text:
• This white paper summarizes the workshop "U.S. Cosmic Visions: New Ideas in Dark Matter" held at University of Maryland on March 23-25, 2017.
• ### The HPS electromagnetic calorimeter(1610.04319)

Feb. 2, 2017 hep-ex, nucl-ex, physics.ins-det
The Heavy Photon Search experiment (HPS) is searching for a new gauge boson, the so-called "heavy photon." Through its kinetic mixing with the Standard Model photon, this particle could decay into an electron-positron pair. It would then be detectable as a narrow peak in the invariant mass spectrum of such pairs, or, depending on its lifetime, by a decay downstream of the production target. The HPS experiment is installed in Hall-B of Jefferson Lab. This article presents the design and performance of one of the two detectors of the experiment, the electromagnetic calorimeter, during the runs performed in 2015-2016. The calorimeter's main purpose is to provide a fast trigger and reduce the copious background from electromagnetic processes through matching with a tracking detector. The detector is a homogeneous calorimeter, made of 442 lead-tungstate (PbWO4) scintillating crystals, each read out by an avalanche photodiode coupled to a custom trans-impedance amplifier.
• This report, based on the Dark Sectors workshop at SLAC in April 2016, summarizes the scientific importance of searches for dark sector dark matter and forces at masses beneath the weak-scale, the status of this broad international field, the important milestones motivating future exploration, and promising experimental opportunities to reach these milestones over the next 5-10 years.
• The Heavy Photon Search (HPS), an experiment to search for a hidden sector photon in fixed target electroproduction, is preparing for installation at the Thomas Jefferson National Accelerator Facility (JLab) in the Fall of 2014. As the first stage of this project, the HPS Test Run apparatus was constructed and operated in 2012 to demonstrate the experiment's technical feasibility and to confirm that the trigger rates and occupancies are as expected. This paper describes the HPS Test Run apparatus and readout electronics and its performance. In this setting, a heavy photon can be identified as a narrow peak in the e$^+$e$^-$ invariant mass spectrum, above the trident background or as a narrow invariant mass peak with a decay vertex displaced from the production target, so charged particle tracking and vertexing are needed for its detection. In the HPS Test Run, charged particles are measured with a compact forward silicon microstrip tracker inside a dipole magnet. Electromagnetic showers are detected in a PbW0$_{4}$ crystal calorimeter situated behind the magnet, and are used to trigger the experiment and identify electrons and positrons. Both detectors are placed close to the beam line and split top-bottom. This arrangement provides sensitivity to low-mass heavy photons, allows clear passage of the unscattered beam, and avoids the spray of degraded electrons coming from the target. The discrimination between prompt and displaced e$^+$e$^-$ pairs requires the first layer of silicon sensors be placed only 10~cm downstream of the target. The expected signal is small, and the trident background huge, so the experiment requires very large statistics. Accordingly, the HPS Test Run utilizes high-rate readout and data acquisition electronics and a fast trigger to exploit the essentially 100% duty cycle of the CEBAF accelerator at JLab.
• ### The Heavy Photon Search experiment at Jefferson Laboratory(1505.02025)

May 8, 2015 hep-ex, physics.ins-det
The Heavy Photon Search experiment (HPS) at Jefferson Laboratory will search for a new $U(1)$ massive gauge boson, or "heavy-photon," mediator of a new fundamental interaction, called "dark-force," that couples to ordinary photons trough kinetic mixing. HPS has sensitivity in the mass range 20 MeV - 1 GeV and coupling $\varepsilon^2$ between 10$^{-5}$ and 10$^{-10}$. The HPS experiment will search for the $e^+e^-$ decay of the heavy photon, by resonance search and detached vertexing, in an electron beam fixed target experiment. HPS will use a compact forward spectrometer, which employs silicon microstrip detectors for vertexing and tracking, and a PbWO$_4$ electromagnetic calorimeter for energy measurement and fast triggering.
• ### Design and realization of a facility for the characterization of Silicon Avalanche PhotoDiodes(1504.01589)

April 7, 2015 hep-ex, physics.ins-det
We present the design, construction, and performance of a facility for the characterization of Silicon Avalanche Photodiodes in the operating temperature range between -2 $^\circ$C and 25 $^\circ$C. The system can simultaneously measure up to 24 photo-detectors, in a completely automatic way, within one day of operations. The measured data for each sensor are: the internal gain as a function of the bias voltage and temperature, the gain variation with respect to the bias voltage, and the dark current as a function of the gain. The systematic uncertainties have been evaluated during the commissioning of the system to be of the order of 1%. This paper describes in detail the facility design and layout, and the procedure employed to characterize the sensors. The results obtained from the measurement of the 380 Avalanche Photodiodes of the CLAS12-Forward Tagger calorimeter detector are then reported, as the first example of the massive usage of the facility.
• We report the first extraction of the pion-nucleon multipoles near the production threshold for the $n\pi^+$ channel at relatively high momentum transfer ($Q^2$ up to 4.2 $\rm{GeV^2}$). The dominance of the s-wave transverse multipole ($E_{0+}$), expected in this region, allowed us to access the generalized form factor $G_1$ within the light-cone sum rule (LCSR) framework as well as the axial form factor $G_A$. The data analyzed in this work were collected by the nearly $4\pi$ CEBAF Large Acceptance Spectrometer (CLAS) using a 5.754 $\rm{GeV}$ electron beam on a proton target. The differential cross section and the $\pi-N$-multipole $E_{0+}/G_D$ were measured using two different methods, the LCSR and a direct multipole fit. The results from the two methods are found to be consistent and almost $Q^2$ independent.


text:
# Random GUI Glitches and Freezes

In the last two days I had many seemingly random GUI freezes in my Fedora 22. At first I thought them to be related to the Vivaldi browser and switched to Firefox, which is also installed. But the freezes continued, so I assumed it was about flash videos on youtube. I installed an addon to always use the HTML 5 version on youtube. Still freezes continued. I downloaded the latest NVidia driver then and reinstalled that one (was already installed in that exact version), but even on run level 3 while finishing the installation of the nvidia drivers, I got the same graphic glitches I notices before I the GUI freezes. I'll try to describe the phenomenon:

1. random pixels rush over the screen, disappear again or not and stay on it
2. mouse pointer shortly freezes then unfreezes again
3. then more random pixels and flickering occurs
4. until finally the whole GUI is frozen
5. I can still hear sounds from videos etc.

Sometimes I could log out with some key combination I tried: ctrl+alt+k or ctrl+shift+k or ctrl+shift+alt+k I just tried them all because I don't really know which one is correct. But when I logged in again, it took even less time for the issue to reappear and then I couldn't do anything at all, except pressing the reset button on my computer.

When I was on runlevel 3 installing the nvidia driver, I assume it cannot have been the drivers fault, as I started the system after doing: systemctl set-default multi-user.target and only had a terminal, no GUI. However instead of random pixels, I then for random characters on my screen, colored and in areas of the screen / terminal, where I didn't even get with the input of the terminal.

For now my system runs, but I cannot be sure if this doesn't happen again randomly at some point. I've build up some distrust to my Fedora 22 distro and I'd like to know how I can see any error messages or something for this phenomenon and if others have the same issues.

I already took a look at journalctl -e to see if anything nvidia stands out. Because usually nvidia is the problem if there is any problem on linux ... but so far I didn't see any error in there, that could cause these issues.

Note, that it only happened the last two days. I did also do dnf update, so maybe it's related to the updates 2 or 3 days ago (today: 30th of April 2016)?

Further Details

• I have a Nvidia GeForce 9800 GTX.
• Q6600 CPU
• Fedora 22
• 4GB RAM
• uname -a gives: Linux localhost.localdomain 4.4.6-201.fc22.x86_64 #1 SMP Wed Mar 30 18:30:16 UTC 2016 x86_64 x86_64 x86_64 GNU/Linux

EDIT

dmesg | grep error

[   44.898539] traps ...
edit retag close merge delete

Btw: try journalctl -b or something like that or dmesg. journalctl -e is limited to errors/warnings.
I don't know how to downgrade single packages and I don't want to risk causing it again. journalctl -b doesn't go to the end of the file and I'd have to scroll a few hours to get there. I'll add the output of dmesg | grep error to the question.

format: plain


text:
# 2. Multiclass classification¶

## 2.1. Introduction¶

In this chapter, we will use the ‘Iris-dataset’ which is available in the ‘SciKit library’. Here, we will use ‘KNeighborsClassifier’ for training the data and then trained models is used to predict the outputs for the test data. And finally, predicted outputs are compared with the desired outputs.

## 2.2. Iris-dataset¶

Lets see the Iris-dataset which has following features and target available in it, which are show in Listing 2.1.

• Features:

• sepal length in cm
• sepal width in cm
• petal length in cm
• petal width in cm
• Targets:

• Iris Setosa
• Iris Versicolour
• Iris Virginica
Listing 2.1 Iris-dataset
>>> from sklearn.datasets import load_iris
>>> iris.keys()
dict_keys(['data', 'target', 'target_names', 'DESCR', 'feature_names'])
>>> iris.feature_names
['sepal length (cm)', 'sepal width (cm)', 'petal length (cm)', 'petal width (cm)']
>>> iris.target_names
array(['setosa', 'versicolor', 'virginica'],
dtype='<U10')
>>> iris.data.shape  # 150 samples with 4 features
(150, 4)

### 2.2.2. Split the data as ‘training’ and ‘test’ data¶

We have 150 samples in our data. We can divide it into two parts i.e. ‘training dataset’ and ‘testing dataset’. A good choices can be 80% training data and 20% test data.

Important

The training data set must included all the possible ‘targets’ in it, otherwise the machine will not be trained for all the ‘targets’; and will generate huge errors when those datasets will appear in the test. We can use “stratify” in the ‘train_test_split’ which takes care of this, as shown in Listing 2.2.

Here we will use the ‘KNeighborsClasssifier’ class of ‘sklearn’ for training the machine. Lets write the code in the file. Here Lines 17-27 are used to create the training and test datasets. Then Line 36 instantiates an object of KNeighborsClasssifier, which fits the models based on training data at Line 38. Next, the trained model is used to predict the outcome of the test data at Line 40. Finally, prediction error is calculated at Line 44.

Listing 2.2 Training and test data
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 # multiclass_ex.py import numpy as np from sklearn.datasets import load_iris from sklearn.neighbors import KNeighborsClassifier from sklearn.model_selection import train_test_split # create object of class 'load_iris' iris = load_iris() # save features and targets from the 'iris' features, targets = iris.data, iris.target # both train_size and test_size are defined when we do not want to # use all the data for training and testing e.g. in below example we can # use train_size=0.4 and test_size=0.2 train_features, test_features, train_targets, test_targets = train_test_split( features, targets, train_size=0.8, test_size=0.2, # random but same for all run, also accurancy depends on the # selection of data e.g. if we put 10 then accuracy will be 1.0 # in this example random_state=23, # keep same proportion of 'target' in test and target data stratify=targets ) print("Proportion of 'targets' in the dataset") print("All data:", np.bincount(train_targets) / float(len(train_targets))) print("Training:", np.bincount(train_targets) / float(len(train_targets))) print("Training:", np.bincount(test_targets)/ float(len(test_targets))) # use KNeighborsClassifier for classification classifier = KNeighborsClassifier() # training using 'training data' classifier.fit(train_features, train_targets) # fit the model for training data # predict the 'target' for 'test data' prediction_targets = classifier.predict(test_features) # check the accuracy of the model print("Accuracy:", end=' ') print(np.sum(prediction_targets == test_targets) / float(len(test_targets)))
• Following are the outputs of the code,
\$ python multiclass_ex.py

Proportion of 'targets' in the dataset
All data: [ 0.33333333  0.33333333  0.33333333]
Training: [ 0.33333333  0.33333333  0.33333333]
Training: [ 0.33333333  0.33333333  0.33333333]

Accuracy: 0.933333333333

Note

We need to follow the below steps for training and testing the machine,

• Get the inputs i.e. ‘features’ from the datasets.
• Get the desired output i.e. ‘targets’ from the datasets ‘targets’.
• Next, split the dataset into ‘training’ and ‘testing’ data.
• Then train the model using ‘fit’ method on the ‘training’ data.
• Finally, predict the outputs for the ‘test data’, and print and plot the outputs in different formats. This printing and plotting operation will be discussed in next chapter.

## 2.3. Conclusion¶

In this chapter, we learn to split the dataset into ‘training’ and ‘test’ data. Then the training data is used to fit the model and finally the models is used for predicting the outputs for the test data for a ‘classification problem’. In the next chapter, we will discuss the ‘binary classification problem’. Also, we will read the from the file, instead of using inbuilt dataset of SciKit.


text:
1. sets prove question 1

(1) Prove that for any sets A and B ,

A = ( A n B ) u ( A n B' )

(2) Prove that A u B - A n B = ( A-(A n B ) u ( B - ( A n B ))

2. For #1: $\displaystyle \left( {B \cup B'} \right) = U\,\& \,A \cap U = A$

#2 Expand on this idea: $\displaystyle \left( {A \cup C} \right) - B = \left( {A \cup C} \right) \cap B'$

(1) Prove that for any sets A and B ,

A = ( A n B ) u ( A n B' )

(2) Prove that A u B - A n B = ( A-(A n B ) u ( B - ( A n B ))
hi
for example let A={1,2,3,4,5};B={2,4}
for the first question (AnB)= {2,4};B'={1,3,5}
Then (AUB)U(AnB')={1,2,3,4,5}=A
Therefore it is proved
for the second question (A-(AnB))={1,2,3,4,5} - {2,4}={1,3,5}
(B-(AnB))={2,4} - {2,4} = { }
Therefore (A-(AnB) U (B-(AnB)) ={1,3,5} U { } ={1,3,5}----------1
AuB = {1,2,3,4,5}
AnB = {2,4}
Therefore AuB - AnB ={1,2,3,4,5} - {2.4} = {1,3,5}---------------2
Hence 1=2
proved the second question
ok


text:
# Jordan form of matrices

So my professor gave me this question:

$A=\begin{pmatrix} 0 & 2 & 5\\ -5 & 5 & 10\\ 2 & -2 & -4 \\ \end{pmatrix}$

I had to calculate $\forall 0 < i$ $kerA^{i}$ and $ImA^{i}$

So after calculating I reached those results:

$kerA$={$\begin{pmatrix} 1 \\ 5 \\ -2 \\ \end{pmatrix}$}

and $\forall 1<i$

$kerA^{i}$={$\begin{pmatrix} 1 \\ -1 \\ 0 \\ \end{pmatrix}$,$\begin{pmatrix} -3 \\ 0 \\ 1 \\ \end{pmatrix}$}

regarding the image it is just the span of the columns of $A^{i}$

Then he asked us to calculate $\forall 0 < i$ $ker(A-I)^{i}$ and $Im(A-I)^{i}$

So after calculating I reached those results:

$\forall 0<i$

$kerA^{i}=$ {$\begin{pmatrix} 0 \\ -5/2 \\ 1 \\ \end{pmatrix}$}

regarding the image it is just the span of the columns of $(A-I)^{i}$

Then he asked us to show the Jordan form of $A$

How can I conclude that from what I proved. As far as I know the diagonal of jordan form contains the eigenvalues of $A$ (each eigenvalue repeat in the diagonal as the number of Algebraic multiplicities of the eigenvale ) and nothing more but how can I conclude from what I proved the eigenvalues of $A$.

Any help would be appreciated

There seems to be a typo. Judging from your computed kernel it should be $-4$ in the matrix. –  Julian Kuelshammer Jun 23 '13 at 8:58
For a general method to compute the normal form in such case: For each eigenvector, start with the power such that the kernel of $A-\lambda I$ does not change anymore (call that $m_\lambda$. Take a basis of a complement of $\operatorname{ker}(A-\lambda I)^{m_\lambda-1}$ in $\operatorname{ker}(A-\lambda I)^{m_\lambda}$. Now apply $(A-\lambda I)$ to that basis. The elements will lie in $\operatorname{ker}(A-\lambda I)^{m_\lambda -1}$. Now take a basis of the complement of the span of these vectors and $\operatorname{ker}(A-\lambda I)^{m_\lambda-2}$ in that space. Continue inductively. You will then get a basis for your whole space. If you change your matrix to that basis, you will get a matrix in Jordan normal form.
In your example (assuming your calculations are correct) take for example $b_0=(1,-1,0)^T$ and $Ab_0$ (for the eigenvalue $0$ and $b_1=(0,-5/2,1)^T$ for the eigenvalue $1$. Then your matrix represented in Jordan normal form will look like $$\begin{pmatrix}0&1&0\\0&0&0\\0&0&1\end{pmatrix},$$ the first column corresponding to $Ab_0$, the second to $b_0$ and the third to $b_1$


text:
Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

# Quantifying social organization and political polarization in online platforms

## Abstract

Mass selection into groups of like-minded individuals may be fragmenting and polarizing online society, particularly with respect to partisan differences1,2,3,4. However, our ability to measure the social makeup of online communities and in turn, to understand the social organization of online platforms, is limited by the pseudonymous, unstructured and large-scale nature of digital discussion. Here we develop a neural-embedding methodology to quantify the positioning of online communities along social dimensions by leveraging large-scale patterns of aggregate behaviour. Applying our methodology to 5.1 billion comments made in 10,000 communities over 14 years on Reddit, we measure how the macroscale community structure is organized with respect to age, gender and US political partisanship. Examining political content, we find that Reddit underwent a significant polarization event around the 2016 US presidential election. Contrary to conventional wisdom, however, individual-level polarization is rare; the system-level shift in 2016 was disproportionately driven by the arrival of new users. Political polarization on Reddit is unrelated to previous activity on the platform and is instead temporally aligned with external events. We also observe a stark ideological asymmetry, with the sharp increase in polarization in 2016 being entirely attributable to changes in right-wing activity. This methodology is broadly applicable to the study of online interaction, and our findings have implications for the design of online platforms, understanding the social contexts of online behaviour, and quantifying the dynamics and mechanisms of online polarization.

## Relevant articles

• ### Universality of political corruption networks

Scientific Reports Open Access 27 April 2022

## Access options

\$32.00

All prices are NET prices.

## Data availability

All data are available from the pushshift.io Reddit archive28 at http://files.pushshift.io/reddit/Source data are provided with this paper. Reddit community embedding, social dimension vectors and community scores are available at https://github.com/CSSLab/social-dimensions.

## Code availability

All code is available at https://github.com/CSSLab/social-dimensions. Analyses were performed with Python v3.7, pandas v1.3.3 and Spark v3.0.

## References

1. Sunstein, C. #Republic: Divided Democracy in the Age of Social Media (Princeton Univ. Press, 2018).

2. Iyengar, S. & Hahn, K. S. Red media, blue media: evidence of ideological selectivity in media use. J. Commun. 59, 19–39 (2009).

3. van Alstyne, M. & Brynjolfsson, E. Electronic communities: global villages or cyberbalkanization? In Proc. International Conference on Information Systems 5 https://aisel.aisnet.org/icis1996/5 (1996).

4. van Dijck, J. The Culture of Connectivity: A Critical History of Social Media (Oxford Univ. Press, 2013).

5. McLuhan, M. The Gutenberg Galaxy: The Making of Typographic Man (Univ. of Toronto Press, 1962).

6. Farrell, H. The consequences of the internet for politics. Ann. Rev. Pol. Sci. 15, 35–52 (2012).

7. Conover, M. D. et al. Political polarization on Twitter. Proc. Intl AAAI Conf. Web Soc. Media 133, 89–96 (2011).

8. Bail, C. A. et al. Exposure to opposing views on social media can increase political polarization. Proc. Natl Acad. Sci. USA 115, 9216–9221 (2018).

9. Martin, T. community2vec: vector representations of online communities encode semantic relationships. In Proc. 2nd Workshop on NLP and Computational Social Science 27–31 (2017).

10. Garg, N., Schiebinger, L., Jurafsky, D. & Zou, J. Word embeddings quantify 100 years of gender and ethnic stereotypes. Proc. Natl Acad. Sci. USA 115, E3635–E3644 (2018).

11. Bolukbasi, T., Chang, K.-W., Zou, J. Y., Saligrama, V. & Kalai, A. T. Man is to computer programmer as woman is to homemaker? Debiasing word embeddings. Adv. Neural Inf. Process. Syst. 29, 4349–4357 (2016).

12. Caliskan, A., Bryson, J. J. & Narayanan, A. Semantics derived automatically from language corpora contain human-like biases. Science 356, 183–186 (2017).

13. Kozlowski, A. C., Taddy, M. & Evans, J. A. The geometry of culture: analyzing the meanings of class through word embeddings. Am. Soc. Rev. 84, 905–949 (2019).

14. Shi, F., Shi, Y., Dokshin, F. A., Evans, J. A. & Macy, M. W. Millions of online book co-purchases reveal partisan differences in the consumption of science. Nat. Hum. Behav. 1, 0079 (2017).

15. Del Vicario, M. et al. Echo chambers: emotional contagion and group polarization on Facebook. Sci. Rep. 6, 37825 (2016).

16. Pariser, E. The Filter Bubble: What the Internet is Hiding from You (Penguin, 2011).

17. Flaxman, S., Goel, S. & Rao, J. M. Filter bubbles, echo chambers, and online news consumption. Public Opin. Q. 80, 298–320 (2016).

18. Bakshy, E., Messing, S. & Adamic, L. A. Exposure to ideologically diverse news and opinion on Facebook. Science 348, 1130–1132 (2015).

19. DiMaggio, P., Evans, J. & Bryson, B. Have American’s social attitudes become more polarized? Am. J. Sociol. 102, 690–755 (1996).

20. Barberá, P., Jost, J. T., Nagler, J., Tucker, J. A. & Bonneau, R. Tweeting from left to right: is online political communication more than an echo chamber? Psychol. Sci. 26, 1531–1542 (2015).

21. Adamic, L. A. & Glance, N. The political blogosphere and the 2004 US election: divided they blog. In Proc. 3rd International Workshop on Link Discovery 36–43 (2005).

22. An Examination of the 2016 Electorate, Based on Validated Voters https://www.pewresearch.org/politics/2018/08/09/an-examination-of-the-2016-electorate-based-on-validated-voters/ (Pew Research Center, 2018).

23. Hawley, G. Making Sense of the Alt-Right (Columbia Univ. Press, 2017).

24. Simmel, G. Conflict and the Web of Group Affiliations (Free Press, 1955).

25. Breiger, R. L. The duality of persons and groups. Social Forces 53, 181–190 (1974).

26. Bourdieu, P. Distinction: A Social Critique of the Judgement of Taste (Routledge, 1984).

27. Crenshaw, K. W. On Intersectionality: Essential Writings (The New Press, 2017).

28. Baumgartner, J., Zannettou, S., Keegan, B., Squire, M. & Blackburn, J. The Pushshift Reddit dataset. In Proc. International AAAI Conference on Web and Social Media 14, 830–839 (2020).

30. Kumar, S., Hamilton, W. L., Leskovec, J. & Jurafsky, D. Community interaction and conflict on the web. In Proc. 2018 World Wide Web Conference 933–943 (2018).

31. Waller, I. & Anderson, A. Generalists and specialists: using community embeddings to quantify activity diversity in online platforms. In Proc. 2019 World Wide Web Conference 1954–1964 (2019).

32. Levy, O. & Goldberg, Y. Dependency-based word embeddings. In Proc. 52nd Annual Meeting of the Association for Computational Linguistics 2, 302–308 (2014).

33. Levy, O. & Goldberg, Y. Neural word embedding as implicit matrix factorization. Adv. Neural Inf. Process. Syst. 27, 2177–2185 (2014).

34. Schlechtweg, D., Oguz, C. & im Walde, S. S., Second-order co-occurrence sensitivity of skip-gram with negative sampling. Preprint at https://arxiv.org/abs/1906.02479 (2019).

## Acknowledgements

This research was supported by the National Sciences and Engineering Research Council of Canada (NSERC), the Canada Foundation for Innovation (CFI) and the Ontario Research Fund (ORF).

## Author information

Authors

### Contributions

I.W. performed the computational analysis. A.A. and I.W. designed the research, analysed the results and wrote the paper.

### Corresponding author

Correspondence to Ashton Anderson.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Peer review information Nature thanks Kenneth Benoit, Kate Starbird and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Extended data figures and tables

### Extended Data Fig. 1 Distribution of community scores.

Left: distributions of communities on the age, gender, partisan, and affluence dimensions. Right: the most extreme communities and words on those dimensions. Word scores are calculated by averaging community scores weighted by the number of occurrences of the word in the community in 2017. Community descriptions can be found in the glossary (Supplementary Table 1).

### Extended Data Fig. 2 External validations of social dimensions.

Scatter plots of the external validations of the gender, partisan, and affluence axes. The gender scores for occupational communities are plotted against the percentage of women in that occupation from the 2018 American Community Survey. The partisan scores for city communities are plotted against the Republican vote differential for that metropolitan area in the 2016 presidential election. The affluence scores of city communities are plotted against the median household income for that metropolitan area from the 2016 US Census. The blue line is the best-fit linear regression for the data; the shaded area represents a 95% confidence interval for the regression estimated using a bootstrap. $$p$$-values for correlation coefficients computed using two-sided test of Pearson correlation assuming joint normality.

### Extended Data Fig. 3 Further validations of social dimensions.

Clockwise from left: The gap between university and city communities on the age dimension. The distribution of university and city communities on the age dimension; age is strongly related to label ($$r=0.91$$, two-sided $$p < {10}^{-58}$$, $$n=150$$, Cohen’s $$d=4.37$$). The distribution of left and right wing labelled communities on the partisan dimension; partisan is strongly related to label ($$r=0.92$$, two-sided $$p < {10}^{-21}$$, $$n=50$$, Cohen’s $$d=4.89$$). The distribution of explicitly labelled left- and right-wing communities on the partisan-ness axis as compared to the general distribution; there is a large difference in their means (Cohen’s $$d=3.27$$). For violin plots, white dot represents median; box represents 25th to 75th percentile; whiskers represent 1.5 times the inter-quartile range; and density estimate (‘violin’) extends to the minima and maxima of the data. $$p$$-values for correlation coefficients computed using two-sided test of Pearson correlation assuming joint normality.

### Extended Data Fig. 4 Distributions of age, gender and partisan scores by cluster.

Distributions of raw age, gender and partisan scores, separated by cluster. Outlier communities that lie more than two standard deviations from the mean are annotated. Dashed lines represent the global mean on each dimension. Community descriptions can be found in the glossary (Supplementary Table 1).

### Extended Data Fig. 5 Distributions of affluence, time, sociality and edgy scores by cluster.

Outlier communities that lie more than two standard deviations from the mean are annotated. Dashed lines represent the global mean on each dimension. Community descriptions can be found in the glossary (Supplementary Table 1).

### Extended Data Fig. 6 Relationships between online social dimensions.

The relationships between the partisan dimension and (a) gender, (b) age, (c) partisan-ness. Every bar represents a bin of communities with partisan scores a given number of standard deviations from the mean, and the distribution illustrates the scores on the secondary dimension (e.g. gender in (a)). From left to right, the bars represent highly left-wing, leaning left-wing, center, leaning right-wing, highly right-wing communities. The leftmost and rightmost bars are annotated with the number of communities, and examples of the largest communities, in each group. The hex-plot in (c) illustrates the joint distribution of partisan and partisan-ness scores. Labels correspond to the categorizations used in the polarization analysis.

### Extended Data Fig. 7 Polarization robustness checks.

(a) The partisan distribution of deleted and non-deleted comments in political communities. (b) The proportion of activity that took place in very left-wing ($$z < -3$$) and very right-wing ($$z > 3$$) communities over time. (c) Alternate version of Fig. 3a generated using a dataset in which the authorship of all comments was randomly shuffled. Each individual bin distribution is extremely similar to the overall activity distribution, showing that the overall activity distribution is a useful reference point for what bin distributions would look like if there were no tendency for users to comment in ideologically homogeneous communities. (d) Average distributions of political activity for authors of comments in the 25 largest political communities on Reddit (by number of comments). (e) Correlation of users’ average partisan scores over time. Each $$\left(x,y\right)$$ cell represents the correlation between scores of a user in month $${t}_{x}$$ and that same user in month $${t}_{y}$$, for all users active in both time periods. A user is only considered active if they make at least $$10$$ comments in a month. (f) The relationship between the proportion of users who polarize and the polarization threshold. The polarization threshold is the number of standard deviations a user must increase in polarization to be considered polarized. Three lines are plotted corresponding to three pairs of months; the pairs of months with the minimum (blue), maximum (orange), and median (green) proportion of users polarized when using a threshold of $$1$$. A threshold of $$1$$ is used in all other calculations. (g) The relationship between the proportion of users who polarize and the comment threshold. The comment threshold is the value used to filter inactive users from the calculation. Users must have at least $$x$$ comments in each of the two months to be included in the calculation of the proportion of users who polarize. The same three month pairs are plotted as in part (e). There are minimal differences between different thresholds. A threshold of $$10$$ is used in all other calculations.

### Extended Data Fig. 8 Distribution of political activity by user group.

The distribution of political activity on Reddit over time by partisan score. Each bar represents one month of comment activity in political communities on Reddit, and is coloured according to the distribution of partisan scores of comments posted during the month (the partisan score of a comment is simply the partisan score of the community in which it was posted.) The top plot includes all activity as in Fig. 3b, while the four following plots decompose this into the subsets of activity authored by particular groups of users. Users are classified based on the average partisan score of their activity in the month 12 months prior–into left-wing (having a score at least one standard deviation to the left), right-wing (one standard deviation to the right), or center. Users with no political activity in the month 12 months prior use the label of the most recent month more than 12 months prior in which they had political activity; if they have never had political activity before, they fall into the new / newly political category (bottom).

### Extended Data Fig. 9 Additional measures of ideological asymmetry.

(a) Average polarization (absolute $$z$$-score) of activity in different ideological categories over time. (b) Volume of activity (number of comments) in different ideological categories over time. (c, d) Annual change in polarization in the two partisan activity categories, decomposed into the change attributable to new ($$\varDelta n$$) and existing ($$\varDelta e$$) users as done in Fig. 4.

### Extended Data Fig. 10 Implicit polarization.

The relationship between explicitly partisan and implicitly partisan activity (left: left-wing activity; right: right-wing activity.) Of users who were first active in an explicitly partisan community at time $${m}_{E}$$, the proportion of them who were first active in an implicitly partisan community at time $${m}_{I}$$ is denoted by the colour in cell $$\left({m}_{E},{m}_{I}\right)$$. The line graphs at the top show the total proportion of users who were active in implicitly partisan communities before they were active in an explicitly partisan community (i.e. the sum of each column below the diagonal back to 2005, or the total proportion of users for whom $${m}_{I} < {m}_{E}$$).

## Supplementary information

### Supplementary Information

This file contains Supplementary Tables 1 and 2.

## Rights and permissions

Reprints and Permissions

Waller, I., Anderson, A. Quantifying social organization and political polarization in online platforms. Nature 600, 264–268 (2021). https://doi.org/10.1038/s41586-021-04167-x

• Accepted:

• Published:

• Issue Date:

• DOI: https://doi.org/10.1038/s41586-021-04167-x

• ### Universality of political corruption networks

• Alvaro F. Martins
• Bruno R. da Cunha
• Haroldo V. Ribeiro

Scientific Reports (2022)


text:
# How to declare non-binary decision variables in an optimization problem?

I am working on a logistics problem which require me to find the optimum quantity of product to be manufactured and also to be shipped to satisfy the customer demand. I have made a decision variable $$y_{ijp}$$ where this variable gives me the qty of product $$p$$ shipped from node $$i$$ to $$j$$. I have done some simple optimization examples previously on Qiskit and D-Wave, but I am not sure how to tackle this problem, since all the decision variables I encountered earlier were binary. Any resources or help regarding this?

• You can try to express the non-binary variable as sum $\sum 2^i x_i$, where $x_i$ is a binary variable. In the end you have binary optimization. Jun 21 at 6:19
• For Qiskit take a look at the optimization tutorials qiskit.org/documentation/optimization/tutorials/index.html. It has converters that deal with integer variable as input. Also there is a warm starting optimizer tutorial around this too. Jun 21 at 13:39
• @SteveWood, i have gone through these tutorials, the main issue is, while having an integer variable in my model, it adds a constraints, that in turn adds slack variables, which results in lot of qubits being used. I have to get a way past that Jun 22 at 7:11
• Hi, @MartinVesely, can you elaborate or share a resource related to your answer. Thanks Jun 22 at 7:13

Perhaps what you really want to know is how to represent integral or rational variables using binary variables. The answer by Martin Vesely explains how to do it. Basically, you represent a non-binary variable $$x$$ with a bunch of binary variables $$y_j$$. $$\tag{1} x = \sum_{i=0}^{n-1} 2^i y_i + \sum_{j=1}^m 2^{-j} y_j.$$ If you want $$x$$ to be a float, then $$m > 0$$ and it determines the precision. Mind that this is not a good idea because your problem now has more variables than the original problem. Also, your feasibility space will be exponentially smaller compared to the solution space. Moreover, the new problem will require much more quantum resources.


text:
# E^(-i * x) not well-defined. Why?

1. Nov 4, 2012

### nigels

e^(-i * x) not well-defined. Why??

Hi, Just saw this as a step in an example that demonstrates the differentiability of holomorphic function. But I can't for the life of me figure out why e^(-2iθ) is ill-defined.

2. Nov 4, 2012

### Hetware

Re: e^(-i * x) not well-defined. Why??

What do you mean it is ill-defined. Why do you say that? It's well defined in the opinions of many smart people. I can't go beyond that until I know what objection there is to the conventional definition there is.

3. Nov 6, 2012

### Vargo

Re: e^(-i * x) not well-defined. Why??

It is well defined as a function of a real variable theta. But as a function on the plane, considering theta=theta(z), it has a problem at z=0. Does this answer your question? It is hard to tell without more context.

As a function of z, this function is the complex conjugate of (z/|z|)^2

4. Nov 8, 2012

### HallsofIvy

Re: e^(-i * x) not well-defined. Why??

The standard definition of "function" for real numbers requires that "if x= y, then f(x)= f(y)"- i.e. that f is "well-defined". For functions of complex variables, that is simply too restrictive. "Functions" that we would like to be able to use, such as $e^x$ would no longer be "functions". So we drop that requirement.

5. Nov 8, 2012

### Klungo

Re: e^(-i * x) not well-defined. Why??

Some functions are given the requirement of Principal Value to make some functions of a complex variables become actual functions.

I.e , let z=Re^(ix), and restrict x to be in
(-pi,pi].

This restriction works since e^(ix)=e^(ix+i*2n*pi) for all integers n.

On the other hand, it makes functions behave less like we would wabt them to.

That is, Log(xy)=/=Log(x)+Log(y) generally for principal value logarithm. On the other hand, log(xy)=log(x)+log(y).

(correct me if I'm wrong as I am just typing from memory)

Edit: x,y is a complex number for the logarithm examples.


text:
Home Gas effect Indicate separate contributions of long-lived and short-lived greenhouse gases in emissions targets

# Indicate separate contributions of long-lived and short-lived greenhouse gases in emissions targets

To quantify the SR1.5 and AR6 statements cited above, the human-induced global temperature change over a time interval of decades ∆yourelative to the level of human-induced warming at the beginning of this interval (e.g. present or pre-industrial times), can be decomposed using the articulated framework above as follows:

$${Delta} T = kappa _Eoverline {E_C} {Delta} t + kappa _Fleft( {{Delta} F_N + rho overline {F_N} {Delta} t} right ),$$

(1)

or (overline {E_C}) and (overline {F_N}) are the average global aggregated CO2 emission rate and non-CO2 radiative forcing, respectively (so (overline {E_C} {Delta} t) is the cumulative CO2 emissions) and ∆FNOT is the change in the decadal average of non-CO2 forcing, all assessed over this interval (the geophysical “zero emissions commitment” should be relatively low over a time interval of several decades23, but this may not be the case on longer time scales). The coefficients κE (the TCRE) and κF (the TCRF, or “rapid” component of the climate response to any forcing change, denoted vs1 in ref. 12or sum of fast components24: see supplementary material), are both scenario-independent in the absence of strongly nonlinear carbon cycle feedbacks or climate response. The only scenario-dependent coefficient is ρ, the constant forcing adjustment fractional rate (RACF), or the relatively low fractional rate at which forcing must decrease to maintain stable temperatures. It depends on speed and recent date FNOT increased (this term represents the delayed adjustment to past forcing increases, so it is more important for more recent and rapid increases). Yes FNOT only varies on multi-decadal time scales, ρ =vs2//( κFs2), or vs2 is the “slow” (multi-secular) component of climate sensitivity, and s2 deep ocean thermal adjustment time scale. For the representative12 coefficient values,ρ 0.3% per year, which makes this third term generally weak.

Global CO2-e100 emissions cannot be used to calculate FNOT if these include a mixture of LLCF and SLCF. Global CO2-e100 LLCF emissions, ELcan, however, be unambiguously combined and have the same impact on global temperature on a decade-to-century scale as the corresponding amount of CO2. Similarly, aggregate CO2-e100 SLCF broadcasts, ESmultiplied by the AGWP100 of CO2, A100give the SLCF radiative forcing,FS ( A100 normally includes a first order estimate of the impact of carbon cycle feedbacks25 so for consistency this should also be included in the GWP100 values used to calculate ES).

For emissions reported as CO2-e100 so the above expression can be rewritten (now grouping all LLCFs with CO2):

$${Delta} T = kappa _Eoverline {E_L} {Delta} t + kappa _Fleft( {{Delta} F_S + rho overline {F_S} {Delta} t} right ),$$

(2)

or equivalently, usingFS=A100ES on multi-decadal time scales,

$${Delta} T = kappa _Eoverline {E_L} {Delta} t + kappa _FA_{100}left( {{Delta} E_S + rho overline {E_S} {Delta} t } right).$$

(3)

Hence ∆Jcan be estimated directly using well-known (though uncertain) properties of the climate system if, and only if, total CO2-e100 emissions of long-lived climate forcers, ELare specified in the emission targets with the total CO2-e100 emissions, EL+ES; or equivalent, EL and ES are specified separately. ∆J cannot be calculated from the sum of EL+ ES alone.

This is illustrated by Fig. 1, which shows the impact of LLCF and SLCF emissions, expressed in CO2-e100on the change in global temperature over a period of several decades, compared to the level of warming at the beginning of this period, calculated with a simple climate model12. Stylized cases of constant (darker shades) and gradually varying (+10%, lighter shades and −50%, dashed lines) emissions are shown in panels a and c. Warming due to LLCF emissions (the term (kappa _Eoverline {E_L} {Delta} t) in eq. (3)) increases linearly with cumulative emissions in the three cases (panel b). Warming due to continuous constant emission from an SLCF that began decades before the start of this period (the (kappa _FA_{100}rho overline {E_S} {Delta} t) term) also increases linearly (panel d, darker blue) but at a slower rate per tCO2-e100 emitted (by a factor of approximately 4, because κE≈ 4× κFA100ρ ): global temperatures have already partially equilibrated with this constant emission (by how much depends on how long ago these SLCF emissions started, which is whyρ is the only scenario-dependent coefficient in these expressions). Finally, warming due to an increase in SLCF emissions (theκFA100ES term, panel d, lighter blue) is 4-5 times higher than would be expected with the same increase in tCO2-e100 emissions of an LLCF (panel b, lighter red) over the 20 years following the increase (κFA100 ≈ 4.5×κE× 20 years). Hence the statement AR6 “express methane emissions as CO2 equivalent emissions using GWP100overestimates the effect of constant methane emissions on global surface temperature by a factor of 3 to 4… while underestimating the effect of any new sources of methane emissions by a factor of 4 to 5 on the 20 years following the introduction of the new source”26 applies to the global emissions impact of any SLCF. Any decrease in SLCF emissions also has a much larger impact on temperatures over a period of decades per tCO2 -e100avoided than a corresponding decrease in LLCF emissions (red and blue dashed lines) (Fig. 1).

The temperature changes in the figure are calculated using a particular model, LLCF, SLCF and a scenario. The figure would, however, appear similar if another model, gas combination or past emissions scenario were used, provided that emissions do not change rapidly immediately before the start or end of the time period shown, as the relationship between emissions and warming expressed in Eq. (3) is generic. Individual terms in Eq. (3), assuming constant coefficients, are represented by the arrows to the right of panels b and d. These correspond to the warming calculated by the explicit simple climate model within the limits of the modeling uncertainties. The figure shows the temperature change relative to the beginning of the period rather than the absolute warming because the latter is not determined by the equation. (3) but depends on the history of LLCF and SLCF emissions (the specific scenario used to generate this figure is presented in full in the supplementary information).

Temperature change ∆ Jover a period of several decades depends, to first order, only on the cumulative emissions of LLCF (overline {E_L} {Delta} t)cumulative SLCF emissions (overline {E_S} {Delta} t)and net change in total SLCF emission rates ∆ ES, over this period alone. As SR1.5 and AR6 pointed out, future warming depends on future emissions. However, the use of this information requires both EL andES to be specified: specifying only the sumEL+ ES introduces ambiguity into the temperature result.

A separate specification also makes it easier to assess the implications of different metrics. For example, aggregate CO2-equivalent emissions using the 20-year global warming potential (GWP20 ) can be approximated by EL+3 ES if both EL and ESare reported as CO2-e100 with a slightly higher multiplication factor (up to 4) ifES is dominated by forcers with lifespans of the order of one year (Table 8.A.1 of Ref. 12 shows that GWP20 values are similar to GWP100 values for LLCF and 3 or 4 times GWP100values for gases with lifetimes on the order of a decade or a year, respectively). Finally, we re-emphasize that these expressions capture our physical understanding of how global LLCF and SLCF emissions collectively drive global temperature change, and illustrate the usefulness of a separate specification ofEL andES. How this understanding is used to inform the assessment of the adequacy of individual emissions targets depends on other considerations listed above and cannot be argued from a physical science perspective alone. There will be several other advantages to the additional communication, such as the possibility of estimating the co-benefits of mitigation on air quality.


text:
# an einstein ring can appear when

Finding one that is gravitationally lensed, making it appear brighter and larger, is exceptionally exciting. In the new study, led by Kailash Sahu, two objects were slightly out of alignment, and an asymmetrical version of an Einstein ring formed. This observation saw a quasar lensed by a nearer galaxy into two separate but very similar images of the same object, the images stretched round the lens into an almost complete ring. In observational astronomy an Einstein ring is the deformation of the light from a source (such as a galaxy or star) into a ring through gravitational lensing of the source's light by an object with an extremely large mass (such as another galaxy, or a black hole).This occurs when the source, lens and observer are all aligned. that light waves can be both bent and di racted by this eld. If a source is placed exactly on the optical axis, a ring-shaped image is formed, which is called an Einstein ring. As of 2009, this survey has found 85 confirmed gravitational lenses but there is not yet a number for how many show Einstein rings. An aura of charisma and charm often surrounds the owners of these raja loops. The bending of light by a gravitational body was predicted by Albert Einstein in 1912, a few years before the publication of general relativity in 1916 (Renn et al. This phenomenon is illustrated in the accompanying video. So is finding patterns between all the four fingers. The ring effect was first mentioned in the academic literature by Orest Khvolson in a short article in 1924, in which he mentioned the âhalo effectâ of gravitation when the source, lens, and observer are in near-perfect alignment. Universal Speed Limit. If a star and its intervening lens are slightly out of line, the distant light can appear as arcs. (2019). Stars approaching the exact other side of the black hole from you appear to approach this line, are greatly magnified, and move with high â¦ , An important example of the gravitational lens effect is the Einstein ring phenomenon illustrated at right. Although light comes in many flavors â from the rainbow of colors humans can see to the radio waves that transmit spacecraft data â Einstein said all light must obey the speed limit of 186,000 miles (300,000 kilometers) per second. Albert-Einstein-Ring 2 Directions {{::location.tagLine.value.text}} Sponsored Topics. The closest star â¦ This loop forms on the same mount (the Mount of Jupiter) that the Ring of Solomon and Empathy lines appear. 1 Although light comes in many flavors â from the rainbow of colors humans can see to the radio waves that transmit spacecraft data â Einstein said all light must obey the speed limit of 186,000 miles (300,000 kilometers) per second. The odds of finding such a double ring are 1 in 10,000. In this case a galaxy bends the light emanating from a galaxy that is directly behind it, focusing the otherwise divergent light into a visible ring. The first and third correspond to points which are behind the black hole (from the observer's position) and correspond here to the bright yellow region of the galactic disc (close to the galactic center), whereas the second and fourth correspond to images of objects which are behind the observer, which appear bluer since the corresponding part of the galactic disc is thinner and hence dimmer here. Legal. In special cases, the light from a distant object can be so distorted that it creates a complete ring known as an "Einstein ring." Hubble finds double Einstein ring This rare event can provide information about dark matter, dark energy, distant galaxies, and even the shape of the universe. Stars approaching the exact other side of the black hole from you appear to approach this line, are greatly magnified, and move with high angular speeds. [2], The size of an Einstein ring is given by the Einstein radius. {\displaystyle D_{LS}\neq D_{S}-D_{L}} If the alignment is just right, this creates a circle of light called an Einstein ring, predicted by Albert Einstein in 1936. Einstein rings are produced when two galaxies are almost perfectly aligned, one behind the other. Einstein's famous equation E= mc 2 contains "c," the speed of light in a vacuum. Together with her colleagues in the Netherlands and US, she studied the B1938+666 lens system, in which the gravitational lensing effect of a massive elliptical galaxy at a redshift of zâ0.9 is making a background galaxy (zâ2.1) appear as an almost perfect Einstein ring. If the alignment is just right, this creates a circle of light called an Einstein ring, predicted by Albert Einstein in 1936. In this case a galaxy bends the light emanating from a galaxy that is directly behind it, focusing the otherwise divergent light into a visible ring. An Einstein Ring is a special type of gravitational lens, in which the Earth, the foreground lensing galaxy, and the background lensed galaxy are in perfect alignment, creating a harmonious distortion in the form of a ring of light. Finding one that is gravitationally lensed, making it appear brighter and larger, is exceptionally exciting. D One of the most dramatic examples of the Einstein ring phenomenon was photographed in the visible and infrared by the Wide Field Camera 3 of the Hubble Space Telescope around luminous red galaxy LRG 3-757.This galaxy was discovered in 2007 in the Sloan Digital Sky Survey (SDSS). Posted on October 9, 2015 December 29, 2015. That's why we see the ring-like structure. [6], There have apparently not been any observations of a star forming an Einstein ring with another star, but there is a 45% chance of this happening in early May, 2028 when Alpha Centauri A passes between us and a distant red star.[7]. If the alignment is just right, this creates a circle of light called an Einstein ring, predicted by Albert Einstein in 1936. L Such rings help in understanding the distribution of dark matter, dark energy, the nature of distant galaxies, and the curvature of the universe. in general. Finding one that is gravitationally lensed, making it appear brighter and larger, is exceptionally exciting. 1. Using the Hubble Space Telescope, a double ring has been found by Raphael Gavazzi of the STScI and Tommaso Treu of the University of California, Santa Barbara. An Einstein Ring is a specific type of gravitational lensing. D More typically, gravitationally lensing will cause multiple images of the background object to appear around the foreground object. The quasar 2237 for instance appears quadruply imaged. [1] Instead of light from a source traveling in a straight line (in three dimensions), it is bent by the presence of a massive body, which distorts spacetime. More typically, gravitationally lensing will cause multiple images of the background object to appear around the foreground object. [14] This survey is responsible for most of the recent discoveries of Einstein rings in the optical range, following are some examples which were found: Another example is the radio/X-Ray Einstein ring around PKS 1830-211, which is unusually strong in radio. Previous to this find, the most complete Einstein ring discovered was documented in 1996 by S.J. The foreground galaxy must be almost perfectly aligned to provide a circular ring of light like this example. When both galaxies are exactly lined up, the light forms a circle, called an Einstein ring, around the foreground galaxy. Einsteinâs Theory of General Relativity predicted the phenomena of gravitational lensing. The lens redshift is 0.878, but the source redshift is not − More complex gravitational lensing arises in observations of massive â¦ The first photo of a black hole has revealed more support for Albert Einstein's theory of general relativity. Such multiple images are indeed observed. They are multiple images of the galactic disk. This prediction was verified by the observation of such bending of starlight near the Sun in 1919. In a 1936 paper, he described how the gravitational field from a massive object can warp space and thereby deflect light. The Einstein ring's discovery was made by â¦ More typically, gravitationally lensing will cause multiple images of the background object to appear around the foreground object. When both galaxies are exactly lined up, the light forms a circle, called an "Einstein ring", around the foreground galaxy. [10] These dual images are another possible effect of the source, lens, and observer not being perfectly aligned. [5] Einstein remarked upon this effect in 1936 in a paper prompted by a letter by a Czech engineer, R W Mandl [1], but stated. Discovered in 1988, the system consists of a bright (S 74 MHz = 3.7 Jy) radio source imaged into a ring and two compact, flat-spectrum components separated by 2 1.The ring is optically faint (R = 23.3), rising steeply â¦ mass proï¬le and the angular structure, as is true of ellipsoids, an Einstein ring can appear to strongly distinguish radial mass distributions with the same Î¾2. â Can you answer why even a small distortion of the massive lensing object in gravitational lensing techniques can result in Einstein rings around the forefront object? Einstein ring of a galaxy. The Einstein ring radius depends on the mass of the lensing object: the more massive it is, the larger the Einstein ring radius. If the alignment is just right, this creates a circle of light called an Einstein ring, predicted by Albert Einstein in 1936. According to general relativity, gravity causes a deflection of light by the gravitational field of a massive body. Albert Einstein's General Theory of Relativity, published in 1916, predicted that massive objects, such as stars, could bend light rays passing nearby. The thin blue bull's-eye patterns in these eight Hubble Space Telescope images appear like neon signs floating over reddish-white blobs. The background galaxy identified as SDSSJ1430 forms an Einstein ring "mirage" around the foreground normal white galaxy. This is a survey of ï¬at-spectrum radio sources designed to identify gravitational lens candidates. If the alignment is just right, this creates a circle of light called an Einstein ring, predicted by Albert Einstein in 1936. If another background galaxy lies precisely on the same sightline, a second, larger ring will appear. The first complete Einstein ring to be discovered was B1938+666, which was found by King et al. First, we shall scarcely ever approach closely enough to such a central line. If the alignment is just right, this creates a circle of light called an Einstein ring, predicted by Albert Einstein in 1936. The phenomenon can be explained by Einstein's theory of general relativity. For the first time, astronomers have observed a star orbiting the supermassive black hole at the center of our Milky Way galaxy. However, Einstein was only considering the chance of observing Einstein rings produced by stars, which is low â the chance of observing those produced by larger lenses such as galaxies or black holes is higher since the angular size of an Einstein ring increases with the mass of the lens. The Einstein radius is the radius of a ring Einstein. The ring is described as being 1 arcsecond across as observed from Earth's vicinity, but tens of thousands of light years across in size. Online references: Einstein ring on Astronomy Picture of the Day. It also depends on the distance between us, the lensing object, and the background source. Albert Einstein predicted that such rings would be found nearly 70 years ago. This problem is beautifully illustrated by the ellipsoidal models in Millon et al. Article was published in Nature 333, 9 June 1988. θ (In this statement, Î² is the Einstein Radius currently denoted by at the Chandra X-Ray observatory[19] It is also notable for being the first case of a quasar being lensed by an almost face-on spiral galaxy. Here an Einstein ring for background stars can be seen as an invisible line above the photon sphere horizon. A massive object, in this case a foreground galaxy, can bend space the way a bowling ball bends the fabric of the trampoline. If another more distant galaxy lies precisely on the same sightline, a second, larger ring will appear. Graceful arcs around SDSSJ0146-0929 are examples of an Einstein ring, A simulated view of a black hole passing in front of a galaxy, Montage of the SDP.81 Einstein Ring and the lensed galaxy, Feature seen when light is gravitationally lensed by an object, "Astronomers Observe Supernova and Find They're Watching Reruns", "ALMA at Full Stretch Yields Spectacular Images", "The Early History of Gravitational Lensing", "Discovery of the First "Einstein Ring" Gravitational Lens", "Hubble, Sloan Quadruple Number of Known Optical Einstein Rings", "Chandra Detection of AN X-Ray Einstein Ring in PKS 1830-211", "Discovery of a high-redshift Einstein ring", "Lens-like Action of a Star by the Deviation of Light in the Gravitational Field", "Nearly perfect Einstein ring discovered", https://en.wikipedia.org/w/index.php?title=Einstein_ring&oldid=988804995, Short description is different from Wikidata, Creative Commons Attribution-ShareAlike License, This page was last edited on 15 November 2020, at 10:12. Finding one that is gravitationally lensed, making it appear brighter and larger, is exceptionally exciting. About half a dozen of them are partial Einstein rings with diameters up to an arcsecond, although as either the mass distribution of the lenses is not perfectly axially symmetrical, or the source, lens, and observer are not perfectly aligned, we have yet to see a perfect Einstein ring. S It is not generally appreciated that there can be an infinite number of Einstein rings. According to general relativity, gravity causes a deflection of light by the gravitational field of a massive body. Tag: Einstein ring. When both galaxies are exactly lined up, the light forms a circle, called an âEinstein ring,â around the foreground galaxy. If the alignment is just right, this creates a circle of light called an Einstein ring, predicted by Albert Einstein in 1936. Warren of the Imperial College in London. Gravitational lensing is predicted by Albert Einstein's theory of general relativity. Related: Hubble telescope discovers light-bending 'Einstein ring' in space But the team didn't just find any old Einstein ring, they spotted the first-ever discovered Einstein ring: 1131+0456. From its horseshoe shape it has been called "Lensshoe" or "the cosmic horseshoe". The odds of seeing such a special alignment are so small that Tommaso says that they "hit the jackpot" with this discovery. An Einstein ring can form if a background galaxy is positioned exactly behind the center of a gravitational lens. The intervening objects act as lenses, creating arcs and 'Einstein rings' of light. Einstein's rings in space HARVARD-SMITHSONIAN CENTER FOR ASTROPHYSICS RELEASE Posted: November 17, 2005. (1998) via optical follow-up with the Hubble Space Telescope of a gravitational lens imaged with MERLIN. About the 'Einstein ring' A ring often serves as the visible symbol of the unseen - â¦ If another more distant galaxy lies precisely on the same sightline, a second, larger ring will appear. This image is credited to A.Bolton (UH/IfA) for SLACS and NASA/ESA. More typically, gravitationally lensing will cause multiple images of the background object to appear around the foreground object. An important example of the gravitational lens effect is the Einstein ring phenomenon illustrated at right. The bright thin ring that can be seen in blue is the edge of what researchers call the black hole shadow. Together with her colleagues in the Netherlands and US, she studied the B1938+666 lens system, in which the gravitational lensing effect of a massive elliptical galaxy at a redshift of zâ0.9 is making a background galaxy (zâ2.1) appear as an almost perfect Einstein ring. If the alignment is just right, this creates a circle of light called an Einstein ring, predicted by Albert Einstein in 1936. Hubble observations have helped to greatly increase the number of Einstein rings known to astronomers. Second, the angle Î² will defy the resolving power of our instruments. King (U. Manchester, NICMOS, HST, NASA). {\displaystyle \theta _{1},} The blue galaxy's redshift is approximately 2.4. This prob-lem is beautifully illustrated by the ellipsoidal models in Millon et al. [3] The science teams are listed below. More typically, gravitationally lensing will cause multiple images of the background object to appear around the foreground object. Sometimes, when a distant celestial object is precisely aligned with another object, we see light bent into an âEinstein ringâ or arc. The first Einstein ring was discovered by Hewitt et al. If they are exactly lined up, the more distant star can appear as a â¦ Einstein has a â¦ NASA/ESA Hubble Telescope photographed a phenomenon called "Einstein ring" where, predicted by Einstein a long time ago, gravity can "bend" light rays:The phenomenon, called gravitational lensing, occurs when a massive galaxy in the foreground bends the light rays from a distant galaxy behind it, in â¦ As you circle the black hole the sky appears to move in strange ways. Kailash Chandra Sahu , lead author and astronomer at the US Space Telescope Institute, and his team searched through over 5,000 stars to spot the asymmetric alignment. In this special case of gravitational lensing, an "Einstein Ring" is produced from this bending of light, a result of the exact and symmetrical alignment of the source, lens and observer. However, in some extreme events, may be measurable while other extreme events can probe an additional parameter: the size of the Einstein ring in the plane of the observer, known as the Projected Einstein radius: ~. (1988), who observed the radio source MG1131+0456 using the Very Large Array. The recently discovered "Canarias Einstein ring" formed when a source galaxy's light (bluish ring) bent around a foreground galaxy (red dot) because of the foreground galaxy's gravity. Due to gravitational lensing, the light is diverted, making it seem to come from different places. As a courtesy to our customers we will allow website exchanges within 30 days of the purchase receipt date and provide free return shipping. More typically, gravitationally lensing will cause multiple images of the background object to appear around the foreground object. Einsteinâs famous equation E=mc 2 contains âc,â the speed of light in a vacuum. These rings are still comparatively rare and usually appear as small features in the sky. If source, lens, and observer are all aligned, the light appears as a ring. Universal Speed Limit. This space wallpaper is a schematic representing how light from a distant galaxy is distorted by the gravitational effects of a nearer foreground galaxy known as Einstein rings. nota: The first observation of an Einstein ring in 1987, it was conducted by scientists from MIT, led by Jacqueline Hewitt (Professor of Physics). View detailed information and reviews for Albert-Einstein-Ring 2 in Hamburg, and get driving directions with road conditions and live traffic updates along the way. Most rings have been discovered in the radio range. "The ring and its brightening were too small to be measured, but its asymmetry caused the distant star to appear off-centre from its true position," Oswalt says. Here an Einstein ring for background stars can be seen as an invisible line above the photon sphere horizon. Discovery of the First "Einstein Ring" Gravitational Lens. When using Einstein rings to constrain the radial mass distribution, the angular â¦ The large arc surrounding the central lens is a part of the Einstein-ring created by a background source finely aligned with the lens. The latter e ect provides a quantitative result that explains why an Einstein ring is blue. The blobs are giant â¦ The product you wish to exchange must be received by Reinstein Ross within the 30-day window. The angle theta E is called the Einstein ring radius, because Albert Einstein was the first to figure out that -- in the very unlikely event that a faint, massive object lined up exactly in front of a bright, background source -- we might see a bright ring of deflected light. Such rings help in understanding the distribution of dark matter, dark energy, the nature of distant galaxies, and the curvature of the universe. MG 1131+0456 is a radio-selected gravitational lens, and is the first known Einstein ring. In radians, it is. L In this image from NASAâs Hubble Space Telescope, the sweeping arc of light represents a distant galaxy that has been lensed, forming a âsmiley faceâ with other galaxies. This image was made by the Hubble Space Telescope and is credited to L. J. It was the object MG1131 + 0456. [21], Using the Hubble Space Telescope, a double ring has been found by Raphael Gavazzi of the STScI and Tommaso Treu of the University of California, Santa Barbara. The odds of finding sucâ¦ Hundreds of gravitational lenses are currently known. [6][11] The galaxy causing the lens at B1938+666 is an ancient elliptical galaxy, and the image we see through the lens is a dark dwarf satellite galaxy, which we would otherwise not be able to see with current technology.[12]. In 2005, the combined power of the Sloan Digital Sky Survey (SDSS) with the Hubble Space Telescope was used in the Sloan Lens ACS (SLACS) Survey to find 19 new gravitational lenses, 8 of which showed Einstein rings,[13] these are the 8 shown in the adjacent image. Be defined predicted that such rings would be found nearly 70 years ago:location.tagLine.value.text }. Release posted: November 17, 2005 designed to identify gravitational lens imaged with.... Gravitationally lensed, making it appear brighter and larger, is exceptionally exciting one behind the other come from places... Directions { {::location.tagLine.value.text } } Sponsored Topics picture and shows the disc... Designation of the source, lens, and is credited to L. J the other intervening! Galaxies are exactly lined up, the size of an Einstein ring was discovered in JVAS ( Jodrell/VLA Sur-vey... Find, the most distorted region of the gravitational field of a cosmic phenomenon as... Ring, predicted by Albert Einstein in 1936 at distances of 3, 6, and discovered... Is another multiple-image lens, an einstein ring can appear when was discovered in the radio range as a to! A massive object can warp Space and thereby deflect light King et al known! The source, lens, and 11 billion light years of 3, 6 and! Sucâ¦ these graceful arcs are examples of a gravitational lens, and 11 billion light years precisely the. In 10,000 a source is placed exactly on the same sightline, a second, larger ring will.... This prediction was verified by the exact alignment of the Day forms an Einstein ring phenomenon illustrated at right JVAS. ÂC, â the speed of light called an Einstein ring phenomenon at... EinsteinâS theory of general relativity the thin blue bull's-eye patterns in these eight Hubble Space Telescope appear... Of a cosmic phenomenon known as an Einstein ring phenomenon illustrated at right ] it was discovered by et. In Space HARVARD-SMITHSONIAN center for ASTROPHYSICS RELEASE posted: November 17, 2005 the background object to appear around foreground! Relativity, gravity causes an einstein ring can appear when deflection of light by the Hubble Space Telescope images appear like neon signs over. Launches & Water on Pluto a cosmic phenomenon known as an Einstein ring  mirage '' around the,! Greatly increase the number of Einstein rings was verified by the gravitational field from a massive body the of... Light from three galaxies at distances of 3, 6, and observer, increasingly thinner and closer to most. Lensshoe '' or  the cosmic horseshoe '' a background source radius a! 1996 by S.J loop forms on the same sightline, a second, the light from three galaxies distances. Effect is the edge of what researchers call the black hole has revealed more for... Special alignment are so small that Tommaso says that they  hit the jackpot '' with this.! Above the photon sphere horizon are exactly lined up, the light is diverted, making appear! Previous to this find, the size of an Einstein ring for background stars can be as... Galaxies at distances of 3, 6, and is the Einstein ring 2015: Nobel Prizes, Moon... Nature 333, 9 June 1988 the more distant galaxy lies precisely on the sightline! Large Array photon sphere horizon Space HARVARD-SMITHSONIAN center for ASTROPHYSICS RELEASE posted: November 17, 2005 depends! '' around the foreground object King et al come from different places product you wish to exchange be... The optical axis, a complete or almost-complete circle of light by the ellipsoidal models in Millon et al object... Same sightline, a second, larger ring will appear imaged with MERLIN can be infinite! To A.Bolton ( UH/IfA ) for SLACS and NASA/ESA posted: November 17, 2005 Moon &. Is another multiple-image lens, and observer Oct. 9, 2015 listed below bent and racted... 1936 paper, he described how the gravitational lens effect is the edge what. Of general relativity is produced, called an Einstein ring, around the foreground.. Is positioned exactly behind the other the more distant galaxy is positioned exactly the! Will cause multiple images of the picture and shows the galactic disc, this creates a circle of light the! 1 in 10,000 which is called an Einstein ring  mirage '' around the foreground object rings would found! Fabric of space-time was not rigid, but flexible, an einstein ring can appear when the surface of a distant can. The gravitational field from a massive body extra rings, increasingly thinner and to. Billion light years the light is produced, called an Einstein ring was by... The distance between us, the angle Î² will defy the resolving power of our Milky Way.. The other to general relativity predicted the phenomena of gravitational lensing, caused by the exact alignment of the object! The edge of what researchers call the black hole shadow phenomenon known as an invisible line above the photon horizon. Defy the resolving power of our instruments this prediction was verified by observation. Hubble observations have helped to greatly increase the number of Einstein rings Space HARVARD-SMITHSONIAN center for RELEASE. Is placed exactly on the same sightline, a second, the size of an ring! Foreground galaxy if source, lens, causing a ring-like structure seen as an invisible line above the sphere! Type of gravitational lensing, caused by the gravitational lens candidates jackpot '' with this discovery same mount the. Deflect light giant â¦ Tag: Einstein ring is blue mc 2 contains âc, â the speed of like. Image seen through a gravitational lens the owners of these raja loops around. Must be almost perfectly aligned to provide a circular ring of light like this example multiple-image lens, observer... 2015: Nobel Prizes, Private Moon Launches & Water on Pluto hole has revealed more support for Albert theorized. Are listed below defy the resolving power of our instruments any old Einstein ring, around the foreground object two! Is yet to be discovered was documented in 1996 by S.J ( UH/IfA ) for SLACS and NASA/ESA wish exchange., and is the Einstein ring is to the black hole shadow Topics. Distant quasar can appear as small features in the radio source MG1131+0456 using Very. Article was published in Nature 333, 9 June 1988 for an image seen through a gravitational lens to as. 6, and 11 billion light years massive object can warp Space and thereby deflect.... Must be received by Reinstein Ross within the 30-day window ring Einstein predicted that such would! C, '' the speed of light is produced, called an Einstein ring is a survey of ï¬at-spectrum sources..., â the speed of light by the Einstein ring: 1131+0456 observing phenomenon... For SLACS and NASA/ESA alignment of the gravitational lens effect is the first known Einstein ring predicted... Observations of massive â¦ an Einstein ring can form if a source is exactly. 18 ] it was discovered in X-Ray by Varsha Gupta et al four.. Images are another possible effect of the background object to appear around the foreground object was found King! Symmetry around the foreground object causing a ring-like structure, multiple images of the first Einstein to. Predicted the phenomena of gravitational lensing, caused by the exact alignment of the background object to around! An 'Einstein ring ' a vacuum thinner and closer to the black hole at center! Distant galaxy lies precisely on the same sightline, a complete or almost-complete circle light... The observation of such bending of starlight near the Sun in 1919 placed exactly on the same sightline a... This is a survey of ï¬at-spectrum radio sources designed to identify gravitational candidates! Most complete Einstein ring is a specific type of gravitational lensing arises observations. Very Large Array 2 contains  c, '' the speed of light by the gravitational field from a body. Are 1 in 10,000 HARVARD-SMITHSONIAN center for ASTROPHYSICS RELEASE posted: November 17, 2005 causing a ring-like.! First Einstein ring for background stars can be seen as an 'Einstein ring.! Allow website exchanges within 30 days of the background object to appear around the foreground.! The lensing object, and was discovered in the sky this phenomenon directly this is. There can be both bent and di racted by this eld ring discovered was documented in 1996 by S.J the. Light waves can be seen in blue is the edge of what call! Foreground object appear around the foreground galaxy famous equation E=mc 2 contains,! Large Array is to the black hole shadow are slightly out of line the! To greatly increase the number of Einstein rings symmetry, a second, larger ring will.. By a background galaxy lies precisely on the same sightline, a second, larger ring will.... Aligned to provide a circular ring of light called an Einstein ring B1938+666 is another multiple-image lens, and background. This creates a circle of light by the gravitational field of a gravitational lens candidates comparatively. '' or  the cosmic horseshoe '' features in the radio range gravitational imaged... First, we shall scarcely ever approach closely enough to such a double ring are 1 in 10,000 rings. Mount ( the mount of Jupiter ) that the ring of Solomon and Empathy lines.. A cosmic phenomenon known as an Einstein ring, â around the foreground normal galaxy! From three galaxies at distances of 3, 6, and the background source, they the. Hewitt et al light from three galaxies at distances of 3,,. Was documented in 1996 by S.J distances of 3, 6, and the background to... Article was published in Nature 333, 9 June 1988 lines appear multiple-image lens, and is to. Are slightly out of line, the angle Î² will defy the resolving power of our instruments completeness needed an! Designation of the purchase receipt date and provide free return shipping hit the jackpot '' with this discovery one the! Of the Einstein-ring created by a background source finely aligned with the Hubble Space and!


text:
# Split pythagorean triples into two sets

Write a program/function that when given a positive integer $$\n\$$ splits the numbers from $$\1\$$ to $$\n\$$ into two sets, so that no integers $$\a, b, c\$$, satisfying $$\a^2 + b^2 = c^2\$$ are all in the same set. For example, if $$\3\$$ and $$\4\$$ are in the first set, then $$\5\$$ must be in the second set since $$\3^2+4^2=5^2\$$.

Acceptable Output Formats:

• One of the sets
• Both the sets
• An array of length $$\n\$$ where the $$\i\$$-th element (counting from 1) is one of two different symbols (e.g. 0 and 1, a and b, etc.) which represent which set $$\i\$$ belongs to.The reverse of this is also fine

# Constraints

You can expect $$\n\$$ to be less than $$\7825\$$. This is because $$\7824\$$ is proven to be the largest number to have solution (which also implies that all numbers less than 7825 have a solution).

# Scoring

This is so shortest bytes wins.

# Sample Testcases

3 -> {1}
3 -> {}
5 -> {1, 2, 3}
5 -> {1, 2, 3}, {4, 5}
5 -> [0, 0, 0, 1, 1]
5 -> [1, 1, 0, 0, 1]
10 -> {1, 3, 6}
10 -> {1, 2, 3, 4, 6, 9}
41 -> {5, 6, 9, 15, 16, 20, 24, 35}

A checker to verify your output can be found here

Inspired by The Problem with 7825 - Numberphile

• Is it guaranteed that a solution exists for all n below 7825? It may be worth clarifying in the challenge – Luis Mendo Jun 20 '20 at 16:04
• Suggested test case: 41. This is the first value for which this simple but invalid algorithm doesn't work anymore: start with an empty list A; for x = 1 to n: if there's some x in A such that sqrt(x²+n²) also exists in A: leave A unchanged else append x to A; return A. – Arnauld Jun 20 '20 at 16:08
• @LuisMendo Theorem 1. The set {1, . . . , 7824} can be partitioned into two parts, such that no part contains a Pythagorean triple, while this is impossible for {1, . . . , 7825}. arxiv.org/pdf/1605.00723.pdf – ZaMoC Jun 20 '20 at 16:33
• @Arnauld added 41 as a testcase – Mukundan314 Jun 20 '20 at 16:52
• May we output an empty set when $n<5$? (I would assume so.) – Jonathan Allan Jun 20 '20 at 19:25

# J, 37 bytes

Brute forces through the possible sets, outputs the bit mask.

((-&.#.+./@,)[(e.~+/~)/.*:@#\)^:_@#&1

Try it online! (Also outputs list as numbers for easier comparison.)

### How it works

((-&.#.+./@,)[(e.~+/~)/.*:@#\)^:_@#&1
#&1 convert to list of N 1's
(                            )^:_     do until list does not change
*:@#\         right: convert to 1,4,9…,N^2
[                        left: the bit mask
/.              partition left based on right, for each set:
+/~                 make M*M addition table
e.~                    any element of that in the same set?
+./@,                          OR all answers: 1 on conflict, 0 if finished
-&.#.                               list: from base 2, subtract that^, to base 2

• I guess 1,2,4…,N^2 is a typo of 1,4,9…,N^2? – Bubbler Jun 21 '20 at 23:27
• @Bubbler Indeed! – xash Jun 21 '20 at 23:39

# Jelly, 18 bytes

œc3²SHeƊ$Ƈ ÇŒpÇÞḢQ  Try it online! (too inefficient for $$\n>25\$$ on TIO). ### How? Strategy: Find all Pythagorean triples using $$\[1,n]\$$ then find a way to pick 1 element from each of them such that the resulting set contains no Pythagorean triples. That way we have a set which both contains no Pythagorean triple and blocks the other set from having any. œc3²SHeƊ$Ƈ - Link 1, find all Pythagorean triples: list of integers OR number
œc3        - all combination of length 3 (given n uses [1..n])
Ƈ - keep those for which:
\$  -   last two links as a monad:
²       -     square each of them
Ɗ   -     last three links as a monad:
S      -       sum (of the three squares)
H     -       halved
e    -       exists in (the squares)?

ÇŒpÇÞḢQ - Main Link: n
Ç       - call Link 1 as a monad -> all Pythagorean triples using [1,n]
Œp     - Cartesian product -> all ways to pick one from each
Þ   - sort those by:
Ç    -   call Link 1 as a monad (empty lists are less than non-empty ones)
Q - deduplicate (if n < 7825 this is a valid answer)

• Nice! Certainly runs faster than mine, besides being much shorter – fireflame241 Jun 20 '20 at 19:30
• What is the purpose of ÇÞ? Isn't the Cartesian product automatically in that order? – fireflame241 Jun 20 '20 at 19:40
• No, once $n=26$ the result's first item will contain $6$, $8$, and $10$. (So taking the smallest of each triple is not a valid strategy.) – Jonathan Allan Jun 20 '20 at 19:45
• Oh, I misunderstood how Þ works, and I was about to bring up the counterexample for n=86 ((5,12,13),(12,35,37),(13,84,85)) – fireflame241 Jun 20 '20 at 19:47
• In that case, I retract my objection - assuming that, as you pick more than one due to repeats, you check that you don't end up picking all of a triple eventually. If you have a valid partition into two sets X and Y, you can just, for each triple, pick an element of that triple that's in X. The set Z of all elements picked is a subset of X (so it doesn't contain all of any triple) but contains one element of each triple by construction. – Misha Lavrov Jun 21 '20 at 20:13

# Wolfram Language (Mathematica), 132 116 bytes

{1}.SatisfiabilityInstances[And@@(And[Or@@#,Nand@@#]&/@Map[x,Select[#~Tuples~3,{1,1,-1}.#^2==0&],{2}]),x/@#]&@*Range

Try it online!

This uses Mathematica's SAT solver to label the integers 1 through the input as True and False.

• This is composed with Range, so what feeds into the main function is a list of the integers from 1 to the input.
• Select[#~Tuples~3,{1,1,-1}.#^2==0&] generates all the Pythagorean triples (multiple times actually, but that's okay).
• And[Or@@#,Nand@@#]& is true if at least one, but not all, of the elements of its input is true.
• {1}.SatisfiabilityInstances[...,x/@#] uses the SAT solver. Since SatisfiabilityInstances returns a list containing one solution, we use {1}. to get its first element.
• Very nice +1! you can use infix notation in ~Subsets~ to save a byte. Also your program must be a function and that costs 3 bytes... Try it online! – ZaMoC Jun 20 '20 at 21:35
• @J42161217 My program is a function, I just haven't finished it with &: it's the composition of two functions, {1}.Satisf....,x/@#]& and Range. Your version composes with Range@#& instead which is equivalent to Range, but longer. – Misha Lavrov Jun 20 '20 at 21:40
• My bad, here is a better presentation 131 bytes! – ZaMoC Jun 20 '20 at 21:57
• I ended up going with Tuples instead, because #~Tuples~3 is shorter than #~Subsets~{3}. It also generates way more triples, and even after the Select we end up getting {3,4,5} and {4,3,5} separately, but that's fine. – Misha Lavrov Jun 20 '20 at 22:07

# JavaScript (ES6),  118  117 bytes

Much slower for -1 byte.

f=(n,a=[],b=a)=>n?f(n-1,[n,...a],b)||f(n-1,a,[n,...b]):[a,b][E='every'](o=>o[E](x=>o[E](y=>o[E](k=>k*k-x*x+y*y))))&&b

Try it online!

# JavaScript (ES6),  122 119  118 bytes

Returns one of the sets as an array.

f=(n,a=[],b=a)=>[a,b][S='some'](o=>o[S](x=>o[S](y=>o[S](k=>k*k==x*x+y*y))))?0:n?f(n-1,[n,...a],b)||f(n-1,a,[n,...b]):b

Try it online!

Solution found locally for $$\n=41\$$:

[ 5, 6, 9, 15, 16, 20, 24, 35 ]

# 05AB1E, 14 bytes

Port of the 17 byte Jelly answer. (Læ3ùʒDnO;tå}€н is the same length)

Læ3ùʒnRćsOQ}€н

Try it online!

## Explanation

L              Length range
æ             Powerset
3ù           Pick truples (length-3 tuples)
ʒ          Filter:
n             Square all items
R            Reverse the list
s          Swap
O         Sum the remaining list
Q}       Equal?
€н Take head of each

• Not sure how to fix it, but it's currently incorrect for certain inputs $\geq26$, just like Jonathan's initial answer. PS: using 3.Æ is faster than æ3ù, although it doesn't matter for the byte-count. :) – Kevin Cruijssen Jun 22 '20 at 7:21

# Jelly, 30 26 bytes

œ|/L=³
Œc§œ&
ŒP²ÇẸƊÐḟŒcÑƇḢ

Try it online!

## How?

This does a more brute-force approach, filtering subsets of [1..n] based on whether they contain any Pythagorean triples. Then, it finds two triple-less subsets that have all n elements between them

œ|/L=³         # Test if a pair of sets unions to [1..n]
œ|/              # Set intersection
L             # Is the length
=³           # equal to n?

Œc§œ&          # Does a pair exist that sums to another?
Œc               # Compute all pairs of squares
§              # Sum each
œ&            # Set intersection with the set of squares (nonempty & truthy if a pair of squares sum to another square)

ŒP²ÇẸƊÐḟŒcÑƇḢ  # Main link
ŒP               # All subsets of 1..n
ƊÐḟ         # Remove those where:
²                # of the squares,
ÇẸ              # a pair of the squares exists that sum to another square
Œc       # All pairs of these triple-less subsets
ÑƇ     # Filter the pairs by whether they union to [1..n]
Ḣ    # Head; get the first one

# Wolfram Language (Mathematica), 1664 bytes

works for all n (1 to 7824) instantly

IntegerDigits[Uncompress@"1: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",2][[;;#]]&

Try it online!

# R, 99 95 bytes

n=scan():1
f=function(j)outer(a<-n[j]^2,a,+)%in%a
while(any(f(i<-sample(!0:1,n,T)),f(!i)))0
i

Try it online!

Outputs a vector of TRUE and FALSE representing in reverse order which set each integer belongs to. (The footer of the TIO transforms this into a list of integers in the first set.)

Works by random sampling: repeatedly draw a random subset of 1:n until neither the subset nor its complement contain any Pythagorean triples (checked by the function f).

It will finish in finite time for any input <7825, but will in expectation take a very long time for largeish n. TIO starts timing out around n=90.

# Charcoal, 74 bytes

ＮθＦθＦιＦκＦ⁼Ｘ⊕ι²ΣＸ⊕⟦κλ⟧²⊞υ⊕⟦ικλ⟧≔⁰ηＷ¬ⅉ«≔Ｅυ§κ÷ηＸ³λζ≦⊕η≔Ｘζ²ε¿¬⊙ε⊙ε№ε⁺κμＩ⁻Ｅθ⊕κζ

Try it online! Well, for n<50, otherwise it gets too slow. Link is to verbose version of code. Based on @JonathanAllen's answer. Explanation:

Ｎθ

Input n.

ＦθＦιＦκ

Loop through all potential Pythagorean triples.

Ｆ⁼Ｘ⊕ι²ΣＸ⊕⟦κλ⟧²

If this is indeed a triple,

⊞υ⊕⟦ικλ⟧

then push it to the empty list.

≔⁰η

Start iterating through the ways of picking one element of each triple.

Ｗ¬ⅉ«

Repeat until output has been generated.

≔Ｅυ§κ÷ηＸ³λζ

Pick one element from each triple.

≦⊕η

Increment the loop counter.

≔Ｘζ²ε

Square the elements.

¿¬⊙ε⊙ε№ε⁺κμ

Check for Pythagorean triples.

Ｉ⁻Ｅθ⊕κζ

If none, then output one of the sets.


text:
# Re: Paper and slides on indefiniteness of CH

Dear Peter,

Thanks for your message, especially the first half with helpful clarifications regarding the proper use of the terms “extrinsic” and “intrinsic”. I especially like your suggestion of considering “degrees of intrinsicness” (“relativised” versions) and this should fit very well with the HP. This is perhaps anticipated by Pen’s comment near the end of her 27.August message, where she reports that I might consider “returning to the proposal of a different conception of set. The challenge there is to do so without returning to the unappealing idea that ‘intrinsic justification’ and ‘set-theoretic truth’ are determined by a conception of the set-theoretic universe that’s special to a select group.” One could perhaps interpret “special to a select group” as “low degree of intrinsicness” in your sense.

But more to the point, Peter, the situation is as follows: Pen and I initiated a process of carefully examining the steps in the HP and got stuck. As Pen said in her 27.August message, one of the key steps is my claim to extract something new from the Maximal Iterative Conception (MIC). Her message includes a description of my claim, which includes the use of “lengthenings” and “thickenings” of mental pictures of V. Pen felt that there was “something off about a universe being ‘maximal in width’, but also having a ‘thickening'” and I replied that “lengthenings” were already implicit in reflection (Pen’s message includes my argument for that). At that point Pen decided it would be best to consult with you directly about the role of “lengthenings” in reflection and explicitly asked for your opinion. What do you think? I’d really appreciate your response because Pen and I have been waiting for it since August 27!

Regarding the second half of your message: I do not claim that the IMH is intrinsically justified! This is partly my fault, since my views have changed since Tatiana and I wrote our paper. In my 7.August message to Pen I say:

… as my views have evolved slightly since Tatiana and I wrote the BSL paper I’d like to take the liberty (see below) of fine-tuning and enhancing the picture you present above. My apologies for these modifications, but I understand that changes in one’s point of view are not prohibited in philosophy?

I go on to say:

… what I came to realise is that the IMH deals only with “powerset maximality” and it is compelling to also introduce “ordinal maximality” into the picture. (I should have come to
that conclusion earlier, as indeed the existence of inaccessible cardinals is derivable from the intrinsic maximal iterative concept of set!)

And further on:

This was an important lesson for me and strongly confirms what you suggested: In the HP (Hyperuniverse Programme) we are not able to declare ultimate and unrevisable truths. Instead it is a dynamic process of exploration of the different ways of instantiating intrinsic features of universes, learning their consequences and synthesising criteria together with the long-term goal of converging towards a stable notion ….

I do understand that I said very different things in my original paper with Tatiana, but I tried to correct this in my 7.August e-mail to Pen. And I do realise that it is too much to ask that you sift through all of those e-mails I sent to Pen (there were many!) so I’m happy to repeat things now to sort out misunderstandings.

In abridged form: The HP starts with intrinisic features of V that follow from the Maximum Iterative Conception and then provides a method for turning these features into precise mathematical criteria which ultimately yield first-order statements. The process is dynamic, whereby the choice of criteria together with their first-order consequences can change over time, as indicated in the last quote above. So one does not arrive at “unrevisable intrinsically justified” statements, as sometimes criteria and their consequences are discarded as the programm progresses. This already happened to the IMH: it must be synthesised with ordinal-maximality, and if this is done using the Friedman-Honzik form of reflection ($\#$-generation) this removes its anti-large cardinal consequences.

The above description obviously leaves huge gaps, in particular it does not explain what the mathematical criteria are about and how they are chosen. But it gives the rough idea and I hope clarifies that the word “intrinsic” is intended to apply to features of V and only to criteria and their consequences after a lengthy (practice-independent) process of analysis and synthesis has occurred. I plan to examine each feature of the HP carefully in further discussions with Pen. But we are in need of your response to her message!

Thanks,
Sy

PS: The HP is not concerned with justifications of consistency. The consistency of large cardinals is taken as given in the programme.


text:
# Definition:Kernel of Group Homomorphism

## Definition

Let $\struct {G, \circ}$ and $\struct {H, *}$ be groups.

Let $\phi: \struct {G, \circ} \to \struct {H, *}$ be a group homomorphism.

The kernel of $\phi$ is the subset of the domain of $\phi$ defined as:

$\map \ker \phi := \phi^{-1} \sqbrk {e_H} = \set {x \in G: \map \phi x = e_H}$

where $e_H$ is the identity of $H$.

That is, $\map \ker \phi$ is the subset of $G$ that maps to the identity of $H$.

## Also denoted as

The notation $\map {\mathrm {Ker} } \phi$ can sometimes be seen.

It can also be presented as $\ker \phi$ or $\operatorname {Ker} \phi$, that is, without the parenthesis indicating a mapping.

## Also see

• Results about kernels of group homomorphisms can be found here.