Many people are frightened of mathematics and thinkthey cannot read it at all. No one is quite sure why this is so.Some psychologists think there is such a thing as "symbolblindness"-the inability to set aside one's dependence on theconcrete and to follow the controlled shifting of symbols.

Mandel’s system was simple — but incredibly complex from a logistical standpoint

so a formula is like a dead equation?

To me this gets to the heart of the teaching of mathematics. There are (at least) two distinct skills: The mechanical system of converting one thing to another using an array of symbols The translation system mapping those moves onto the world. We can emphasise the mechanical (the approach of most school systems) or we can emphasise the translation (the approach of Conrad Wolfram and others who argue that computers should do the calculation).

Both the cult of learning around Dante and the cult ofignorance around Newton are phenomena of the vicious spe-cialization of scholarship.

Small Collision Probabilities

There is just one little nit for mathematicians to pick: morphisms don’t have to form a set. In the world of categories there are things larger than sets. A category in which morphisms between any two objects form a set is called locally small.

Is composition associative? Check!

Many authors noted that generations tended to fall into clichés, especially when the system was confronted with scenarios less likely to be found in the model's training data. For example, Nelly Garcia noted the difficulty in writing about a lesbian romance — the model kept suggesting that she insert a male character or that she have the female protagonists talk about friendship. Yudhanjaya Wijeratne attempted to deviate from standard fantasy tropes (e.g. heroes as cartographers and builders, not warriors), but Wordcraft insisted on pushing the story toward the well-worn trope of a warrior hero fighting back enemy invaders.

A. Watermelon

TREE(3) is surprisingly large.TREE(1) = 1 and TREE(2) = 3, but then TREE(3) is suddenly vastly beyond comprehension.But once you know that TREE(3) is too big to grok, there’s not a lot left to be said about TREE(n) for specific n > 3; surely they’re bigger. In fact, they are probably so much bigger that even TREE(4) will dwarf TREE(3) to a much greater extent than TREE(3) dwarfs our familiar numbers. But that isn’t surprising once you know about TREE(3).In principle, of course, there is no reason you have to stop. You could consider, for example, TREETREE(TREE(3)) (3), where you start by plugging 3 into TREE, then plug that into TREE, and keep going through a total of TREE(TREE(3)) iterations. That would definitely be a (disturbingly, in my opinion) large number. But it’s not particularly interesting, as large numbers go.

sum(1 to n) = n(n + 1) / 2

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.”

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.

The symbol dx has different interpretations depending on the theory being used. In Leibniz's notation, dx is interpreted as an infinitesimal change in x and his integration notation is the most common one in use today. If the underlying theory of integration is not important, dx can be seen as strictly a notation indicating that x is a dummy variable of integration; if the integral is seen as a Riemann integral, dx indicates that the sum is over subintervals in the domain of x; in a Riemann–Stieltjes integral, it indicates the weight applied to a subinterval in the sum; in Lebesgue integration and its extensions, dx is a measure, a type of function which assigns sizes to sets; in non-standard analysis, it is an infinitesimal; and in the theory of differentiable manifolds, it is often a differential form, a quantity which assigns numbers to tangent vectors. Depending on the situation, the notation may vary slightly to capture the important features of the situation. For instance, when integrating a variable x with respect to a measure μ, the notation dμ(x) is sometimes used to emphasize the dependence on x. Source: http://en.wikipedia.org/wiki/Integral#Terminology_and_notation

In this case, for a test to be statistically significant, p-value must be lower than 0.05.

We are 99% confident that the true average “attitude” difference betweenliving environments is between 1.32 and 7.88. At a significance level of 0.01we can say that living in a minority environment is associated with higherscore

While it was once regarded as a low-level, “primitive” instinct, researchers arecoming to recognize that imitation—at least as practiced by humans, includingvery young ones—is a complex and sophisticated capacity. Although non-humananimals do imitate, their mimicry differs in important ways from ours. Forexample, young humans’ copying is unique in that children are quite selectiveabout whom they choose to imitate. Even preschoolers prefer to imitate peoplewho have shown themselves to be knowledgeable and competent. Researchshows that while toddlers will choose to copy their mothers rather than a personthey’ve just met, as children grow older they become increasingly willing tocopy a stranger if the stranger appears to have special expertise. By the time achild reaches age seven, Mom no longer knows best.

Learning to "factor an equation" is the process of arranging your teepee. In this case: If x=−3 then Component A falls down. If x=2, Component B falls down.

We invented the decimal point to handle the crazy idea of a number more than zero but less than one.

using the symbol 0 to represent nothingness

about 0.6180340

618034

The current mass media such as t elevision, books, and magazines are one-directional, and are produced by a centralized process. This can be positive, since respected editors can filter material to ensure consistency and high quality, but more widely accessible narrowcasting to specific audiences could enable livelier decentralized discussions. Democratic processes for presenting opposing views, caucusing within factions, and finding satisfactory compromises are productive for legislative, commercial, and scholarly pursuits.

Since any powerful tool, such as a genex, can be used for destructive purposes, the cautions are discussed in Section 5.

Hmm...this page doesn’t exist. Try searching for something else.

猜想的艺术

Alicia Boole Stott

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.

Coaching with SEL in mind

logₑ-1 is iπ

Sanity check = Does this even make sense? For example, if your application only outputs integers, sqrt(-1) and log(-1) are undefined.

"dividing n-dimensional space with a hyperplane."

cheap trick

In computer science, a tree is a widely used abstract data type that simulates a hierarchical tree structure

Model theory recognizes and is intimately concerned with a duality: it examines semantical elements (meaning and truth) by means of syntactical elements (formulas and proofs) of a corresponding language

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.

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.

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.

The key idea is that the equation captures not just the ingredients of the formula, but also the relationship between the different ingredients.

In fact, the Product comonad is just the dual of the Writer monad and effectively the same as the Reader monad (both discussed below)

(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.

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

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.

A big reason for GPUs popularity in Machine Learning is that they are essentially fast matrix multiplication machines. Deep Learning is essentially matrix multiplication.

A matrix is a linear map but linear maps are far more intuitive to think about than matrices

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.

New Quantum Algorithms Finally Crack Nonlinear Equations

Ergodic theory is a forbiddingly technical branch of mathematics.

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.

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

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

两个向量的外积，又叫向量积、叉乘等

The Visual Display of Quantitative Information 2nd Edition by Edward R. Tufte

Alexs Adventures In Numberland

How Not to Be Wrong: The Power of Mathematical Thinking Paperback – Illustrated, May 26, 2015 by Jordan Ellenberg

A Mind for Numbers: How to Excel at Math and Science (Even If You Flunked Algebra) Audible Audiobook – Unabridged Barbara Oakley (Author), Grover Gardner (Narrator), Gildan Media, LLC (Publisher) & 2 more

Espen Slettnes 3rd degree connection3rd Espen has a account BMC-Upper Monthly contest designer and local coordinator at Berkeley Math Circle

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.

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.

“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.”

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,

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

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.

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.

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%;}

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.”

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

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.”

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.

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.

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.”

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.

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 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.

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/.

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.

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.

Math Problem Solving Skills

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