70 Matching Annotations
  1. Last 7 days
    1. It would allow end users to determine their own tolerances for different types of speech but make it much easier for most people to avoid the most problematic speech, without silencing anyone entirely or having the platforms themselves make the decisions about who is allowed to speak.

      But platforms are making huge decisions about who is allowed to speak. While they're generally allowing everyone to have a voice, they're also very subtly privileging many voices over others. While they're providing space for even the least among us to have a voice, they're making far too many of the worst and most powerful among us logarithmic-ally louder.

      It's not broadly obvious, but their algorithms are plainly handing massive megaphones to people who society broadly thinks shouldn't have a voice at all. These megaphones come in the algorithmic amplification of fringe ideas which accelerate them into the broader public discourse toward the aim of these platforms getting more engagement and therefore more eyeballs for their advertising and surveillance capitalism ends.

      The issue we ought to be looking at is the dynamic range between people and the messages they're able to send through social platforms.

      We could also analogize this to the voting situation in the United States. When we disadvantage the poor, disabled, differently abled, or marginalized people from voting while simultaneously giving the uber-rich outsized influence because of what they're able to buy, we're imposing the same sorts of problems. Social media is just able to do this at an even larger scale and magnify the effects to make their harms more obvious.

      If I follow 5,000 people on social media and one of them is a racist-policy-supporting, white nationalist president, those messages will get drowned out because I can only consume so much content. But when the algorithm consistently pushes that content to the top of my feed and attention, it is only going to accelerate it and create more harm. If I get a linear presentation of the content, then I'd have to actively search that content out for it to cause me that sort of harm.

    2. Meanwhile, politicians from the two major political parties have been hammering these companies, albeit for completely different reasons. Some have been complaining about how these platforms have potentially allowed for foreign interference in our elections.3 3. A Conversation with Mark Warner: Russia, Facebook and the Trump Campaign, Radio IQ|WVTF Music (Apr. 6, 2018), https://www.wvtf.org/post/conversation-mark-warner-russia-facebook-and-trump-campaign#stream/0 (statement of Sen. Mark Warner (D-Va.): “I first called out Facebook and some of the social media platforms in December of 2016. For the first six months, the companies just kind of blew off these allegations, but these proved to be true; that Russia used their social media platforms with fake accounts to spread false information, they paid for political advertising on their platforms. Facebook says those tactics are no longer allowed—that they've kicked this firm off their site, but I think they've got a lot of explaining to do.”). Others have complained about how they’ve been used to spread disinformation and propaganda.4 4. Nicholas Confessore & Matthew Rosenberg, Facebook Fallout Ruptures Democrats’ Longtime Alliance with Silicon Valley, N.Y. Times (Nov. 17, 2018), https://www.nytimes.com/2018/11/17/technology/facebook-democrats-congress.html (referencing statement by Sen. Jon Tester (D-Mont.): “Mr. Tester, the departing chief of the Senate Democrats’ campaign arm, looked at social media companies like Facebook and saw propaganda platforms that could cost his party the 2018 elections, according to two congressional aides. If Russian agents mounted a disinformation campaign like the one that had just helped elect Mr. Trump, he told Mr. Schumer, ‘we will lose every seat.’”). Some have charged that the platforms are just too powerful.5 5. Julia Carrie Wong, #Breaking Up Big Tech: Elizabeth Warren Says Facebook Just Proved Her Point, The Guardian (Mar. 11, 2019), https://www.theguardian.com/us-news/2019/mar/11/elizabeth-warren-facebook-ads-break-up-big-tech (statement of Sen. Elizabeth Warren (D-Mass.)) (“Curious why I think FB has too much power? Let's start with their ability to shut down a debate over whether FB has too much power. Thanks for restoring my posts. But I want a social media marketplace that isn't dominated by a single censor. #BreakUpBigTech.”). Others have called attention to inappropriate account and content takedowns,6 6. Jessica Guynn, Ted Cruz Threatens to Regulate Facebook, Google and Twitter Over Charges of Anti-Conservative Bias, USA Today (Apr. 10, 2019), https://www.usatoday.com/story/news/2019/04/10/ted-cruz-threatens-regulate-facebook-twitter-over-alleged-bias/3423095002/ (statement of Sen. Ted Cruz (R-Tex.)) (“What makes the threat of political censorship so problematic is the lack of transparency, the invisibility, the ability for a handful of giant tech companies to decide if a particular speaker is disfavored.”). while some have argued that the attempts to moderate discriminate against certain political viewpoints.

      Most of these problems can all fall under the subheading of the problems that result when social media platforms algorithmically push or accelerate content on their platforms. An individual with an extreme view can publish a piece of vile or disruptive content and because it's inflammatory the silos promote it which provides even more eyeballs and the acceleration becomes a positive feedback loop. As a result the social silo benefits from engagement for advertising purposes, but the community and the commons are irreparably harmed.

      If this one piece were removed, then the commons would be much healthier, fringe ideas and abuse that are abhorrent to most would be removed, and the broader democratic views of the "masses" (good or bad) would prevail. Without the algorithmic push of fringe ideas, that sort of content would be marginalized in the same way we want our inane content like this morning's coffee or today's lunch marginalized.

      To analogize it, we've provided social media machine guns to the most vile and fringe members of our society and the social platforms are helping them drag the rest of us down.

      If all ideas and content were provided the same linear, non-promotion we would all be much better off, and we wouldn't have the need for as much human curation.

  2. Dec 2019
    1. Computers can only natively store integers, so they need some way of representing decimal numbers. This representation comes with some degree of inaccuracy. That's why, more often than not, .1 + .2 != .3

      Computers make up their way to store decimal numbers

    1. I literally couldn’t remember when I’d last looked at my RSS subscriptions. On the surface, that might seem like a win: Instead of painstakingly curating my own incoming news, I can effortlessly find an endless supply of interesting, worthwhile content that the algorithm finds for me. The problem, of course, is that the algorithm isn’t neutral: It’s the embodiment of Facebook and Twitter’s technology, data analysis, and most crucial, business model. By relying on the algorithm, instead of on tags and RSS, I’m letting an army of web developers, business strategists, data scientists, and advertisers determine what gets my attention. I’m leaving myself vulnerable to misinformation, and manipulation, and giving up my power of self-determination.
  3. Sep 2019
    1. Safiya Noble, Algorithms of Oppression (New York: New York University Press, 2018). See also Mozilla’s 2019 Internet Health Report at https://internethealthreport.org/2019/lets-ask-more-of-ai/.
  4. Jul 2019
    1. One major idea in mathematics is the idea of “closure”. This is the ques-tion: What is the set of all things that can result from my proposed oper-ations? In the case of vectors: What is the set of vectors that can result bystarting with a small set of vectors, and adding them to each other andscaling them? This results in a vector space

      closure in mathematics. sounds similar to domain of a function

    1. In Hardy's words, "Exposition, criticism, appreciation, is work for second-rate minds. [...] It is a melancholy experience for a professional mathematician to find himself writing about mathematics. The function of a mathematician is to do something, to prove new theorems, to add to mathematics, and not to talk about what he or other mathematicians have done."

      similar to Nassim Taleb's "History is written by losers"

    1. A spokeswoman for Summit said in an e-mail, “We only use information for educational purposes. There are no exceptions to this.” She added, “Facebook plays no role in the Summit Learning Program and has no access to any student data.”

      As if Facebook needed it. The fact that this statement is made sort of goes to papering over the idea that Summit itself wouldn't necessarily do something as nefarious or worse with it than Facebook might.

  5. May 2019
    1. Meta co-founder and CEO Sam Molyneux writes that “Going forward, our intent is not to profit from Meta’s data and capabilities; instead we aim to ensure they get to those who need them most, across sectors and as quickly as possible, for the benefit of the world.”

      Odd statement from a company that was just acquired by Facebook founder's CVI.

    1. If you do the math

      Here is the math: 2020-2023 4 years (Jan 1-Dec 31). 1 million annual is 4 million. That is way more than the 1.7 million

    2. four

      3 million vehicles divided by 4 years divided by 500,000 per factory makes less than two factories in my book.

    1. “It sure smells like the prescreening provisions of the FCRA,” Reidenberg told The Intercept. “From a functional point of view, what they’re doing is filtering Facebook users on creditworthiness criteria and potentially escaping the application of the FCRA.”
    2. In an initial conversation with a Facebook spokesperson, they stated that the company does “not provide creditworthiness services, nor is that a feature of Actionable Insights.” When asked if Actionable Insights facilitates the targeting of ads on the basis of creditworthiness, the spokesperson replied, “No, there isn’t an instance where this is used.” It’s difficult to reconcile this claim with the fact that Facebook’s own promotional materials tout how Actionable Insights can enable a company to do exactly this. Asked about this apparent inconsistency between what Facebook tells advertising partners and what it told The Intercept, the company declined to discuss the matter on the record,
    3. How consumers would be expected to navigate this invisible, unofficial credit-scoring process, given that they’re never informed of its existence, remains an open question.
    4. But these lookalike audiences aren’t just potential new customers — they can also be used to exclude unwanted customers in the future, creating a sort of ad targeting demographic blacklist.
  6. Apr 2019
    1. A 2015 clip about vaccination from iHealthTube.com, a “natural health” YouTube channel, is one of the videos that now sports a small gray box.

      Does this box appear on the video itself? Apparently not...

      Examples:

      But nothing on the embedded version:

      A screengrab of what this looks like:

    2. YouTube doesn’t give an exact recipe for virality. But in the race to one billion hours, a formula emerged: Outrage equals attention.

      Talk radio has had this formula for years and they've almost had to use it to drive any listenership as people left radio for television and other media.

      I can still remember the different "loudness" level of talk between Bill O'Reilly's primetime show on Fox News and the louder level on his radio show.

    3. When Wojcicki took over, in 2014, YouTube was a third of the way to the goal, she recalled in investor John Doerr’s 2018 book Measure What Matters.“They thought it would break the internet! But it seemed to me that such a clear and measurable objective would energize people, and I cheered them on,” Wojcicki told Doerr. “The billion hours of daily watch time gave our tech people a North Star.” By October, 2016, YouTube hit its goal.

      Obviously they took the easy route. You may need to measure what matters, but getting to that goal by any means necessary or using indefensible shortcuts is the fallacy here. They could have had that North Star, but it's the means they used by which to reach it that were wrong.

      This is another great example of tech ignoring basic ethics to get to a monetary goal. (Another good one is Marc Zuckerberg's "connecting people" mantra when what he should be is "connecting people for good" or "creating positive connections".

    4. The conundrum isn’t just that videos questioning the moon landing or the efficacy of vaccines are on YouTube. The massive “library,” generated by users with little editorial oversight, is bound to have untrue nonsense. Instead, YouTube’s problem is that it allows the nonsense to flourish. And, in some cases, through its powerful artificial intelligence system, it even provides the fuel that lets it spread.#lazy-img-336042387:before{padding-top:66.68334167083543%;}

      This is a great summation of the issue.

    5. Somewhere along the last decade, he added, YouTube prioritized chasing profits over the safety of its users. “We may have been hemorrhaging money,” he said. “But at least dogs riding skateboards never killed anyone.”
  7. Mar 2019
    1. pi = 3.1415 ... e = 2.718 ... Euler's constant, gamma = 0.577215 ... = lim n -> infinity > (1 + 1/2 + 1/3 + 1/4 + ... + 1/n - ln(n)) (Not proven to be transcendental, but generally believed to be by mathematicians.) Catalan's constant, G = sum (-1)^k / (2k + 1 )^2 = 1 - 1/9 + 1/25 - 1/49 + ... (Not proven to be transcendental, but generally believed to be by mathematicians.) Liouville's number 0.110001000000000000000001000 ... which has a one in the 1st, 2nd, 6th, 24th, etc. places and zeros elsewhere. Chaitin's "constant", the probability that a random algorithm halts. (Noam Elkies of Harvard notes that not only is this number transcendental but it is also incomputable.) Chapernowne's number, 0.12345678910111213141516171819202122232425... This is constructed by concatenating the digits of the positive integers. (Can you see the pattern?) Special values of the zeta function, such as zeta (3). (Transcendental functions can usually be expected to give transcendental results at rational points.) ln(2). Hilbert's number, 2(sqrt 2 ). (This is called Hilbert's number because the proof of whether or not it is transcendental was one of Hilbert's famous problems. In fact, according to the Gelfond-Schneider theorem, any number of the form ab is transcendental where a and b are algebraic (a ne 0, a ne 1 ) and b is not a rational number. Many trigonometric or hyperbolic functions of non-zero algebraic numbers are transcendental.) epi pie (Not proven to be transcendental, but generally believed to be by mathematicians.) Morse-Thue's number, 0.01101001 ... ii = 0.207879576... (Here i is the imaginary number sqrt(-1). Isn't this a real beauty? How many people have actually considered rasing i to the i power? If a is algebraic and b is algebraic but irrational then ab is transcendental. Since i is algebraic but irrational, the theorem applies. Note also: ii is equal to e(- pi / 2 ) and several other values. Consider ii = e(i log i ) = e( i times i pi / 2 ) . Since log is multivalued, there are other possible values for ii. Here is how you can compute the value of ii = 0.207879576... 1. Since e^(ix) = Cos x + i Sin x, then let x = Pi/2. 2. Then e^(iPi/2) = i = Cos Pi/2 + i Sin Pi/2; since Cos Pi/2 = Cos 90 deg. = 0. But Sin 90 = 1 and i Sin 90 deg. = (i)*(1) = i. 3. Therefore e^(iPi/2) = i. 4. Take the ith power of both sides, the right side being i^i and the left side = [e^(iPi/2)]^i = e^(-Pi/2). 5. Therefore i^i = e^(-Pi/2) = .207879576... Feigenbaum numbers, e.g. 4.669 ... . (These are related to properties of dynamical systems with period-doubling. The ratio of successive differences between period-doubling bifurcation parameters approaches the number 4.669 ... , and it has been discovered in many physical systems before they enter the chaotic regime. It has not been proven to be transcendental, but is generally believed to be.)
  8. Jan 2019
    1. A rooted binary tree is full if every vertex has either two children or no children.

      Example of Catalan Numbers use case.

    1. Catalan numbers notation and short explanation of it.

      Use - LookUP: Combinatorics (non crossing combinations) ex: ((())), ()(()), ()()(), (())(), (()())

      ref: https://www.geeksforgeeks.org/program-nth-catalan-number/

  9. Dec 2018
    1. With Alphabet Inc.’s Google, and Facebook Inc. and its WhatsApp messaging service used by hundreds of millions of Indians, India is examining methods China has used to protect domestic startups and take control of citizens’ data.

      Governments owning citizens' data directly?? Why not have the government empower citizens to own their own data?

    1. computer owned by Patrick Laroche of Ocala, Fla., discovered the number on Dec. 7,

      Newest Mersenne Prime was discovered in Ocala, FL, very close to us.

      Interesting tidbit :)

  10. Nov 2018
    1. Начните с нескольких основных книг, которые изменят ваши представления о математикеЛучше всего «ставит голову» книга «Начала теории множеств» Николая Верещагина и Александра Шеня. Она даст основу для понимания (а в дальнейшем и самостоятельного построения!) логических рассуждений. С нее же начнется и понимание теории множеств, лежащей в основе современной математики.Охватить больше разделов математики поможет книга Рихарда Куранта и Герберта Роббинса «Что такое математика?». Как и книгу Верещагина и Шеня, эту книгу нужно читать внимательно, делая все упражнения. Если первые две книги окажутся сложными, можете начать с моей «Математики для гуманитариев». Ее также следует читать с самого начала, страницу за страницей, не стоит браться за чтение с середины. Она не очень простая, но предварительных сведений и математической культуры не предполагает.Чтобы понять, каким образом математика входит в нашу жизнь, можно прочитать «Кому нужна математика?» Андрея Райгородского и Нелли Литвак или «Математическую составляющую» (сборник сюжетов под редакцией Николая Андреева).

      nice books

  11. Oct 2018
    1. A more active stance by librarians, journalists, educators, and others who convey truth-seeking habits is essential.

      In some sense these people can also be viewed as aggregators and curators of sorts. How can their work be aggregated and be used to compete with the poor algorithms of social media?

  12. Sep 2018
  13. Jun 2018
    1. Having low scores posted for all coworkers to see was “very embarrassing,” said Steph Buja, who recently left her job as a server at a Chili’s in Massachusetts. But that’s not the only way customers — perhaps inadvertently — use the tablets to humiliate waitstaff. One diner at Buja’s Chili’s used Ziosk to comment, “our waitress has small boobs.”According to other servers working in Ziosk environments, this isn’t a rare occurrence.

      This is outright sexual harrassment and appears to be actively creating a hostile work environment. I could easily see a class action against large chains and/or against the app maker themselves. Aggregating the data and using it in a smart way is fine, but I suspect no one in the chain is actively thinking about what they're doing, they're just selling an idea down the line.

      The maker of the app should be doing a far better job of filtering this kind of crap out and aggregating the data in a smarter way and providing a better output since the major chains they're selling it to don't seem to be capable of processing and disseminating what they're collecting.

    2. Systems like Ziosk and Presto allow customers to channel frustrations that would otherwise end up on public platforms like Yelp — which can make or break a restaurant — into a closed system that the restaurant controls.

      I like that they're trying to own and control their own data, but it seems like they've relied on a third party company to do most of the thinking for them and they're not actually using the data they're gathering in the proper ways. This is just painfully deplorable.

  14. May 2018
    1. eight years after release, men are 43% more likely to be taken back under arrest than women; African-Americans are 42% more likely than whites, and high-school dropouts are three times more likely to be rearrested than college graduates.

      but are these possibly the result of external factors (like racism?)

  15. Apr 2018
    1. we found advanced math and science courses advertised but not actually offered and specialized STEM programs eroded or discontinued within a few years of inception.

      Some interesting data points in this article.

    1. Yet, across the country, 2 in 5 high schools don't offer physics, according to an Education Week Research Center analysis of data from the U.S. Department of Education's office for civil rights.

      How widely known is this figure?

  16. arxiv.org arxiv.org
    1. n this note we determine thetorsion subgroup of the additive group of (1.1) in the case whereLhas rank 2, thatisXis a set of two elements, andcis a prime number

      Wow!

    1. Of particular interest are the lower central series: G1=GG1=GG_1 =G, Gi+1=[Gi,G]Gi+1=[Gi,G]G_{i+1} = [G_i, G], i≥1i≥1i \ge 1, and, for a fixed prime number p, the Zassenhaus series (see [7, 8]).

      This is interesting.

    1. Leonhard Euler, who adopted it in 1737

      Actually, he first used π in 1727, and it meant 6.28.., not 3.14..

  17. Mar 2018
      • R/(gg)plots visualise => linguists detect interesting phenomena to analyse further
      • Wang2Vec: train backprogagation network
  18. Feb 2018
    1. Let’s use some common units as examples: gram (g), erg (erg), and solar mass per cubic megaparsec (Msun / Mpc33^3). g is an atomic, CGS base unit, erg is an atomic unit in CGS, but is not a base unit, and Msun/Mpc33^3 is a combination of atomic units, which are not in CGS, and one of them even has an SI prefix. The dimensions of g are mass and the cgs factor is 1. The dimensions of erg are mass * length$^2$ * time−2−2^{-2} and the cgs factor is 1. The dimensions of Msun/Mpc33^3 are mass / length33^3 and the cgs factor is about 6.8e-41.
  19. Jul 2017
    1. satisfy the two usual distributive laws. Singlesemirings as well as classes of semirings form important structures in Automataand Formal Languages Theories [5

      Like so.

  20. May 2017
    1. 1st order Eulerian numbers:Permutations

      Annotate math with math!

      $$\varepsilon = \frac{2}{h^3} \int_0^{p_F} \sqrt{p^2 c^2 + m^2 c^4} \cdot 4 \pi p^2 dp=$$

      $$\frac{8 \pi}{h^3} \frac{m c^2}{\lambda^3} \int_0^x \sqrt{1+y^2} \cdot y^2 dy $$

    1. The first review, by C. Hendricks Brown et al., poses the issues raised by the growingrecognition

      $$\varepsilon = \frac{2}{h^3} \int_0^{p_F} \sqrt{p^2 c^2 + m^2 c^4} \cdot 4 \pi p^2 dp=$$

      $$\frac{8 \pi}{h^3} \frac{m c^2}{\lambda^3} \int_0^x \sqrt{1+y^2} \cdot y^2 dy$$

  21. Apr 2017
    1. For detailed study of various properties, generalization and application of Wrightfunction and generalized Wright function,
  22. Feb 2017
    1. it encourages a “growth” mindset: the belief that your abilities can improve with your efforts.

      I'll be this also helps with their feeling of "flow" too.

    2. “Many thought, okay to get from A to B there are these three steps, but it turns out there are really five or six,”

      Sounds a lot like the mathematicians who came after Perelman to show that his proof of Poincare was correct--they needed help in getting from A to B too.

  23. Jan 2017
    1. Whether you're a student, parent, or teacher, this book is your key to unlocking the aha! moments that make math click -- and learning enjoyable.

      You had me already at the Coffee Cup picture over the equations! :)

  24. Nov 2016
  25. Oct 2016
    1. Sunil Singh asks us to stop promoting mathematics based on its current applications in business and science. Math is an art that should be enjoyed for its own sake.

      This reminded me of A Mathematician's Lament by Paul Lockhart. This is a 25-page essay which was later worked into a 140-page book. (And Sunil Singh has read at least one of them. He credits Lockhart in one of the replies.)

      It also reminds me of this article on the history of Gaussian elimination and the birth of matrix algebra. Newton's algebra text included instructions for solving systems of equations -- but it didn't have much practical use until later. (Silly word problems are as old as mathematics.)

  26. Sep 2016
  27. Jul 2016
    1. The first set, called Math Instructional, was for apps that would make math relevant for students by linking it to their lives and enabling students at different ability levels to work together
  28. Mar 2016
    1. New property of prime numbers discovered. Primes greater than 5 can end with 1, 3, 7, or 9. The next prime is less likely to end with the same digit, and biased toward one of the remaining three. For instance, a prime ending in 3 is most likely to be followed by a prime ending in 9. The bias evens out as the primes get larger, but only very slowly.

      http://arxiv.org/abs/1603.03720

  29. Feb 2016
    1. Free courses and tutorials from the Santa Fe Institute on subjects related to complex systems science.

  30. Jan 2016
    1. P(B|E) = P(B) X P(E|B) / P(E), with P standing for probability, B for belief and E for evidence. P(B) is the probability that B is true, and P(E) is the probability that E is true. P(B|E) means the probability of B if E is true, and P(E|B) is the probability of E if B is true.
    2. The probability that a belief is true given new evidence equals the probability that the belief is true regardless of that evidence times the probability that the evidence is true given that the belief is true divided by the probability that the evidence is true regardless of whether the belief is true. Got that?
    1. I would like to see an accurate array of photographs of these tasty lunch options. What does a a "Princess Sandwich" even look like? Is a "Celery Sandwich" satisfying? I'd be pleased to see precise measurements of the ideal "Tea Biscuit" Sandwich.

    1. Category Theory for the Sciences by David I. Spivak<br> Creative Commons Attribution-NonCommercial-ShareAlike 4.0<br> MIT Press.

  31. Mar 2015
    1. θ dμ ≥ p 16 π | Σ |

      Qual a relação dessa desigualdade com a dita desigualdade de Penrose Riemanniana provada por Huisken-Ilmanen e Bray?

    2. GIBBONS-PENROSE INEQUALITY

      Qual a relação dessa desigualdade com a dita desigualdade de Penrose Riemanniana provada por Huisken-Ilmanen e Bray?

    3. θ dμ ≥ p 16 π | Σ |

      Isso significa que a taxa expansão nula futura (para fora) \( \theta \) é no mínimo $$ \sqrt{16 \pi |\Sigma|} $$

  32. Jan 2015
    1. A function like f(x,y)=x+y is a function of two variables. It takes an element of R2, like (2,1), and gives a value that is a real number (i.e., an element of R), like f(2,1)=3. Since f maps R2 to R, we write f:R2→R. We can also use this “mapping” notation to define the actual function. We could define the above f(x,y) by writing f:(x,y)↦x+y. To contrast a simple real number with a vector, we refer to the real number as a scalar. Hence, we can refer to f:R2→R as a scalar-valued function of two variables or even just say it is a real-valued function of two variables. Everything works the same for scalar valued functions of three or more variables. For example, f(x,y,z), which we can write f:R3→R, is a scalar-valued function of three variables.

      f:R^2 \rightarrow R demek f(x,y)=z | Skalar-Değerli f f:R \rightarrow R^2 demek f(x)=(y,z) | VektörelDeğ f

    2. f:R→R as a shorthand way of expressing that f is a function from R onto R.
  33. Sep 2013
    1. A computable Dedekind cut is a computable function which when provided with a rational number as input returns or ,

      This definition of computable Dedekind cut is wrong. The correct definition is that the lower and the upper cut be computably enumerable.