198 Matching Annotations
  1. Sep 2021
    1. We teachers can help our students with this. Let them know when the most difficult work is coming. Help them prepare for that work, then admit that the challenge is real and it is difficult.

      But let's also be aware of the all-too-prevalent math shaming that occurs when we say "math is difficult". That definitely isn't productive.

  2. Aug 2021
  3. Jun 2021
    1. cheap trick

      Does Morningstar think that math too suffers from the same issues he finds in critical theory, or just Godel's incompleteness theorem (I'm assuming that's what Morningstar is alluding to)? Explore a deep discussion about whether Godel's incompleteness theorem is a cheap trick.

  4. May 2021
  5. Apr 2021
  6. Mar 2021
    1. In computer science, a tree is a widely used abstract data type that simulates a hierarchical tree structure

      a tree (data structure) is the computer science analogue/dual to tree structure in mathematics

    1. I'd say an equation is anything with an equals sign in it; a formula is an equation of the form A= stuffA= stuffA={\rm\ stuff} where AAA does not appear among the stuff on the right side.
    2. I think that over time the distinction is lost. My math teacher, 35 years ago stated "formulas are used in chemistry, in math we have equations". To this day, the word 'formula' in math seems wrong, but I'd accept it's used commonly.
    3. An equation is meant to be solved, that is, there are some unknowns. A formula is meant to be evaluated, that is, you replace all variables in it with values and get the value of the formula.
    4. The key idea is that the equation captures not just the ingredients of the formula, but also the relationship between the different ingredients.
  7. Feb 2021
    1. In fact, the Product comonad is just the dual of the Writer monad and effectively the same as the Reader monad (both discussed below)
      • Establish the number of Rubik's cube combinations.
      • Establish "algorithm" capabilities based on the x12 "impossible" cubes.
      • "Signmaster notation" for describing algorithm moves.
      • x2 links to youtube tutorials.
    1. (The forms !=, /= or <> are generally used in programming languages where ease of typing and use of ASCII text is preferred.) x ≈ y means x is approximately equal to y. This may also be written ≃, ≅, ~, ♎ (Libra Symbol), or ≒. G ≈ H means that group G is isomorphic (structurally identical) to group H.

      what does that have to do with this game?

    1. Critical and creative thinkers engage in active planning and forethought to set goals, outline strategies, and determine the best methods through which they can achieve their goals

      Head Scratcher: How are we promoting critical and creative thinkers in our instruction?

      As a high school math teacher this can be easier at times, and more difficult at times depending on the class and the course material. At times it is easy to promote creative when dealing with honors classing and higher math courses. But when working with remedial Algebra classes it can be more difficult to promote creativity and critical thinking because of high levels of apathy and prior knowledge. Sometimes the best way to promote success in those classes is through repetition and memeorization of steps to solve common test promblems.

    1. All those names of things - topology, complex analysis, and differential geometry - might not sound like much to you now, but you'll soon learn that they're really just describing the shapes of things in our Universe, and the way those shapes change in time and space are explained by things like calculus and chaos theory.
  8. Jan 2021
    1. A big reason for GPUs popularity in Machine Learning is that they are essentially fast matrix multiplication machines. Deep Learning is essentially matrix multiplication.

      Deep Learning is mostly about matrix multiplication

    2. A matrix is a linear map but linear maps are far more intuitive to think about than matrices
    3. I often get asked by young students new to Machine Learning, what math do I need to know for Deep Learning and my answer is Matrix Multiplication and Derivatives of square functions.

      Deep Neural Networks are a composition of matrix multiplications with the occasional non-linearity in between

    1. New Quantum Algorithms Finally Crack Nonlinear Equations
      • We can’t predict the weather, among many other complex issues, because computers still can’t solve nonlinear equations.
      • But this might change soon, as two different research teams created algorithms that can be used for nonlinear modelling on quantum computers.
      • Their techniques still need refining, and won’t be real-world ready for years, but these studies are another stepping stone towards truly useful quantum algorithms.
    1. We don’t know how many media outlets have been run out of existence because of brand safety technology, nor how many media outlets will never be able to monetize critical news coverage because the issues important to their communities are marked as “unsafe.”
  9. Dec 2020
    1. Ergodic theory is a forbiddingly technical branch of mathematics.

      It's supremely sad that a paper in Nature should "math shame" ergodic theory this way. What the hell is going on?

    1. The company’s early mission was to “give people the power to share and make the world more open and connected.” Instead, it took the concept of “community” and sapped it of all moral meaning. The rise of QAnon, for example, is one of the social web’s logical conclusions. That’s because Facebook—along with Google and YouTube—is perfect for amplifying and spreading disinformation at lightning speed to global audiences. Facebook is an agent of government propaganda, targeted harassment, terrorist recruitment, emotional manipulation, and genocide—a world-historic weapon that lives not underground, but in a Disneyland-inspired campus in Menlo Park, California.

      The original goal with a bit of moderation may have worked. Regression to the mean forces it to a bad place, but when you algorithmically accelerate things toward our bases desires, you make it orders of magnitude worse.

      This should be though of as pure social capitalism. We need the moderating force of government regulation to dampen our worst instincts, much the way the United State's mixed economy works (or at least used to work, as it seems that raw capitalism is destroying the United States too).

  10. Nov 2020
    1. We say i (lowercase) is 1.0 in the imaginary dimension Multiplying by i is a 90-degree counter-clockwise turn, to face “up” (here’s why). Multiplying by -i points us South It’s true that starting at 1.0 and taking 4 turns puts us at our starting point: And two turns points us negative: which simplifies to: so

      Great explanation of why \(i=\sqrt-1\)

    2. Imaginary numbers seem to point North, and we can get to them with a single clockwise turn. Oh! I guess they can point South too, by turning the other way. 4 turns gets us pointing in the positive direction again It seems like two turns points us backwards

      Imaginary numbers explained in plain-english

    3. Imaginary numbers let us rotate around the number line, not just move side-to-side.

      Imaginary numbers Another graph:

  11. Oct 2020
    1. 两个向量的外积,又叫向量积、叉乘等


    1. Espen Slettnes 3rd degree connection3rd Espen has a account BMC-Upper Monthly contest designer and local coordinator at Berkeley Math Circle
    1. 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.
    2. 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.
    3. “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.”
    4. 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,
    1. 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.

    2. 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...


      But nothing on the embedded version:

      A screengrab of what this looks like:

    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.”
    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?

    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. 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. 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.

    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.

    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. 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.

    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.
    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/.
    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?)

    1. MATH LESSONS SheLovesMath.com is a free math website that explains math in a simple way, and includes lots of examples, from Counting through Calculus. Here are the topics that She Loves Math covers, as expanded below:  Basic Math, Pre-Algebra, Beginning Algebra, Intermediate Algebra, Advanced Algebra, Pre-Calculus, Trigonometry, and Calculus.
    1. Math Problem Solving Skills
    2. I was just helping a student with his AoPS homework, when I came across the following related problems: Eight people, including Fred, are in a club.  They decide to form a 3 person committee.  How many possible committees can be formed? So we are choosing 3 people out of 8 or . How many possible committees include Fred? Since Fred is taking one committee seat, that means we need to choose 2 more people from the remaining seven, or . How many possible committees do not include Fred? Since we can’t choose Fred, we need to choose 3 members from the remaining 7 or . Since the total number of committees is equal to the number of of committees with Fred plus the number of committees without Fred, then we can say

      Pascal's rule and is Fred on the committee

  12. Sep 2020
    1. 4.… 9.7

      You know, they say, all Supreme Courts are created equal, but you look at the court now, and you look at the court in 1868, and you can see that statement is NOT TRUE. See, normally, you propose something like an Expanded Supreme Court, you got a fifty-fifty chance of passing it. But Majority Leader McConnell is a genetic freak, and the Congress as it stands is not normal.

      So, right now, you got a 25% chance AT BEST at beating this do-nothing Congress and passing something. But then you add Election Day to the mix? You the chances of passing the expanded court drastically go up. See, at the three way on Election Day (where the Senate will either be Republican Majority, Democratic Majority, or tied, because of course that can happen), you got a 33 and a third chance of the Republicans winning! But Democrats! Democrats got a 66 and two thirds chance of winning, 'cause President Trump KNOOOOOWS he can't win fairly, and he's not even gonna try, and he might take some of his party's senators down with him. So, DAVID, you take Republicans 33 and a 3rd percent chance of winning minus Democrats 25% chance of passing something through the Senate (if they was to go one-on-one right now) and you've got an 8 1/3 percent chance of Republicans defeating court expansion! But THEN, you take Democrats 75% chance of passing something through the House, plus the 66 and 2/3 percents, they got a 141 2/3 chance of expanding the court!

      Senator McConnell! The numbers don't lie, and they spell 17 1 5 4 5 7 3 12 for you at Sacrifice!

  13. Aug 2020
  14. mgonzalezm.keybase.pub mgonzalezm.keybase.pub
    1. (a?—a + 1) (a*—a? + 1) (a% + a + 1). Comprobar el resultado ha- ciendo a = 2.
    2. (a2—ab+b2+a+b+1)(a+b—1).
    3. (a* + a% + a®b? +ab® + b*) (a—b).
    4. (x2—x—1)2(x2 + x + 1).
    5. (2 + xy—6)*) + (x+29).
    6. 2a*mx?y) + 6a?nyz?) + (20”y)

      Typo: Sobra el pimer parentesis ')'


    7. (4abx — 8b%x?y) + (2bx?).
    8. (2 + 322 + 5) (x*—1 + 4x).
    9. (m3 —m? + m-—-1) (—m3 -+ m—m +1).
    10. (x2 4 y2 + 22— ay — az — yz) (x + y + 2).
    11. (a? — 2ab + 4b?) (a + 2b). Comprobar el resultado haciendo a = 2 3.
    12. (x2—3xy + y*) (2x— 3y + 2).
    13. (a? + 2ab — 2b*) (3a—7b).
    14. (2x2—5y) (4x + 2y*).
    15. xy(x2— 2y + 4).
    16. (—ab2c) (3a2bc) (2abc?).
    17. (8a2b) (—2ab2).
    18. Demostrar que la suma de cualquier número negativo con su valor abso- luto es igual a cero.
    19. Demostrar que la suma de todas las expresiones en los ejercicios 11-15 es igual a la expresión en el ejercicio 11.
    20. Calcular 4 —B—C.
    21. Calcular B— A4 —C.
    22. Calcular 4 —B + C.
    23. Calcular B—4 + C.
    24. Calcular 4 + B—C.
    25. m* + 6m$ — 7m? + 8m — 9, 2m? + Im?—4m — 3.
    26. 2a + 4by — 2ey? + dys, 2dy> — 2by —a + 30y”.
    27. a — 3a”b + 3ab? —b, a5 — 4a*b + 2ab? + b.
    28. x3—4x? + 2x—5, —a3 + 2x2—3x —3.
    29. 3a—2b + 4c — d, 20 + b—3c—d.
    30. E2 + 2cd — 2d, 3c — 3cd — 2d?, e? 4+ 4d — 2c + 2d?.
    31. 3x3 — 8x? + 9x, —a3 + 3x2—8, 2x3 — 2x?—7x 5.
    32. x2-—4xy + 3y, 2x? + 2xy-—2y?, 2xy — y? —22.
    33. 4m? — 3mn + 2n?, Gmn-—2n? + 5, 3n2 — 3—2m.
    34. 2a> — 2a?b + 2b5, Yu?b — 4ab? — 4b3, dab* —m.
    35. (x+a)(y +a)(z +a).
    36. Demostrar el Teorema 3 del Art. 2.4.
  15. Jul 2020
    1. f.real numbers.

      f. real numbers. \( \{ -9, -1.3, 0, 0.\overline{3}, \frac{\pi}{2}, \sqrt{9}, \sqrt{10} \} \)

    2. e.irrational numbers

      e. irrational numbers. \( \{ \frac{\pi}{2} , \sqrt{10} \} \)

    3. d.rational numbers

      d. rational numbers. \( \{ -9, -1.3, 0, 0.\overline{3}, \sqrt{9} \} \)

      \( 1.3 = \frac{13}{10} \)

      \( 0.\overline{3} = \frac{1}{3} \)

      \( 0 = \frac{0}{1} \)

      \( \sqrt{9} = \frac{3}{1} \)

    4. c.integers.

      c. integers \( \{ -9, 0, \sqrt{9} \} \)

    5. b.whole numbers

      b. whole numbers. \( \{ 0, \sqrt{9} \} \)

      0 is a whole number

    6. a.natural numbers

      a. natural numbers. \( \{ \sqrt{9} \} \)

      \( \{ \sqrt{9} \} = 3 \)

    7. Find the union:{3, 4, 5, 6, 7}∪{3, 7, 8, 9}.

      \( \{7,8,9,10,11\}\cup\{6,7,8,9,10,11,12\}=\{7,8,9,10,11,6,12\} \)

    8. Find the intersection:{3, 4, 5, 6, 7} ̈{3, 7, 8, 9}.

      \( \{3,4,5,6,7\}\cap\{3,7,8,9\}=\{3,7\} \)

    9. b.By how much does the formula underestimate or overestimate the actual cost shown in Figure P.1?

      \( 8893 - 8714 = 179 \)

      It underestimates by 179

    10. a.Use the formula T=4x2+330x+3310, described in Example 2, to find the average cost of tuition and fees at public U.S. colleges for the school year ending in 2014.

      \( T=4x^2 +330x +3310 \)

      Because 2014 is 14 years after 2000, we substitute 14 for x in the given formula.

      \( T=4(14)^2 + 330(14) +3310 \)

      \( T=4(196) +4620 +3310 \)

      \( T=784 +7930 \)

      \( T=8714 \)

    11. Evaluate 8+6(x-3)2 for x=13.

      Evaluate \( 8+6(x-3)^2 \) for \( x=13 \)

      \( 8 + 6 (13 - 3)^2 \)

      \( 8 + 6 (10)^2 \)

      \( 8 + 6 * 100 \)

      \( 8 + 600 \)

      \( 608 \)

    1. In logic, functions or relations A and B are considered dual if A(¬x) = ¬B(x), where ¬ is logical negation. The basic duality of this type is the duality of the ∃ and ∀ quantifiers in classical logic. These are dual because ∃x.¬P(x) and ¬∀x.P(x) are equivalent for all predicates P in classical logic
    2. the ∧ and ∨ operators are dual in this sense, because (¬x ∧ ¬y) and ¬(x ∨ y) are equivalent. This means that for every theorem of classical logic there is an equivalent dual theorem. De Morgan's laws are examples
  16. May 2020
    1. Related concepts in other fields are: In natural language, the coordinating conjunction "and". In programming languages, the short-circuit and control structure. In set theory, intersection. In predicate logic, universal quantification.

      Strictly speaking, are these examples of dualities (https://en.wikipedia.org/wiki/Duality_(mathematics))? Or can I only, at strongest, say they are analogous (a looser coonection)?

    1. Mathematically speaking, necessity and sufficiency are dual to one another. For any statements S and N, the assertion that "N is necessary for S" is equivalent to the assertion that "S is sufficient for N".
    1. In mathematical contexts, duality has numerous meanings[1] although it is "a very pervasive and important concept in (modern) mathematics"[2] and "an important general theme that has manifestations in almost every area of mathematics".[3]
  17. Apr 2020
    1. the phrase up to is used to convey the idea that some objects in the same class — while distinct — may be considered to be equivalent under some condition or transformation
    2. "a and b are equivalent up to X" means that a and b are equivalent, if criterion X, such as rotation or permutation, is ignored
    1. If solutions that differ only by the symmetry operations of rotation and reflection of the board are counted as one, the puzzle has 12 solutions. These are called fundamental solutions; representatives of each are shown below
    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. Suppose you have only two rolls of dice. then your best strategy would be to take the first roll if its outcome is more than its expected value (ie 3.5) and to roll again if it is less.

      Expected payoff of a dice game:

      Description: You have the option to throw a die up to three times. You will earn the face value of the die. You have the option to stop after each throw and walk away with the money earned. The earnings are not additive. What is the expected payoff of this game?

      Rolling twice: $$\frac{1}{6}(6+5+4) + \frac{1}{2}3.5 = 4.25.$$

      Rolling three times: $$\frac{1}{6}(6+5) + \frac{2}{3}4.25 = 4 + \frac{2}{3}$$

    1. Therefore, En=2n+1−2=2(2n−1)

      Simplified formula for the expected number of tosses (e) to get n consecutive heads (n≥1):


      For example, to get 5 consecutive heads, we've to toss the coin 62 times:


      We can also start with the longer analysis of the 5 scenarios:

      1. If we get a tail immediately (probability 1/2) then the expected number is e+1.
      2. If we get a head then a tail (probability 1/4), then the expected number is e+2.
      3. If we get two head then a tail (probability 1/8), then the expected number is e+2.
      4. If we get three head then a tail (probability 1/16), then the expected number is e+4.
      5. If we get four heads then a tail (probability 1/32), then the expected number is e+5.
      6. Finally, if our first 5 tosses are heads, then the expected number is 5.



      We can also generalise the formula to:

      $$e_n=\frac{1}{2}(e_n+1)+\frac{1}{4}(e_n+2)+\frac{1}{8}(e_n+3)+\frac{1}{16}\\(e_n+4)+\cdots +\frac{1}{2^n}(e_n+n)+\frac{1}{2^n}(n) $$

    1. In math, idempotence describes only unary functions that you can call on their own output. Math-idempotence is, “If you take the absolute value of a number, and then you take the absolute value of that, the result doesn’t change on the second (or subsequent) operations.” Math.abs is math-idempotent. Math-idempotence only applies to functions of one parameter where the parameter type and return type are the same. Not so useful in programming.
    2. Programming-idempotence is about side effects. It’s about stuff that happens to the outside world when you call a function. Idempotence says “If you’ve called me once, it doesn’t matter whether you called me again.”
    1. Tau Day is an annual celebration of the circle constant τ=6.283185…\tau = 6.283185\ldots, which takes place every June 28 (6/28 in the American calendar system).

      Tau (τ) = 2π radians = 360°

      It's simply a more intuitive way of representing a full circular rotation. Possibly Euler could've changed it 100 years ago, but too many textbooks already applied the concept of 2π radians instead




  18. Mar 2020
    1. This denotes the factorial of a number. It is the product of numbers starting from 1 to that number.

      Exclamation in Python: $$x!$$ is written as:

      x = 5
      fact = 1
      for i in range(x, 0, -1):
          fact = fact * i

      it can be shortened as:

      import math

      and the output is:

      # 5*4*3*2*1
    2. The hat gives the unit vector. This means dividing each component in a vector by it’s length(norm).

      Hat in Python: $$\hat{x}$$ is written as:

      x = [1, 2, 3]
      length = math.sqrt(sum([e**2 for e in x]))
      x_hat = [e/length for e in x]

      This makes the magnitude of the vector 1 and only keeps the direction:

      math.sqrt(sum([e**2 for e in x_hat]))
      # 1.0
    3. It gives the sum of the products of the corresponding entries of the two sequences of numbers.

      Dot Product in Python: $$X.Y$$ is written as:

      X = [1, 2, 3]
      Y = [4, 5, 6]
      dot = sum([i*j for i, j in zip(X, Y)])
      # 1*4 + 2*5 + 3*6
      # 32
    4. It means multiplying the corresponding elements in two tensors. In Python, this would be equivalent to multiplying the corresponding elements in two lists.

      Element wise multiplication in Python: $$z=x\odot y$$ is written as:

      x = [[1, 2], 
          [3, 4]]
      y = [[2, 2], 
          [2, 2]]
      z = np.multiply(x, y)

      and results in:

      [[2, 4]],
      [[6, 8]]
    5. This is basically exchanging the rows and columns.

      Transpose in Python: $$X^T$$ is written as:

      X = [[1, 2, 3], 
          [4, 5, 6]]

      and results in:

      [[1, 4], 
       [2, 5],
       [3, 6]]
    6. This denotes a function which takes a domain X and maps it to range Y. In Python, it’s equivalent to taking a pool of values X, doing some operation on it to calculate pool of values Y.

      Function in Python: $$f:X \rightarrow Y$$ is written as:

      def f(X):
          Y = ...
          return Y

      Using R instead of X or Y means we're dealing with real numbers: $$f:R \rightarrow R$$ then, R^2 means we're dealing with d-dimensional vector of real numbers (in this case, example of d=2 is X = [1,2]

    7. This symbol checks if an element is part of a set.

      Epilson in Python: $$3 \epsilon X$$ is written as:

      X = {1,2,3}
      3 in X
    8. The norm is used to calculate the magnitude of a vector. In Python, this means squaring each element of an array, summing them and then taking the square root.

      Norm of vector in Python (it's like Pythagorean theorem): $$| x|$$ is written as:

      x = [1, 2, 3]
      math.sqrt(x[0]**2 + x[1]**2 + x[2]**2)
    9. This symbol denotes the absolute value of a number i.e. without a sign.

      Absolute value in Python: $$|x|$$ is written as:

      x = 10
      y = -20
      abs(x)  #10
      abs(y) #20
    10. In Python, it is equivalent to looping over a vector from index 0 to index N-1 and multiplying them.

      PI in Python is the same as sigma, but you multiply (*) the numbers inside the for loop. $$\prod_{i=1}^Nx^i$$

    11. we reuse the sigma notation and divide by the number of elements to get an average.

      Average in Python: $$\frac{1}{N}\sum_{i=1}^Nx_i$$ is written as:

      x = [1, 2, 3, 4, 5]
      result = 0
      N = len(x)
      for i in range(n):
       result = result + x[i]
      average = result / N

      or it can be shortened:

      x = [1, 2, 3, 4, 5]
      result = sum(x) / len(x)
    12. In Python, it is equivalent to looping over a vector from index 0 to index N-1

      Sigma in Python: $$\sum_{i=1}^Nx_i$$ is written as:

      x = [1, 2, 3, 4, 5]
      result = 0
      N = len(x)
      for i in range(N):
       result = result + x[i]

      or it can be shortened:

      x = [1, 2, 3, 4, 5]
      result = sum(x)
    13. 2D vectors

      2D vectors in Python: $$x_{ij}$$ are written as:

      x = [ [10, 20, 30], [40, 50, 60] ]
      i = 0
      j = 1
      print(x[i][j]) # 20
    1. This volume of paper should be the same as the coaxial plug of paper on the roll.

      Calculating volume of the paper roll: $$\mathbf{Lwt = \pi w(R^2 - r^2)} \~\ L = \text{length of the paper} \ w = \text{width of the paper} \ t = \text{thickness} \ R = \text{outer radius} \ r = \text{inner radius}$$ And that simplifies into a formula for R: $$\color{red} {\bf R = \sqrt{\frac{Lt}{\pi}+r^2}}$$

  19. Feb 2020
    1. During our lives we learn that numbers are linear, that the spaces between them are all the same. If you count from one to nine, each number is one more than the previous one. If you ask someone what number is halfway between one and nine, they will say five—but only because they have been taught to. Wake up, sheeple! Humans instinctively perceive numbers logarithmically, not linearly. A young child or someone who has not been indoctrinated by education will place three halfway between one and nine. 

      Logarithmic middle: middle with respect to multiplication rather than addition --> why do we do this by default

  20. Jan 2020
  21. Sep 2019
  22. 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"

  23. May 2019
    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.

  24. 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.)
  25. 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/

  26. Dec 2018
    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 :)

  27. 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

  28. Sep 2018
  29. 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?