $\Gamma(s)$ is the gamma function.

So when I manipulate the LHS of the equation I get the following:

$$\int_{0}^{\infty}e^{-t}t^{\frac{s}{2}-1}(\sum_{n=1}^{\infty}(\frac{1}{\pi^{-s/2}n^s}))dt$$

The thing is What can I do next, and How do I am going to get the negative part of the theta function?.

Can someone help me to prove this identity please?, Thanks a lot in advance :)

Theta function:

$$\nu(s)=\sum_{n=-\infty}^{\infty}e^{-\pi n^{2}s}$$

• Is the one that goes with the sum of the exponential to the -$\pi (n^{2})s$ from -infinity to infinity – user162343 Aug 25 '15 at 13:58

To start with let's write (following the wikipedia convention) $$\nu(t):=\vartheta(0,it)=\sum_{n=-\infty}^{\infty}e^{-\pi n^2 t }=\sum_{n=-\infty}^{-1}e^{-\pi n^2 t }+1+\sum_{n=1}^{\infty}e^{-\pi n^2 t }=2\sum_{n=1}^{\infty}e^{-\pi n^2 t }+1$$
Therefore: $$I=\frac{1}{2}\int_{0}^{\infty}dt \left(\nu(t)-1\right)t^{s/2-1}=\sum_{n=1}^\infty\int_{0}^{\infty}dt e^{-\pi n^2 t }t^{s/2-1}$$
The integral is easily be calculated in terms of $\Gamma$-functions (substituting $\pi n^2 t \rightarrow x$ and use the defintion of $\Gamma$ )
$$I=\pi^{-s/2}\Gamma(s/2)\sum_{n=1}^\infty\frac{1}{n^s}=\pi^{-s/2}\Gamma(s/2)\zeta(s)$$ Q.E.D.

format: plain


text:
Recent Submissions

• Grupos de tranças de superfícies finitamente perfuradas e grupos cristalográficos

(Universidade Federal de São Carlos, UFSCar, Programa de Pós-Graduação em Matemática - PPGM, Câmpus São Carlos, 07/12/2020)
The link between braid groups on surfaces and crystallographic groups has become such an interesting topic. In the last years some advances were found in the studies of this relation, specially in the case of Artin braid ...
• Boa postura analítica e Gevrey da “boa” equação de Boussinesq

(Universidade Federal de São Carlos, UFSCar, Programa de Pós-Graduação em Matemática - PPGM, Câmpus São Carlos, 11/03/2021)
In both the line and the circle, we shall to prove that the Cauchy problem for the good'' Boussinesq equation is locally well-posed in a class of Gevrey functions, which includes a class of analytic functions that can ...
• Existência de pullback atrator exponencial para uma família de problemas dominados por uma perturbação do p(x)-Laplaciano com difusão grande localizada, via método das l-trajetórias

(Universidade Federal de São Carlos, UFSCar, Programa de Pós-Graduação em Matemática - PPGM, Câmpus São Carlos, 22/12/2020)
In this work, we use the method of l-trajectories to prove the existence of a family of exponential attractors for autonomous nonlinear reaction-diffusion equations, involving the p(x)-Laplacian with large localized ...
• Sobre fibrados e poliedros de dimensão finita

(Universidade Federal de São Carlos, UFSCar, Programa de Pós-Graduação em Matemática - PPGM, Câmpus São Carlos, 01/02/2021)
The objective of this work is to establish the existence of universal bundles and to relate this existence to finite dimension polyhedra. As a consequence, we present examples of universal bundles for the case of classic groups.
• Teoremas ergódicos para sistemas multívocos e aleatórios com aplicações a semifluxos generalizados

(Universidade Federal de São Carlos, UFSCar, Programa de Pós-Graduação em Matemática - PPGM, Câmpus São Carlos, 05/02/2021)
This work focuses on four major themes: semiflows; generalized semiflows; invariant measure for set-valued maps; and random dynamical systems. The study of semiflows aims to understand the asymptotic behavior of solutions ...
• Sobre teoremas do tipo Tverberg

(Universidade Federal de São Carlos, UFSCar, Programa de Pós-Graduação em Matemática - PPGM, Câmpus São Carlos, 17/12/2020)
The main objective of this work is to prove Tverberg-type theorems and related results. More specifically, we will present an alternative proof for the Bárány-Larman conjecture for the case $r = p-1$ with $p$ a prime, ...
• Non-autonomous Klein-Gordon-Zakharov system: pullback dynamics in the continuous and impulsive approaches

(Universidade Federal de São Carlos, UFSCar, Programa de Pós-Graduação em Matemática - PPGM, Câmpus São Carlos, 22/12/2020)
This work is dedicated to study a non-autonomous formulation of the Klein-Gordon-Zakharov system, which is a coupled system consisting of two non-autonomous evolution equations, where each one is of second order in time. ...
• Involuções fixando RP(6)URP(2n) e variedades compatíveis com o ponto com respeito à involuções

(Universidade Federal de São Carlos, UFSCar, Programa de Pós-Graduação em Matemática - PPGM, Câmpus São Carlos, 25/11/2020)
In this work, we have two objectives: the first lives in the context of the classification, up to equivariant cobordism, of the pairs (M, T) , where M is a closed and smooth manifold and T is a smooth involution ...
• Identidades e polinômios centrais com involução para a álgebra de Grassmann e álgebra das matrizes triangulares de ordem 3

(Universidade Federal de São Carlos, UFSCar, Programa de Pós-Graduação em Matemática - PPGM, Câmpus São Carlos, 30/11/2020)
Let F be an infinite field of characteristic different from 2. In this thesis we will study the polynomial identities and central polynomials with involution for two important classes of algebras. More precisely, we describe ...
• Três resultados de sistemas dinâmicos em dimensões baixas

(Universidade Federal de São Carlos, UFSCar, Programa de Pós-Graduação em Matemática - PPGM, Câmpus São Carlos, 18/12/2020)
This work contains three independent results that can be framed within the research area named low-dimensional dynamics. The first result is a lower bound for the maximal number of inseparable leaves of non-singular ...
• Boa postura para alguns sistemas de equações dispersivas e para uma equação dispersiva

(Universidade Federal de São Carlos, UFSCar, Programa de Pós-Graduação em Matemática - PPGM, Câmpus São Carlos, 30/10/2020)
The main aim of this work is to establish the well posedness for a dispersive partial differential equations systems and for a partial differential equation, with initial data belonging to Gevrey space. The proof ...


text:
goodness of fit? how to test if a set of simulated data distributions are well-fitted to an empirical data distribution in R or Python?

I have a data set, whose elements (say 1,000 elements) are numeric with values between 0 and 1. It is safe to think the elements are independent from each other, i.e, the order of these elements does not matter at all. Also, from the histogram of the data set, I can tell it does not follow any parametric distribution (normal or others).

I used a modeling tool (not in R) to simulate 100 realizations of this data set given a set of parameters. In each realization, the number of elements is not necessarily equal to 1,000.

My goal is simple: Does the model fit the data well?

I plotted the histograms of the simulated data against the empirical data one (using both counts and probability density) and was sort of able to tell the quality of the fit, but would like to have a solid statistical way to quantitatively reach my goal (say showing the p-value of this fit is > 10% or so).

Is there a good way in R or Python I can do so? I looked up the standard R fitting methods, such as lm() and glm() in R, but they are for fitting a line to the data, not doing something I would like to have. I believe that this is simply because I am not familiar with R and statistics given this question does not seem to be weird and difficult.

Thanks, Ben

• ?ks.test in R (Kolmogorov-Smirnov) – Ben Bolker Feb 24 '12 at 20:39
• Your question seems to contradict itself because you say your original data does not follow a parametric distribution, but then you go on to say somehow you have simulated similar data using a set of parameters. Probably what you mean is it does not follow any simple parametric distribution, but perhaps it can be seen as a mixture eg of two or three different beta distributions? – Peter Ellis Feb 25 '12 at 1:09
• @Peter.. My so-called simulation data were obtained from a partial differentiatial equation based modeling approach based on some a priori knowledge (parameters). My goal is not to estimate these parameters, but to tell if the model fits the data well (quantitatively). I have no idea what the true distribution of the data (either empirical or simulation) would be. – Ben Feb 25 '12 at 3:16

You can use the qqplot function like @Peter Flom suggests as one way to compare visually. Another visual approach is to create a set of graphs where one is the real data and the others are simulated data, but you don't know which is the real data (or a sample of the real data), then see if you can pick out the real data. The vis.test function in the TeachingDemos package for R helps with this. This gives more of a close enough test than an exact match test.

The ks.test function in R can take 2 vectors of numbers and tests the null hypothesis that they come from the same distribution. Note however that with large sample sizes this test can find significance for differences so small that most would not care about and for small sample sizes it may not have the power to differentiate between distributions that we would care about the difference of.

• @Greg... Thanks! I will take a look at the ks.test and vis.test. – Ben Feb 24 '12 at 21:56

Let me see if I understand what you've got: One set of "real data" with 1000 values and 100 sets of "model data" each with about 1000 values.

You could combine all 100 sets and then, in R, run something like qqplot(real, fake).

Or you could do 100 such plots (one for each set).

I don't think the concept of a p-value makes much sense here, as you don't have a null hypothesis that you are attempting to reject, or a test-statistic that you expect not to be large given a null.

• @Peter...I think my null hypothesis is that the real and model data come from the same (though unknown) distribution. A p-value (or any statistical evidence) can give a solid quantitative judgement on the fit, rather than simply rely on a graphically qualitative judgement. :) – Ben Feb 24 '12 at 21:46
• @Peter... most importantly, thanks for the tips on qqplot. surely will give it a try! – Ben Feb 24 '12 at 21:55
• Unfortunately, Ben, when you create the model in response to your review of the data, then--unless you are really bad at modeling--any p-values for goodness of fit you subsequently compute will be hugely over-optimistic. One way around this problem is to retain some of the original data for confirmatory testing later, but in this case that horse left the barn long ago... – whuber Feb 27 '12 at 16:46
• Ben if you want something "solid" or "quantitative" then define "well" in your original question in a solid, quantitative way. Even beyond the valid point that @whuber raised, there is no case when a p-value tells you if a model fits "well". A p-value tells you if a test statistic as extreme as that you get in a sample is likely if the null hypothesis is true in the population. There are very few cases where this question is of much interest, and this isn't one of them, even if you could define it. – Peter Flom Feb 27 '12 at 19:20

For example if you have data x<-c(1,2,3,5,4,6,9) and you have so many distributions to see which one is best. you can do it in R for example, I will consider first Poisson distribution.

dburr=function(obs, c =2.1000001) exp(-c)*c^x/factorial(x)
set.seed(45)
#obs <- c(1,1,0,0,0,8,0,5,0,0)
#obs<-c(rep(0,17),rep(1,9),rep(2,7),rep(3,3),4,5,6,6)
obs<-c(rep(0,64),rep(1,17),rep(2,10),rep(3,9))
#View(obs)
library(MASS)
jk=fitdistr(x = obs,densfun = dburr,start = list(c=0.001), # need to provide named list of starting values
lower = list(c=0.02)) # and named list of lower bounds since c, k > 0
jk$loglik jk$vcov
jk$estimate jk$sd
jk\$n
AIC(jk)
BIC(jk)

Similarly define other distribution density functions in the above code and calculate AIC, BIC, log likelihood....the distribution with minimum AIC, BIC will be good as compared to one having higher value for AIC, BIC.


text:
# mechanics help,

• December 4th 2008, 01:04 PM
Tweety
mechanics help,
A woman and a man support a non-uniform plank, AB of mass 10kg and length 4m. The woman holds the plank 1m from A and the man holds the plank at B. The vertical reaction provided by the woman is 56N find;

A) the vertical reaction force provided my the man

b) The position at which at which the weight of the plank acts measured from A.
• December 5th 2008, 05:28 AM
Tweety
is this moments equation correct...

$2g \times 1m + 3g (xm-3m) =0$ ?
this gives x=2.3m and my book says the answer is $\frac{1}{3}$


text:
## what does it do?

$\ln K = -\frac{\Delta H^{\circ}}{RT} + \frac{\Delta S^{\circ}}{R}$

connie 2C
Posts: 106
Joined: Thu Jul 11, 2019 12:17 am

### what does it do?

what is the difference between this equation and the other equations we know? what does it relate?

Samuel G Rivera - Discussion 4I
Posts: 61
Joined: Sat Aug 17, 2019 12:16 am
Been upvoted: 1 time

### Re: what does it do?

The Van't Hoff equation helps us find how K (the equilibrium constant) changes when temperature changes.

(P.S. I was confused about what equation you were referring to in your post. Even though it was in the Van't Hoff equation section, I recommend being specific in the posts just to avoid confusion)

Chris Tai 1B
Posts: 102
Joined: Sat Aug 24, 2019 12:16 am

### Re: what does it do?

When using the Van't Hoff equation, we assume that the difference in enthalpy is considered to be the same, and we assume that the difference in entropy between reactants and products is also the same at these two different temperatures. The reason this is valuable is because it enables us to show the dependence of temperature on the equilibrium constant, K; it's an indirect way to measure enthalpy as well. If the deltaH is known, Van't Hoff can be used to calculate K at a different temperature.

Alison Trinh 1E
Posts: 104
Joined: Thu Jul 25, 2019 12:16 am

### Re: what does it do?

It helps us calculate the K at different temperatures (if we know the standard DeltaH)

Noe BM 1J
Posts: 50
Joined: Sat Sep 07, 2019 12:17 am

### Re: what does it do?

Could you elaborate a little more on how it's an indirect way to measure enthalpy, please?

Chris Tai 1B wrote:When using the Van't Hoff equation, we assume that the difference in enthalpy is considered to be the same, and we assume that the difference in entropy between reactants and products is also the same at these two different temperatures. The reason this is valuable is because it enables us to show the dependence of temperature on the equilibrium constant, K; it's an indirect way to measure enthalpy as well. If the deltaH is known, Van't Hoff can be used to calculate K at a different temperature.

Trent Yamamoto 2J
Posts: 111
Joined: Sat Aug 17, 2019 12:18 am

### Re: what does it do?

The Van't Hoff Equation helps relate K to the different values of T, since T affects K.

Chris Tai 1B
Posts: 102
Joined: Sat Aug 24, 2019 12:16 am

### Re: what does it do?

Noe BM 1J wrote:Could you elaborate a little more on how it's an indirect way to measure enthalpy, please?

Chris Tai 1B wrote:When using the Van't Hoff equation, we assume that the difference in enthalpy is considered to be the same, and we assume that the difference in entropy between reactants and products is also the same at these two different temperatures. The reason this is valuable is because it enables us to show the dependence of temperature on the equilibrium constant, K; it's an indirect way to measure enthalpy as well. If the deltaH is known, Van't Hoff can be used to calculate K at a different temperature.

The reason we can use the VH equation to indirectly measure enthalpy is because of the presence of enthalpy in the equation itself. The equation is as follows:
ln(K2/K1) = deltaH/R * (1/T1 - 1/T2). If we know two different equilibrium constants at two different temperatures, then we're able to calculate the enthalpy in a more indirect manner (as opposed to the typical method of calculating enthalpy by subtracting enthalpies of formation of the reactants from the products, or doing the same with bond enthalpies etc.)

Mitchell Koss 4G
Posts: 128
Joined: Sat Jul 20, 2019 12:17 am

### Re: what does it do?

The equ can also be used/written in several forms

Noe BM 1J
Posts: 50
Joined: Sat Sep 07, 2019 12:17 am

### Re: what does it do?

Thank you! I see how it is related and it's definitely useful.

Chris Tai 1B wrote:
Noe BM 1J wrote:Could you elaborate a little more on how it's an indirect way to measure enthalpy, please?

Chris Tai 1B wrote:When using the Van't Hoff equation, we assume that the difference in enthalpy is considered to be the same, and we assume that the difference in entropy between reactants and products is also the same at these two different temperatures. The reason this is valuable is because it enables us to show the dependence of temperature on the equilibrium constant, K; it's an indirect way to measure enthalpy as well. If the deltaH is known, Van't Hoff can be used to calculate K at a different temperature.

The reason we can use the VH equation to indirectly measure enthalpy is because of the presence of enthalpy in the equation itself. The equation is as follows:
ln(K2/K1) = deltaH/R * (1/T1 - 1/T2). If we know two different equilibrium constants at two different temperatures, then we're able to calculate the enthalpy in a more indirect manner (as opposed to the typical method of calculating enthalpy by subtracting enthalpies of formation of the reactants from the products, or doing the same with bond enthalpies etc.)

Veronica Lu 2H
Posts: 89
Joined: Wed Sep 18, 2019 12:18 am
Been upvoted: 1 time

### Re: what does it do?

it helps you find the K when there are changes in temperature

Jacob Villar 2C
Posts: 105
Joined: Sat Aug 17, 2019 12:18 am

### Re: what does it do?

The Van’t Hoff equation can be used to calculate K when the reaction undergoes temperature changes.

Rosa Munoz 2E
Posts: 105
Joined: Wed Sep 18, 2019 12:21 am

### Re: what does it do?

Allows you to find K at different tempratures

Philip
Posts: 100
Joined: Sat Sep 07, 2019 12:16 am

### Re: what does it do?

Van't Hoff Equation is used to calculate the new equilibrium constant based on temperature change

Ayushi2011
Posts: 101
Joined: Wed Feb 27, 2019 12:17 am

### Re: what does it do?

It relates equilibrium constant to temperature.


text:
# Math Help - Removable or Non Removable?

1. ## Removable or Non Removable?

"Find the x-values(if any) at which f is not continuous. Which of the discontinuites are removable?"

the funtion is f(x)=x-3/x^2-9

Ok so I know its discontinuous at 3. But is it removable or non removable? and why? Do you need to know the graph to determine this? Thanks.

2. Notice the difference of two squares in the denominator?.

$\frac{x-3}{(x-3)(x+3)}$

Now, see it?. You got a hole at x=3

3. would that be removable or non? and why? Thanks.

4. It is removeable.
$f(x) = \left\{ {\begin{array}{lr}
{\frac{{x - 3}}{{x^2 - 9}}} & {x \ne 3} \\
{\frac{1}{6}} & {x = 3} \\
\end{array}} \right.

$

5. Thanks


text:
## 10.41 Risk Variable

20180723 With some knowledge of the data we observe risk_mm captures the amount of rain recorded tomorrow. We refer to this as a risk variable, being a measure of the impact or risk of the target we are predicting (rain tomorrow). The risk is an output variable and should not be used as an input to the modelling—it is not an independent variable. In other circumstances it might actually be treated as the target variable.

# Note the risk variable - measures the severity of the outcome.

risk <- "risk_mm"

For this risk variable note that we expect it to have a value of 0 for all observations when the target variable has the value No.

# Review the distribution of the risk variable for non-targets.

ds %>%
filter(rain_tomorrow == "No") %>%
select(risk_mm) %>%
summary()
##     risk_mm
##  Min.   :0.00000
##  1st Qu.:0.00000
##  Median :0.00000
##  Mean   :0.07302
##  3rd Qu.:0.00000
##  Max.   :1.00000

Interestingly, even a little rain (defined as 1mm or less) is regarded as no rain. That is useful to keep in mind and is a discovery of the data that we might not have expected. As data scientists we should be expecting to find the unexpected.

A similar analysis for the target observations is more in line with expectations.

# Review the distribution of the risk variable for targets.

ds %>%
filter(rain_tomorrow == "Yes") %>%
select(risk_mm) %>%
summary()
##     risk_mm
##  Min.   :  1.10
##  1st Qu.:  2.40
##  Median :  5.20
##  Mean   : 10.16
##  3rd Qu.: 11.60
##  Max.   :474.00

Your donation will support ongoing development and give you access to the PDF version of this book. Desktop Survival Guides include Data Science, GNU/Linux, and MLHub. Books available on Amazon include Data Mining with Rattle and Essentials of Data Science. Popular open source software includes rattle, wajig, and mlhub. Hosted by Togaware, a pioneer of free and open source software since 1984.


text:
m
• E

F Nous contacter

0

# Documents  Crovisier, Sylvain | enregistrements trouvés : 41

O

P Q

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

## Perturbative techniques of the dynamics in the $C^1$-topology Crovisier, Sylvain | CIRM H

Post-edited

Research talks;Dynamical Systems and Ordinary Differential Equations

These lectures will address the dynamics of vector fields or diffeomorphisms of compact manifolds. For the study of generic properties or for the construction of examples, it is often useful to be able to perturb a system. This generally leads to delicate problems: a local modification of the dynamics may cause a radical change in the behavior of the orbits. For the $C^1$-topology, various techniques have been developed which allow to perturb while controlling the dynamics: closing and connection of orbits, perturbation of the tangent dynamics... We derive various applications to the description of $C^1$-generic diffeomorphisms. These lectures will address the dynamics of vector fields or diffeomorphisms of compact manifolds. For the study of generic properties or for the construction of examples, it is often useful to be able to perturb a system. This generally leads to delicate problems: a local modification of the dynamics may cause a radical change in the behavior of the orbits. For the $C^1$-topology, various techniques have been developed which allow to perturb ...

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

## Singular hyperbolicity and homoclinic tangencies of 3-dimensional flows Crovisier, Sylvain | CIRM H

Post-edited

Research talks;Dynamical Systems and Ordinary Differential Equations

The notion of singular hyperbolicity for vector fields has been introduced by Morales, Pacifico and Pujals in order to extend the classical uniform hyperbolicity and include the presence of singularities. This covers the Lorenz attractor. I will present a joint work with Dawei Yang which proves a dichotomy in the space of three-dimensional $C^{1}$-vector fields, conjectured by J. Palis: every three-dimensional vector field can be $C^{1}$-approximated by one which is singular hyperbolic or by one which exhibits a homoclinic tangency. The notion of singular hyperbolicity for vector fields has been introduced by Morales, Pacifico and Pujals in order to extend the classical uniform hyperbolicity and include the presence of singularities. This covers the Lorenz attractor. I will present a joint work with Dawei Yang which proves a dichotomy in the space of three-dimensional $C^{1}$-vector fields, conjectured by J. Palis: every three-dimensional vector field can be \$C^{1}...

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

## Additive combinatorics methods in fractal geometry - lecture 1 Shmerkin, Pablo | CIRM H

Multi angle

Research schools

In the last few years ideas from additive combinatorics were applied to problems in fractal geometry and led to progress on some classical problems, particularly on the smoothness of Bernoulli convolutions and other self-similar measures. We will introduce some of these tools from additive combinatorics and present some of the main applications, including the smoothness of Bernoulli convolutions outside of a small set of exceptions, and for explicit values of the parameter. In the last few years ideas from additive combinatorics were applied to problems in fractal geometry and led to progress on some classical problems, particularly on the smoothness of Bernoulli convolutions and other self-similar measures. We will introduce some of these tools from additive combinatorics and present some of the main applications, including the smoothness of Bernoulli convolutions outside of a small set of exceptions, and for ...

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

## Additive combinatorics methods in fractal geometry - lecture 1 Varju, Peter | CIRM H

Multi angle

Research schools

In the last few years ideas from additive combinatorics were applied to problems in fractal geometry and led to progress on some classical problems, particularly on the smoothness of Bernoulli convolutions and other self-similar measures. We will introduce some of these tools from additive combinatorics and present some of the main applications, including the smoothness of Bernoulli convolutions outside of a small set of exceptions, and for explicit values of the parameter. In the last few years ideas from additive combinatorics were applied to problems in fractal geometry and led to progress on some classical problems, particularly on the smoothness of Bernoulli convolutions and other self-similar measures. We will introduce some of these tools from additive combinatorics and present some of the main applications, including the smoothness of Bernoulli convolutions outside of a small set of exceptions, and for ...

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

## Additive combinatorics methods in fractal geometry - lecture 2 Shmerkin, Pablo | CIRM H

Multi angle

Research schools

In the last few years ideas from additive combinatorics were applied to problems in fractal geometry and led to progress on some classical problems, particularly on the smoothness of Bernoulli convolutions and other self-similar measures. We will introduce some of these tools from additive combinatorics and present some of the main applications, including the smoothness of Bernoulli convolutions outside of a small set of exceptions, and for explicit values of the parameter. In the last few years ideas from additive combinatorics were applied to problems in fractal geometry and led to progress on some classical problems, particularly on the smoothness of Bernoulli convolutions and other self-similar measures. We will introduce some of these tools from additive combinatorics and present some of the main applications, including the smoothness of Bernoulli convolutions outside of a small set of exceptions, and for ...

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

## Additive combinatorics methods in fractal geometry - lecture 2 Varju, Peter | CIRM H

Multi angle

Research schools

In the last few years ideas from additive combinatorics were applied to problems in fractal geometry and led to progress on some classical problems, particularly on the smoothness of Bernoulli convolutions and other self-similar measures. We will introduce some of these tools from additive combinatorics and present some of the main applications, including the smoothness of Bernoulli convolutions outside of a small set of exceptions, and for explicit values of the parameter. In the last few years ideas from additive combinatorics were applied to problems in fractal geometry and led to progress on some classical problems, particularly on the smoothness of Bernoulli convolutions and other self-similar measures. We will introduce some of these tools from additive combinatorics and present some of the main applications, including the smoothness of Bernoulli convolutions outside of a small set of exceptions, and for ...

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

## Additive combinatorics methods in fractal geometry - lecture 3 Shmerkin, Pablo | CIRM H

Multi angle

Research schools

In the last few years ideas from additive combinatorics were applied to problems in fractal geometry and led to progress on some classical problems, particularly on the smoothness of Bernoulli convolutions and other self-similar measures. We will introduce some of these tools from additive combinatorics and present some of the main applications, including the smoothness of Bernoulli convolutions outside of a small set of exceptions, and for explicit values of the parameter. In the last few years ideas from additive combinatorics were applied to problems in fractal geometry and led to progress on some classical problems, particularly on the smoothness of Bernoulli convolutions and other self-similar measures. We will introduce some of these tools from additive combinatorics and present some of the main applications, including the smoothness of Bernoulli convolutions outside of a small set of exceptions, and for ...

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

## Additive combinatorics methods in fractal geometry - lecture 3 Varju, Peter | CIRM H

Multi angle

Research schools

In the last few years ideas from additive combinatorics were applied to problems in fractal geometry and led to progress on some classical problems, particularly on the smoothness of Bernoulli convolutions and other self-similar measures. We will introduce some of these tools from additive combinatorics and present some of the main applications, including the smoothness of Bernoulli convolutions outside of a small set of exceptions, and for explicit values of the parameter. In the last few years ideas from additive combinatorics were applied to problems in fractal geometry and led to progress on some classical problems, particularly on the smoothness of Bernoulli convolutions and other self-similar measures. We will introduce some of these tools from additive combinatorics and present some of the main applications, including the smoothness of Bernoulli convolutions outside of a small set of exceptions, and for ...

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

## Beyond Bowen specification property - lecture 1 Climenhaga, Vaughn | CIRM H

Multi angle

Research schools

Rufus Bowen introduced the specification property for uniformly hyperbolic dynamical systems and used it to establish uniqueness of equilibrium states, including the measure of maximal entropy. After reviewing Bowen's argument, we will present our recent work on extending Bowen's approach to non-uniformly hyperbolic systems. We will describe the general result, which makes precise the notion of "entropy (orpressure) of obstructions to specification" using a decomposition of the space of finite-length orbit segments, and then survey various applications, including factors of beta-shifts, derived-from-Anosov diffeomorphisms, and geodesic flows in non-positive curvature and beyond. Rufus Bowen introduced the specification property for uniformly hyperbolic dynamical systems and used it to establish uniqueness of equilibrium states, including the measure of maximal entropy. After reviewing Bowen's argument, we will present our recent work on extending Bowen's approach to non-uniformly hyperbolic systems. We will describe the general result, which makes precise the notion of "entropy (orpressure) of obstructions to s...

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

## Beyond Bowen specification property - lecture 1 Thompson, Daniel J. | CIRM H

Multi angle

Research schools

These lectures are a mostly self-contained sequel to Vaughn Climenhaga’s talks in week 1. The focus of the week 2 lectures will be on uniqueness of equilibrium states for rank 1 geodesic flows, and their mixing properties. Burns, Climenhaga, Fisher and myself showed recently that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. I will discuss the proof of this result. With this result in hand, the question of when the “pressure gap” hypothesis can be verified becomes crucial. I will sketch our proof of the “entropy gap”, which is a new direct constructive proof of a result by Knieper. I will also describe new joint work with Ben Call, which shows that all the unique equilibrium states provided above have the Kolmogorov property. When the manifold has dimension at least 3, this is a new result even for the Knieper-Bowen-Margulis measure of maximal entropy. The common thread that links all of these arguments is that they rely on weak orbit specification properties in the spirit of Bowen. These lectures are a mostly self-contained sequel to Vaughn Climenhaga’s talks in week 1. The focus of the week 2 lectures will be on uniqueness of equilibrium states for rank 1 geodesic flows, and their mixing properties. Burns, Climenhaga, Fisher and myself showed recently that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. I will discuss the proof of this result. With this result in hand, ...

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

## Beyond Bowen specification property - lecture 2 Climenhaga, Vaughn | CIRM H

Multi angle

Research schools

Rufus Bowen introduced the specification property for uniformly hyperbolic dynamical systems and used it to establish uniqueness of equilibrium states, including the measure of maximal entropy. After reviewing Bowen's argument, we will present our recent work on extending Bowen's approach to non-uniformly hyperbolic systems. We will describe the general result, which makes precise the notion of "entropy (orpressure) of obstructions to specification" using a decomposition of the space of finite-length orbit segments, and then survey various applications, including factors of beta-shifts, derived-from-Anosov diffeomorphisms, and geodesic flows in non-positive curvature and beyond. Rufus Bowen introduced the specification property for uniformly hyperbolic dynamical systems and used it to establish uniqueness of equilibrium states, including the measure of maximal entropy. After reviewing Bowen's argument, we will present our recent work on extending Bowen's approach to non-uniformly hyperbolic systems. We will describe the general result, which makes precise the notion of "entropy (orpressure) of obstructions to s...

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

## Beyond Bowen specification property - lecture 2 Thompson, Daniel J. | CIRM H

Multi angle

Research schools

These lectures are a mostly self-contained sequel to Vaughn Climenhaga’s talks in week 1. The focus of the week 2 lectures will be on uniqueness of equilibrium states for rank 1 geodesic flows, and their mixing properties. Burns, Climenhaga, Fisher and myself showed recently that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. I will discuss the proof of this result. With this result in hand, the question of when the “pressure gap” hypothesis can be verified becomes crucial. I will sketch our proof of the “entropy gap”, which is a new direct constructive proof of a result by Knieper. I will also describe new joint work with Ben Call, which shows that all the unique equilibrium states provided above have the Kolmogorov property. When the manifold has dimension at least 3, this is a new result even for the Knieper-Bowen-Margulis measure of maximal entropy. The common thread that links all of these arguments is that they rely on weak orbit specification properties in the spirit of Bowen. These lectures are a mostly self-contained sequel to Vaughn Climenhaga’s talks in week 1. The focus of the week 2 lectures will be on uniqueness of equilibrium states for rank 1 geodesic flows, and their mixing properties. Burns, Climenhaga, Fisher and myself showed recently that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. I will discuss the proof of this result. With this result in hand, ...

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

## Beyond Bowen specification property - lecture 3 Climenhaga, Vaughn | CIRM H

Multi angle

Research schools

Rufus Bowen introduced the specification property for uniformly hyperbolic dynamical systems and used it to establish uniqueness of equilibrium states, including the measure of maximal entropy. After reviewing Bowen's argument, we will present our recent work on extending Bowen's approach to non-uniformly hyperbolic systems. We will describe the general result, which makes precise the notion of "entropy (orpressure) of obstructions to specification" using a decomposition of the space of finite-length orbit segments, and then survey various applications, including factors of beta-shifts, derived-from-Anosov diffeomorphisms, and geodesic flows in non-positive curvature and beyond. Rufus Bowen introduced the specification property for uniformly hyperbolic dynamical systems and used it to establish uniqueness of equilibrium states, including the measure of maximal entropy. After reviewing Bowen's argument, we will present our recent work on extending Bowen's approach to non-uniformly hyperbolic systems. We will describe the general result, which makes precise the notion of "entropy (orpressure) of obstructions to s...

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

## Beyond Bowen specification property - lecture 3 Thompson, Daniel J. | CIRM H

Multi angle

Research schools

These lectures are a mostly self-contained sequel to Vaughn Climenhaga’s talks in week 1. The focus of the week 2 lectures will be on uniqueness of equilibrium states for rank 1 geodesic flows, and their mixing properties. Burns, Climenhaga, Fisher and myself showed recently that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. I will discuss the proof of this result. With this result in hand, the question of when the “pressure gap” hypothesis can be verified becomes crucial. I will sketch our proof of the “entropy gap”, which is a new direct constructive proof of a result by Knieper. I will also describe new joint work with Ben Call, which shows that all the unique equilibrium states provided above have the Kolmogorov property. When the manifold has dimension at least 3, this is a new result even for the Knieper-Bowen-Margulis measure of maximal entropy. The common thread that links all of these arguments is that they rely on weak orbit specification properties in the spirit of Bowen. These lectures are a mostly self-contained sequel to Vaughn Climenhaga’s talks in week 1. The focus of the week 2 lectures will be on uniqueness of equilibrium states for rank 1 geodesic flows, and their mixing properties. Burns, Climenhaga, Fisher and myself showed recently that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. I will discuss the proof of this result. With this result in hand, ...

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

## Big mapping class groups - lecture 1 Calegari, Danny | CIRM H

Multi angle

Research schools

Part I - Theory : In the "theory" part of this mini-course, we will present recent objects and phenomena related to the study of big mapping class groups. In particular, we will define two faithful actions of some big mapping class groups. The first is an action by isometries on a Gromov-hyperbolic graph. The second is an action by homeomorphisms on a circle in which the vertices of the graph naturally embed. We will describe some properties of the objects involved, and give some fruitful relations between the dynamics of the two actions. For example, we will see that loxodromic elements (for the first action) necessarily have rational rotation number (for the second action). Using these relations, we will explain how to construct non trivial quasimorphisms on subgroups of big mapping class groups. This includes joint work with Alden Walker.
Part II - Examples : In this part we will discuss a number of natural examples in which big mapping class groups and their subgroups arise. These include the inverse limit constructions of de Carvalho-Hall, the theory of finite depth (taut) foliations of 3-manifolds, the theory of “Artinization” of Thompson-like groups, two dimensional smooth dynamics, one dimensional complex dynamics (topology of the shift locus, Schottky spaces) and several other contexts. We will try to indicate how viewing these examples from the perspective of (big) mapping class groups is a worthwhile approach.
Part I - Theory : In the "theory" part of this mini-course, we will present recent objects and phenomena related to the study of big mapping class groups. In particular, we will define two faithful actions of some big mapping class groups. The first is an action by isometries on a Gromov-hyperbolic graph. The second is an action by homeomorphisms on a circle in which the vertices of the graph naturally embed. We will describe some properties of ...

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

## Big mapping class groups - lecture 2 Calegari, Danny | CIRM H

Multi angle

Research schools

Part I - Theory : In the "theory" part of this mini-course, we will present recent objects and phenomena related to the study of big mapping class groups. In particular, we will define two faithful actions of some big mapping class groups. The first is an action by isometries on a Gromov-hyperbolic graph. The second is an action by homeomorphisms on a circle in which the vertices of the graph naturally embed. We will describe some properties of the objects involved, and give some fruitful relations between the dynamics of the two actions. For example, we will see that loxodromic elements (for the first action) necessarily have rational rotation number (for the second action). Using these relations, we will explain how to construct non trivial quasimorphisms on subgroups of big mapping class groups. This includes joint work with Alden Walker.
Part II - Examples : In this part we will discuss a number of natural examples in which big mapping class groups and their subgroups arise. These include the inverse limit constructions of de Carvalho-Hall, the theory of finite depth (taut) foliations of 3-manifolds, the theory of “Artinization” of Thompson-like groups, two dimensional smooth dynamics, one dimensional complex dynamics (topology of the shift locus, Schottky spaces) and several other contexts. We will try to indicate how viewing these examples from the perspective of (big) mapping class groups is a worthwhile approach.
Part I - Theory : In the "theory" part of this mini-course, we will present recent objects and phenomena related to the study of big mapping class groups. In particular, we will define two faithful actions of some big mapping class groups. The first is an action by isometries on a Gromov-hyperbolic graph. The second is an action by homeomorphisms on a circle in which the vertices of the graph naturally embed. We will describe some properties of ...

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

## Big mapping class groups - lecture 3 Calegari, Danny | CIRM H

Multi angle

Research schools

Part I - Theory : In the "theory" part of this mini-course, we will present recent objects and phenomena related to the study of big mapping class groups. In particular, we will define two faithful actions of some big mapping class groups. The first is an action by isometries on a Gromov-hyperbolic graph. The second is an action by homeomorphisms on a circle in which the vertices of the graph naturally embed. We will describe some properties of the objects involved, and give some fruitful relations between the dynamics of the two actions. For example, we will see that loxodromic elements (for the first action) necessarily have rational rotation number (for the second action). Using these relations, we will explain how to construct non trivial quasimorphisms on subgroups of big mapping class groups. This includes joint work with Alden Walker.
Part II - Examples : In this part we will discuss a number of natural examples in which big mapping class groups and their subgroups arise. These include the inverse limit constructions of de Carvalho-Hall, the theory of finite depth (taut) foliations of 3-manifolds, the theory of “Artinization” of Thompson-like groups, two dimensional smooth dynamics, one dimensional complex dynamics (topology of the shift locus, Schottky spaces) and several other contexts. We will try to indicate how viewing these examples from the perspective of (big) mapping class groups is a worthwhile approach.
Part I - Theory : In the "theory" part of this mini-course, we will present recent objects and phenomena related to the study of big mapping class groups. In particular, we will define two faithful actions of some big mapping class groups. The first is an action by isometries on a Gromov-hyperbolic graph. The second is an action by homeomorphisms on a circle in which the vertices of the graph naturally embed. We will describe some properties of ...

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

## Big mapping class groups - lecture 4 Calegari, Danny | CIRM H

Multi angle

Research schools

Part I - Theory : In the "theory" part of this mini-course, we will present recent objects and phenomena related to the study of big mapping class groups. In particular, we will define two faithful actions of some big mapping class groups. The first is an action by isometries on a Gromov-hyperbolic graph. The second is an action by homeomorphisms on a circle in which the vertices of the graph naturally embed. We will describe some properties of the objects involved, and give some fruitful relations between the dynamics of the two actions. For example, we will see that loxodromic elements (for the first action) necessarily have rational rotation number (for the second action). Using these relations, we will explain how to construct non trivial quasimorphisms on subgroups of big mapping class groups. This includes joint work with Alden Walker.
Part II - Examples : In this part we will discuss a number of natural examples in which big mapping class groups and their subgroups arise. These include the inverse limit constructions of de Carvalho-Hall, the theory of finite depth (taut) foliations of 3-manifolds, the theory of “Artinization” of Thompson-like groups, two dimensional smooth dynamics, one dimensional complex dynamics (topology of the shift locus, Schottky spaces) and several other contexts. We will try to indicate how viewing these examples from the perspective of (big) mapping class groups is a worthwhile approach.
Part I - Theory : In the "theory" part of this mini-course, we will present recent objects and phenomena related to the study of big mapping class groups. In particular, we will define two faithful actions of some big mapping class groups. The first is an action by isometries on a Gromov-hyperbolic graph. The second is an action by homeomorphisms on a circle in which the vertices of the graph naturally embed. We will describe some properties of ...

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

## Big mapping class groups - lecture 5 Calegari, Danny | CIRM H

Multi angle

Research schools

Part I - Theory : In the "theory" part of this mini-course, we will present recent objects and phenomena related to the study of big mapping class groups. In particular, we will define two faithful actions of some big mapping class groups. The first is an action by isometries on a Gromov-hyperbolic graph. The second is an action by homeomorphisms on a circle in which the vertices of the graph naturally embed. We will describe some properties of the objects involved, and give some fruitful relations between the dynamics of the two actions. For example, we will see that loxodromic elements (for the first action) necessarily have rational rotation number (for the second action). Using these relations, we will explain how to construct non trivial quasimorphisms on subgroups of big mapping class groups. This includes joint work with Alden Walker.
Part II - Examples : In this part we will discuss a number of natural examples in which big mapping class groups and their subgroups arise. These include the inverse limit constructions of de Carvalho-Hall, the theory of finite depth (taut) foliations of 3-manifolds, the theory of “Artinization” of Thompson-like groups, two dimensional smooth dynamics, one dimensional complex dynamics (topology of the shift locus, Schottky spaces) and several other contexts. We will try to indicate how viewing these examples from the perspective of (big) mapping class groups is a worthwhile approach.
Part I - Theory : In the "theory" part of this mini-course, we will present recent objects and phenomena related to the study of big mapping class groups. In particular, we will define two faithful actions of some big mapping class groups. The first is an action by isometries on a Gromov-hyperbolic graph. The second is an action by homeomorphisms on a circle in which the vertices of the graph naturally embed. We will describe some properties of ...

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

## Ergodic optimization - lecture 1 Contreras, Gonzalo | CIRM H

Multi angle

Research schools

We will show the proof that for generic Lipschitz functions on an expanding map there is a unique maximizing measure, and it is supported on a periodic orbit.

37D35

#### Filtrer

##### Codes MSC

Ressources Electroniques (Depuis le CIRM)

Books & Print journals

Recherche avancée

0
Z


text:
# Go's Glass Ceiling

Go is becoming increasingly popular and I find it troubling. It’s taken me quite a while to work out why that is, and quite a few drafts to try and express it in a way that gets my point across in what I hope is a fair and balanced way. This is not a “Go is bad” post. That’s been done to death and I don’t want to rehash it. I also have no intention of denigrating people who like Go; doing so would be self-defeating given it’s them this post is really aimed at. This is a post about reaching your potential.

## Go’s design goals

To discuss Go we need to understand the environment that it was designed for. In the Go FAQ there’s a fairly long document that describes the design goals of the language but this quote from one of the author’s talks really elucidates the most important consideration:

The key point here is our programmers are Googlers, they’re not researchers. They’re typically, fairly young, fresh out of school, probably learned Java, maybe learned C or C++, probably learned Python. They’re not capable of understanding a brilliant language but we want to use them to build good software. So, the language that we give them has to be easy for them to understand and easy to adopt. — Rob Pike

This shows us the target audience for the language, but although young and inexperienced engineers is the obvious point it’s not the whole story. Google hires thousands of engineers straight out of college every year and, as is common in this industry, they don’t tend to stay around for long before moving to another company.

Short tenure combined with the necessity to hire thousands of engineers means that software needs to be written in a basic way that people can understand and change without having to spend time improving their skills or knowledge. You can’t assume that the people who originally wrote the software will be around to help the next intake work on it, or pass on the skills they learned while doing so.

It isn’t really worth investing in people because you know they won’t be around for a long time, and they’ll quite possibly be going to a competitor when they leave. Instead you want to get the most out of them now without worrying about their future development.

Given this context, it seems reasonable to hypothesise that the key design goal of Go is not only to make it easy for inexperienced programmers to get to grips with it, but also to constrain what they can do with the language in order to ensure that software stays written in a basic manner. Can we find evidence to support this?

## Go’s omissions

Go is often criticised for lacking many features of modern languages such as data structures, generics, exceptions, pattern matching, enumerations, immutability, macros, and so on.

Although Go doesn’t have user-definable generics, it does have some built-in ones: map, slice (list), array and channel (thread-safe queue). The question I find most interesting is why they don’t have any of the higher-order functions you’d normally expect to find such as Filter or Map or Fold. Although syntax doesn’t exist to define these functions, they could have been provided and special-cased just like the built-in generic types.

You can’t write them yourself either, as without user-definable generics they would have to operate on the interface {} type and thus sacrifice type safety, which isn’t something you tend to want in statically typed languages. Rob Pike has actually got an example library of this but dismisses it with the following comment:

I haven’t had occasion to use it once. Instead, I just use “for” loops. You shouldn’t use it either.

Even if you’re prepared to believe that there is no implementation of generics that’s suitable for Go because they’re either too complex, too bloated, or too slow, the omission of higher-order functions on generics looks to be a decision rather than something that just wasn’t thought of.

Error handling also receives a lot of criticism for the repetitive nature of checking whether an error object is returned. I also prefer not to use exceptions, but arguing about return codes vs exceptions rather misses the point. There are much better options that could be implemented without significant changes to the language.

Go only has three return patterns by convention (result, result, error or error) so the compiler could easily implement some syntactic sugar. You could take inspiration from Rust and introduce an early return operator ? so a statement like result := Foo().? would expand to:

result, err := Foo()
if err != nil {
return _, err
}

Alternatively you could take inspiration from F# or Elixir and introduce a pipeline operator |> so a statement like result, err := Foo() |> Bar() would expand to:

res0, err0 := Foo()
if err0 != nil {
return _, err0
}

result, err := Bar(res0)

These wouldn’t need any significant changes to the core language, would barely slow compilation due to the very simple heuristics needed, wouldn’t inflate the binary size at all, and would work with all existing code. Why hasn’t something like this been implemented?

There’s a theme here.

We could go on to look at other omitted features, but they all follow the same pattern. Any feature that allows meaningful abstraction is omitted so that the mechanics of how the code works must be spelled out in full. This means that even the most inexperienced of programmers can follow how the code works, but at the cost of much repetition.

(The C approach.) Leave them [generics] out. This slows programmers. But it adds no complexity to the language. — Russ Cox

The evidence appears to support the hypothesis that Go is deliberately constrained to ensure software can only be written in a basic manner with the mechanics firmly on display.

Does it matter if we have to build software in a more basic way though? Isn’t it a good thing if anybody can just pick software up, read it, follow the code, and work on it? Well, yes and no.

It matters because the complex features in more powerful languages allow you to build abstractions.

## Why abstraction matters

Rather than just explain, I’ll demonstrate with something that Go ought to be good at given its positioning as a language for building microservices: A parallel scatter/gather. This isn’t a contrived example; it’s the kind of thing that’s extremely common in distributed architectures, for example collecting user data from one service and order data from another in order to present an account page.

Imagine we have three functions fetchUser, fetchOrders and renderPage that return types User, []Order and Page respectively. fetchUser and fetchOrders perform IO (e.g. a HTTP GET) and thus may be slow and may fail (e.g. network error) but are independent and so can be called in parallel. renderPage is fast but needs the results from fetchUser and fetchOrders.

In Go, assuming we want to keep type safety, and handle errors properly by passing them up the call stack rather than just logging and ignoring them, modelling this is going to look something like:

var wg sync.WaitGroup

chanU := make(chan User, 1)
chanO := make(chan []Order, 1)
errors := make(chan error, 1)
finished := make(chan bool, 1)

go func() {
if res, err := fetchUser(); err == nil {
chanU<- res
} else {
errors<- err
}
wg.Done()
}()

go func() {
if res, err := fetchOrders(); err == nil {
chanO<- res
} else {
errors<- err
}
wg.Done()
}()

go func() {
wg.Wait()
close(finished)
}()

var user User
var orders []Order
for {
select {
case u := <-chanU:
user = u
case o := <-chanO:
orders = o
case <-finished:
return renderPage(user, orders)
case err := <-errors:
return _, err
}
}

There’s a lot of code here, but there’s nothing complex. Once you know that chan is just a thread-safe queue and select is a special form of switch that matches whichever channel has something available first you can probably work out what’s happening even if you’ve never seen Go before.

Now let’s see the same thing modelled in a higher level language, Scala:

Apply[Future].map2(fetchUser, fetchOrders)(renderPage)

You might not be able to understand this code just by looking at it. This code uses many abstractions including higher-kinded generic types, immutable types, sum types, implicits, typeclasses, higher-order functions, currying and method values. The code is much shorter, but conceptually there’s much more going on here.

However, once you do understand some of the underlying concepts, it’s much more obvious what’s going on because you only have to scan one line, and it’s much more obviously correct because there aren’t any mechanics to get wrong.

This is the power of abstraction. This just shows a single example in isolation, but when you start building larger programs it starts to matter more. By building abstractions and composing them the program stays manageable overall because you can keep the number of concepts you have to reason about at any point relatively small, which matters to humans. The fewer or worse your abstractions, the more state you have to maintain to understand the overall flow of the code.

So how can you learn to write code at a higher level of abstraction?

## How programmers learn

I’ve been writing software for over two decades, and professionally for seventeen years as an engineer, as a lead, and as a manager. I’ve worked with hundreds of people, so I’ve seen how a broad section of industry programmers learn. The answer shouldn’t surprise you: they learn on the job.

I’ll tell you my story.

Twenty years ago I’d done a lot of programming in Visual Basic 6 and was fairly happy with it as a language. The lack of true object-orientation bothered me a bit (you couldn’t inherit from other classes, only interfaces) but it was fine; I just dealt with it. The lack of generics and concurrency? I’d never heard of either of them.

Then I discovered C# and it was mindblowing! There weren’t any generics yet but there was true object orientation and threading. With actual threads and mutexes and volatiles and mutable collections (this was still the old days). I read and I experimented, and I started to understand all these concepts I’d never even heard of before.

Generics came along and—wow—we didn’t need to copy/paste collection templates for type-safe code any more. They were just built into the language. The next year came generic variance, iterators and then Linq which let you filter or transform collections without needing to write for loops, and chain all the methods together to express your intent more clearly.

Eventually I got to the point where I found C#’s type system frustrating because it just couldn’t express many of the things I wanted to. Later, when I started using Scala on a daily basis I’d finally understand higher-kinded types and typeclasses and unlock a new set of abstractions I could use to solve problems.

As my interest in languages increased I spent my spare time learning others. Scheme, Haskell and Rust are among some of the more interesting, but I’ve studied just about every popular language and other far more esoteric ones. I’ve never got good at any of them though, because like many developers I’m pragmatic and struggle to get too invested in something I can’t use in my day job.

Of course, that’s just my story, and it differs for everybody depending on your philosophy. However, the important point is that everybody learns by doing and there are very few people who understand programming concepts well unless they’ve used them repeatedly in their day job. That’s because programming is difficult. It takes a lot of work to be able to understand these concepts, and a lot of practice to be able to apply them to build nontrivial abstractions.

Let me repeat that: Programming is difficult.

## Go’s comfort blanket

Rewind back to 2001 and imagine that instead of C#, Go was the hot new language on the block.

It would have been similarly mindblowing to my younger self. Better object-orientation than Visual Basic 6. Concurrency! Generic lists and maps! What more could I have wanted? It would have taken everything I knew and then added a whole bunch of things I didn’t.

It would have had everything I needed.

But it wouldn’t have had any of the things I didn’t know I needed. It wouldn’t have had generics. It wouldn’t have had higher-order functions. It wouldn’t have had typeclasses. It wouldn’t have had pattern matching. It wouldn’t have had metaprogramming. It wouldn’t have had macros. It wouldn’t have had any of the tools that I now take for granted to build abstractions.

And here’s the rub. Go’s documentation, in its justification of why the language omits so many features, does its best to explain that they are too difficult or complex for you. In fact, the FAQ includes the words “difficult” or “complex” sixteen times and “simple” or “simplifies” twenty:

Programming had become too difficult and the choice of languages was partly to blame.

Generics are convenient but they come at a cost in complexity in the type system.

Concurrency and multi-threaded programming have a reputation for difficulty.

Your favorite feature may be missing because […] it would make the fundamental system model too difficult.

It’s comforting to believe it.

You’re getting along perfectly fine in your day job. You can do everything you need to do with Go. Sure, it may be clunky and repetitive but as the documentation says, that’s better than having to learn anything difficult. And concepts like “monads” and “applicatives” can’t be expressed in Go anyway so there’s no practical point to learning them. Plus they have scary names. Best to leave all that difficult stuff to those silly type astronauts who want to make life hard for themselves with their category theory nonsense.

It’s pretty easy for most programmers to convince themselves of this. Paul Graham wrote about the phenomenon all the way back in 2001, using a hypothetical language called Blub for illustration:

As long as our hypothetical Blub programmer is looking down the power continuum, he knows he’s looking down. Languages less powerful than Blub are obviously less powerful, because they’re missing some feature he’s used to. But when our hypothetical Blub programmer looks in the other direction, up the power continuum, he doesn’t realize he’s looking up. What he sees are merely weird languages. He probably considers them about equivalent in power to Blub, but with all this other hairy stuff thrown in as well. Blub is good enough for him, because he thinks in Blub.

But here’s the thing. Those of us who have invested huge amounts of time and effort into learning higher level languages that have these complicated features generally prefer to use them over languages like Go. We have the capability to work in pretty much any language we want, and we choose higher level ones like Scala or Ruby or Rust instead of basic ones like Go.

Do you really think it’s just because we want to make life hard for ourselves, or could there be something deeper than that? Could it be that learning these languages and concepts makes you a better programmer, and lets you build software in a way that’s ultimately simpler by creating high level abstractions and composing them?

If you’re happy with Go, you’ll probably never know the answer. And, more tragically, you’ll probably never want to.

That is what troubles me.

An entire generation robbed of the chance to reach their potential.


text:
# Two objects moving towards each other.

1. Mar 11, 2014

### travis51

So im trying to understand some of the ideas of relativity but there is one thing i don't get. I understand that if one object were sent at .99 c and from that object another were launched at .99c the object would appear to be at .99c for the moving object but below the speed of light for an outside observer because time is just slowed down for the first bullet so it appears to be at .99c for the bullet but just below c and above .99c for and outside observer. The thing i do not understand is why if two objects were moving in opposite directions near the speed of light an observer on one of the objects would see the other going slower than the speed of light. Shouldn't it appear much faster because time is slowed in the fast object. Or does it have to do with the time dilation factor on the object. Also if object A and B were going at .99c towards each other why would an observer see it below the speed of light. I just want and explanation, no math or questioning the speed of the objects please because that is the easy way out.
I've heard that in the time, time would be slowed so much on the too intersecting that it would be impossible to measure, does this make sense? It sounds wrong to me, what if the same scenerio happened with two objects of speeds just above .5c

Last edited: Mar 11, 2014
2. Mar 11, 2014

### abbott287

If you were on earth looking up at the two objects, you would see them approaching each other at far above the speed of light. Just below 2c I believe. The people on board due to time dilation, length contration and other distorting factors would measure things totally differently than you, and below light speed. Thats what their clocks and tape measures would show. Whats odd is you would both be correct!

3. Mar 11, 2014

### Staff: Mentor

If A is approaching me from the left at speed $u$ and B is approaching me from the right at speed $v$ (both speeds less than $c$, of course), I will report that the distance between them is shrinking at the speed $u+v$, just as you would expect. However, I'm not seeing anything moving faster than light; I'm seeing two things both moving at less than the speed of light.

A will report he is at rest, I'm moving towards him at speed $u$, and B is moving towards him at speed:$$w=\frac{u+v}{1+\frac{uv}{c^2}}$$
Try plugging in some numbers and you'll see that this always comes out less than $c$. This is the "relativistic velocity addition" formula, and googling for that phrase will find you much more information.

B will report that he is at rest, I'm moving towards him at speed $v$, and A is moving towards him at the same speed $w$, again calculated according to that formula.

Note that my motion is completely irrelevant as long we're careful about defining $u$ and $v$ as speeds relative to me.

4. Mar 11, 2014

### travis51

So if object A and B are going towards each other at .99c and their is one stationary object, the stationary object will seem to be at .99c but object B traveling at .99c towards object A which is going the other way at .99c will seem to be only slightly faster because time is slowed significantly and the object only appears to be slightly faster because of this? Sounds kind of unassuring but makes some sense.

5. Mar 11, 2014

### abbott287

But the objects are approaching each other at almost twice the speed of light from the middle persons frame of reference. So you CAN move two objects in certain ways faster than the speed of light, depending on your frame of reference. Away from each other or towards each other. (While neither object is actually exceeding c.)


text:
# Ultrafast THz-driven Electron Gun

## Overview

TL;DR We built an electron gun powered exclusively by THz radiation, marking the first demonstration of coherent electron acceleration from rest using laser pulses. We achieve state-of-the-art accelerating field, jitter, and produce electron bunches of suitable charge and energy spectra for low-energy diffraction experiments.

The ultimate goal of this project is to develop an end-to-end tabletop x-ray free electron laser (FEL) based purely on optical beams. Such a compact, economical FEL would bring the benefits of rarefied billion-dollar accelerator facilities to the local hospital (medical therapy with X-rays and electron beams) and university laboratory (understanding state transitions in metals, semiconductors, and molecules). This linear accelerator would be integrated with the THz-driven linear accelerator

## The need for ultrafast electron guns

An electron gun is a device which emits and accelerates a narrow beam of electrons at a precise kinetic energy. They are used to ionize particles from an atom via electron ionization, a process commonly used in mass spectrometers. They are used to probe the dynamics of metals, semiconductors, and molecular crystals undergoing phase transitions via ultrafast electron diffraction. They are used in linear accelerators, producing the electron pulse which is accelerated to relativistic energies and converted to x-rays. The figure shows these latter two important applications.

The performance of these systems often relies on having extremely good beam quality, energy spread, and jitter of the electron bunch generated from the gun. Low quality beams blur out the molecular structure captured by diffraction experiments, while low energy spread and high jitter leads to poor spectral- or time-resolution. Typically the higher the electric field on the electron bunch at the time of emission, the better the electron bunch will be, since it more quickly accelerates out of the nonrelativistic (slow) velocity regime where electron-electron Coulombic repulsion forces (space charge) ruin the beam’s integrity.

## The need for THz fields

### Performance

Conventional DC and RF electron guns have been hindered from producing high quality electron bunches due to their inability to produce high surface electric fields. Laser pulses have a significantly higher electric fields, but previous attempts to accelerate electron bunches using infrared lasers were limited in the volume of electrons that they can accelerate, due the short wavelength of the infrared spectral regime. Laser pulses in the THz regime, however, strike the best of both worlds. Optical THz pulses have extremely high field (GV/m), long accelerating wavelength, and zero jitter, making the THz spectral regime ideally suited for scientific electron gun applications. The figure below highlights the advantages of using THz radiation for electron acceleration.

### Compactness

Further, due to the relative compactness of laser THz sources versus RF sources (large energy-hungry klystrons) and DC sources (bulky high voltage equipment), THz electron guns are significantly smaller and energy-efficient than conventional RF and DC based guns. In fact, the THz can be generated from the same laser as that of the photoemitter (the laser which ionizes electrons from the cathode), which is present in every electron For example, our electron gun demonstrated here is about the size of a matchbox.

## A micrometer-scale structure for electron acceleration

### Schematic

Before diving into the emission and acceleration mechanism, we show here an overview schematic of our setup. THz is generated by optical rectification of an infrared (IR) pulse in a lithium niobate (LN) crystal. The THz beam is coupled into the electron gun, which is about the size of a matchbox. The accelerated electrons exit the gun through a slit and are spectrally characterized by a retarded field analyzer (RFA).

The challenge of accelerating electrons with an optical field is that the oscillating field accelerates and decelerates the electrons at a very high frequency (THz frequency, or a trillion cycles per second). If the electrons stay in place, they will gain zero net energy from the oscillating field since each accelerating cycle will cancel out each decelerating cycle. The trick is to design a structure that allows electrons to experience the accelerating half-cycle—then moves out of the way before the decelerating half-cycle which arrives less than one trillionth of a second later. That is the objective of our micrometer-scale structure shown in the figure above.

### Sub-wavelength confinement of the THz wave

The structure acts as a waveguide for THz and a gun for electrons. We choose a waveguide gap of 75 $\mu$m in order to tightly confine the THz (by an order of magnitude below its wavelength) and hence enhance its field strength. Another reason for such a small gap is that the emitted electrons don’t have much time to escape the waveguide before the decelerating phase. The THz pulse is coupled into a THz waveguide by a butterfly-shaped tapering structure. Once inside the waveguide, the THz pulse travels in the TM0 mode with electric field orthogonal to the waveguide walls. Our finite-element simulation of the THz coupling into the waveguide is shown below.

### Emission and acceleration of electron bunch

The electron bunch is emitted when a UV pulse back-illuminates one of these walls, which is only 30 nm thick. The bunch emits right at the beginning of the THz electric half-cycle and gets accelerated toward the other wall. It escapes the waveguide through a small slit at that wall just before the onset of the decelerating half-cycle. The figure below shows our particle tracking simulation of the dynamics.

## Results

### Finding the best operating point

An electron’s final momentum gain, $p_e$, depends on the time at which it is emitted with respect to the THz pulse, and it follows

$$p_e=q\int_{t_{emit}}^{t_{exit}} E_{thz}(t) \mathrm{d}t.$$

where $t_{emit}$ and $t_{exit}$ are the times that the electron enters (through emission) and exits the waveguide, respectively. Thus, if $t_{exit}$ is much greater than the pulse duration of the THz pulse, the electron momentum gain will be a sinusoid 90 degree phase shifted from the THz wave. negative net momentum gain will result in the electron being recombined with the cathode, and such regions should result in low accelerated electron bunch charge and energy gain. We observe these results in the figure below, where we plot the electron energy gain and bunch charge as function of $t_{emit}$, or in practice the delay between THz and UV photoemitter pulse. The electron energy (bottom left) leads the THz wave (top left) by about 90 degrees as a function of $t_{emit}$.

Note that the THz accelerating field exceeds 350 MV/m, which is 1.75 times higher than the best operating RF gun at SLAC National Lab. Two operating points are chosen for the electron gun, namely those at delays -1.5 ps and 0.8 ps. The operating point at 0.8 ps has especially nice electron spectra with a energy spread of only 5%. Our particle tracking simulations, which include the crucial effect of space charge, corroborate well with the observed spectrum (top right), and the bottom right plot shows a simulation of the acceleration dynamics. The raw spectra and bunch charge measurements as a function of delay are shown in the figure below.

### Scaling up the THz energy

We turn our attention to seeing how the electron energy scales with increasing THz energy. Theoretically, the kinetic energy of the electrons $U_e$ follows the simple scaling law

$$U_e = \frac{p_e^2}{2m} \propto E_{thz}^2 \propto W_{thz}$$

where the proportionality to $E_{thz}$ comes from the previous equation for the momentum gain of an electron. Thus, the electron energy should be proportional to the THz energy. We observe this is the case at both operating points of our electron gun as shown in the figure below.

### All-optical low jitter

Because the UV pulse which emits the electrons comes from the same laser as the THz pulse which accelerates them, the entire system is all optical and achieves extremely low timing jitter without any no electronic synchronization of the pulses. We measure this jitter by using a infrared probe pulse to measure the THz field strength in an electro-optic sampling setup. We park the probe pulse at the zero-crossing of the THz field and collect 2500 measurements of the THz field over several seconds. After filtering out the noise spikes at the usual harmonics of 60 Hz and 110 Hz, we measured an RMS jitter of about 5 femtoseconds, which is significantly lower than that achievable by RF guns (about 100 fs at best).

## Conclusion

This work shows the first demonstration of a THz electron gun. The all-optical low jitter gun has an accelerating field of 350 MV/m, which is higher than that achievable by RF guns. The accelerated electron bunch has an energy spread of about 5% and energy of about 0.4 keV which already suitable for low-energy electron diffraction experiments.

##### Ronny Huang
###### Machine Learning Researcher

My interests include adversarial methods, generalization, and sequence modeling.

## Publications

(2015). Toward a terahertz-driven electron gun. Nature Scientific Reports.


text:
2 edited body

Hi Vaughn,

It is an old result of Karl Sigmund that the space of ergodic measures of a subshift of finite type is path connected in weak* topology. The proof is very neat and takes only one a page or so. Here is the paper:

Sigmund, Karl "On the connectedness of ergodic systems. systems." Manuscripta Math. 22 (1977), no. 1, 27–32.

I don't know about generalizations. Sigmund's proof does not generalize directly.

1

Hi Vaughn,

It is an old result of Karl Sigmund that the space of ergodic measures of a subshift of finite type is path connected in weak* topology. The proof is very neat and takes only one page or so. Here is the paper:

Sigmund, Karl On the connectedness of ergodic systems. Manuscripta Math. 22 (1977), no. 1, 27–32.

I don't know about generalizations. Sigmund's proof does not generalize directly.