ROI(tax lawyers) > taxes to pay without them.
870 karma · joined December 8, 2012
The first example is a formula. An instance of magic, in the sense that you use it to compute, without knowing what it does. The $x_i$'s are not quantified. What are they? Are they real numbers? Matrices? Elements of some semi-group? How can you expect to understand the "formula" if the summand is not explained? At best, I can say that it is a formal sum of something. We can forget discussing convergence or it being well-defined. You can cook up arbitrarily 'nice' notation. It won't help. This notation is absolutely fine for someone who can infer that the support of the distribution of X is some denumerable set {x_i}, equipped with p.m.f. p.
A suitable definition of the expected value (as an operator) would have cleared up all the confusing with the variance and E[X^2] vs (E[X])^2. This confusion is not the notation's fault. It is the user's fault for not knowing what E[f(x)] means (for some appropriate meaning of the symbol f).
>> Only the first xixi is squared. p(xi)p(xi) isn't, because it doesn't make any sense in the first place. It should really be just PXiPXi or something, because it's a discrete value, not a function!
Functions are not algebraic expressions by which we associate one real number with another. In fact, we call p(x_i) the probability mass function. It seems to be a common flaw in many undergrad programs. Formulas and functions are never made distinct. The vast majority of functions f : R -> R do not admit an expression in a formula.
The example with the different notation for "derivatives" is a good non-example. The so-called Leibniz notation is used because it allows people to make statements with differential forms, without needing to invoke exterior algebra. If this is done correctly, statements such as "dy = f'(x)dx" can be made fully rigorous, if need be. Students are told that dy/dx is not a fraction, and yet it is used exactly as though it were. This confuses people - because they don't know what is going on. The dot-notation for derivatives is extremely useful in classical mechanics.
Notation is a clutch for succinct and meaningful writing amongst the initiated. One cannot expect to be able to use these tools without knowing what is going on, or by suspending a great deal of questions.
>> There must be other ways we can explain math without having to explain the extraordinarily dense, outdated notation that we use.
My final gripe with this post. We typically use clean and modern notation. It could be so much worse! Also, if we didn't re-use symbols, then we would run out, very quickly. Mathematics exists independently of the symbols we use to communicate it.
>> Instead of indexing the geofences using R-tree or the complicated S2, we chose a simpler route based on the observation that Uber’s business model is city-centric; the business rules and the geofences used to define them are typically associated with a city. This allows us to organize the geofences into a two-level hierarchy where the first level is the city geofences (geofences defining city boundaries), and the second level is the geofences within each city.
I only really started to understand linear algebra when I was forced to in a differential geometry class. The opening chapters were a review intended to fix my university's notoriously broken linear algebra training. As I said, it comes with the territory. You can't design a course based on Halmos' FDVS and expect students coming in, that is, students who've scraped through calculus 1, to manage. So, the recipe book / cookbook style abounds. Granted, there were inklings of mathematics in my linear algebra course. I don't think anyone really appreciated it however. It's hard to grok "vector space over a field" when you've never been introduced to the abstract concept of a field.
When my second year stats lecturer told me that a determinant of a 2x2 matrix was an area, I almost didn't believe him.
Fun exercise I was told about just the other day. Every invertible matrix with integer coefficients has determinant +-1. I would never have known how to solve that after my linear algebra course.
It is false to assume that the state of the electrical supply is either on or off. This may come as a surprise, but not to me. In 2008, Eskom (South Africa's electricity suppliers) experienced similar faults. The mains supply voltage is 220v here. At one point, some devices started to fail in my house, and others, such as lights, continued to work, but significantly dimmer. We measured 180v at the plugs. There were similar outages in my area last year, where an outright cut-off was preceded by voltage drops. This outage is interesting because it is an example of a bug owing to false assumptions!
There have also been incidences where certain cables have been stolen [1] and that has caused the opposite: voltage spikes.
[1] I couldn't tell you which, or what kind, but I remember it has something to do with "the neutral"
Example:
Security: Are you a member of a Hebrew congregation?
Person: Yes, such and such a synagogue
Security: Who is the Rabbi there?
A legit person might say "Oh it's Rabbi so-and-so", but if a person hesitates, and has to think, and blurts out an answer like "Rabbi Cohen" then it is less likely they are being forthcoming.
In some sense, this is a very rough mental-Bayesian process of probability updating. A person arrives at the border / El-Al check-in. Security assigns some prior probability p_0 that a person should be allowed access, and via a string of questions, amongst other techniques, a posterior probability p_n that that person should be on that flight / in Israel is established.
Yes, "the best" is a transparent lie. It doesn't matter though, because it serves the interests of companies who espouse it. DHH is calling a spade a spade here - but I'd hazard that the vast majority of the HN crowd already know.
Also, globally, there are lot of extraordinarily shit software people out there. If we define quality by "gets things done and doesn't break things and act in a crappy and deleterious manner", then it is not difficult for "the best" to mean the upper 60% of software people. It is completely reasonable to assume that the companies with this mantra do indeed hire from the top 60%, if only because of how many rubbish people there are.
The system for gun/bomb detection is only as good as the agents responsible for it. For example, I saw my friend accidentally sneak a knife onto an plane two years ago. The metal detector flagged him, and he confidently told the official present: "Oh, it's my belt!" and after taking it off he walked through, with no further probing. He literally forgot about his EDC knife and wound up with it on the flight. It must be noted that this was in South Africa, but having travelled to both the USA and Europe, I know it could have happened there too.
The current system is not robust to human error, and this is its fundamental flaw. Con artists and psychologists know how to systematically induce human error [1, 2]. It is conceivable that malicious actors (or TSA people pretending to be) can do exactly that and reliably subvert a human detection system. As an aside, it would be interesting to see TSA people try sneak weapons onto an El-Al flight [see 3, 4, 5 for why]. A humorous anecdote: once when returning from Israel, I had my bag full of dirty clothes thoroughly man-handled by an irate woman from El-Al security. I had two books inside, each with a metal inset in the spine. As soon as this was found out, I went from being treated like a terrorist to a non-threat in a few seconds. It was remarkably efficient.
[1] https://en.wikipedia.org/wiki/Confidence_trick
[2] https://en.wikipedia.org/wiki/Cognitive_bias
[3] http://www.wsj.com/articles/SB1001458258397595640
[4] http://edition.cnn.com/2010/OPINION/01/11/yeffet.air.securit...
[5] http://www.tabletmag.com/the-roll/102229/the-gatekeepers
[1] http://www.haaretz.com/opinion/.premium-1.617280
[2] http://www.vice.com/en_uk/read/gay-palestinians-are-being-bl...
[0] I'm looking at you, three years of 'numerical methods', which reduced to memorising algorithms to perform by hand.
[1] I count myself amongst the absolutely useless 'applied mathematicians' from my class.
Number theorists (amongst others) are seemingly obsessed with bounding things. There are entire books written about obtaining and then refining bounds - which appear to be nothing more than inequalities. There is great real-world value to be derived from seeing 'inequalities' as tools to leverage. Brian Kernighan once commented that controlling software complexity is the essence of programming [0]. I believe similar thinking applies to other aspects of software engineering, and product and business development. If you can take a hard problem, and bound its complexity, then you can say "the problem is no more complex than this". This is very useful. The chief value proposition of many SaaS businesses is the trivialisation of the upper bounds of complexity of hard problems. For instance, for many developers, Heroku makes the complexity of deployment very low.
On the contrary, the purpose of an ORM is to let your code be a schema for your database. This is my personal use case for an ORM; I want persistence but am not overly concerned with the database. There are plenty of use cases [1] where a schema update from outside is unlikely or impossible. In these cases, there is a very low cost for (by way of an ORM) tightly coupling your code to your schema.
Also, an ORM lets somebody else worry about sanitisation, and database specific-code. It can abstract away the choice of database, another very valuable feature. I think that the example you quote is actually a poor demonstration of the value of an ORM. The other ones on the linked page are far better :)
[1] A program running on only one machine, like an Android app or a music player on Linux desktops etc
The main activity of research mathematicians is theorem proving. That is, turning conjectures into theorems. In the jargon, proven theorems (and equations, bounds, counterexamples etc.) are often called 'results'. Essentially, the author is saying that with very few constructs and tools, number theorists have been able to pose a conjecture that is comparatively simple to state and understand, but is frustratingly as yet unproven, in spite of decades of effort by many, very good mathematicians. So far, many special cases ('partial results') have been proven, but the highly desired goal that is the proof of the BSD conjecture, in full generality, has not yet been achieved. The consensus is that mathematicians are quite far away from resolving the BSD conjecture, but very strongly believe it is true.
To me this suggests a very simple solution. Legislate that each ingredient parcel must come with an audit trail. Assign to each parcel a reference number - a UUID or hash - that can be used to trace back its entire life and history. The solution seems so obvious that there are probably significant obstacles against doing so that I am not aware of. If such a system is implemented, then the next time a contamination breaks out, the source can be quickly identified and eliminated, then apologies can be made, and business can resume as normal.
>> The spread of norovirus in Simi Valley and Boston was caused by breaches of protocol, Ells says
Eliminating contamination entirely is a highly desirable goal, but outbreaks will likely continue occur regardless of anti-contamination measures. As a result, it makes sense to make it as easy as possible to shut down outbreaks before they become crises.
>> If your use of the system decreases Fun for other players, it is a violation of the Zeroth Rule. If it doesn't, we have no strong objection to it.
>> Illegal market manipulation wait, we're not the SEC -- that's very clearly Fun.
>> [Fun] means what we say it means, and our decisions on it are final.
What constitutes 'fun' seems arbitrary. It might frustrating and very un-Fun to be told "Patrick, Thomas, and Erin say no" because the rules aren't well-defined. For example, I don't think that illegal market manipulation is fun - it's just deleterious - and part of the reason it's illegal in the first place. To what extent are you planning on taking user feedback on what 'fun' is? It's not clear to me whether or not it's 'fun' to craftily infer your internal network topology, for instance. Your game has a global audience, so you have to assume that linguistic/cultural/whatever norms aren't shared by all your players.
Also, the contrast on the code samples is way too low. Please make the text stand out more? I can barely read it.
>> “I think this is state of the art for predictive policing,” Lewin says. >> How will this form of predictive policing be received?
Use of the word "predictive" here is utterly inaccurate because the police are not predicting anything, but rather identifying at-risk individuals. The inferential leap from "people with these n properties have historically been party to gun violence" to "prediction" is enormous. As soon as one mentions "prediction" and police in the same breath, it suggests Minority Report-style impingement on personal freedoms of some kind. The reason I stress this is because such a program might very well work to reduce gun violence, and labelling it as though it were the genesis of a Big Brother is not only disingenuous, but may also harm the program's legitimacy in the eyes of people who have the power to shut it down.
>> Established in 2008, Samasource is a non-profit business that connects marginalized women and youth to dignified work via the Internet
>> Launched in 2012, Samahope is the first crowdfunding platform for medical treatments.
>> Samaschool (previously SamaUSA) prepares people for success in the digital economy.
However, we should restrict our attention to flows that we very strongly believe are correctly described by NS. From the original paper:
>> The main problem of our method is the same as the weakness of all machine learning approaches; the learning methods are not capable to extrapolate the model far outside the data observed during training.
The training data are existing numerical simulations of specific fluid flows. The learning algorithm learned fluid flow dynamics from simulations. This is extremely impressive. It also achieves a significant speed up in simulation time - which is also very impressive. However, as the authors say, it does not generalise well. So, in short the answer to your first question is probably "sometimes yes, but in general not really".
edit: I should add that the simulations in the paper are decidedly based on Navier-Stokes.
[1] https://en.wikipedia.org/wiki/Hagen%E2%80%93Poiseuille_flow_...
[2] https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_existenc...
Having said that, I am going to adopt a somewhat contrarian viewpoint. The math department fired Coward for deliberately and repeatedly subverting their requests to conform to departmental standards [1]. It is not surprising that behaving in a fashion that continuously pissed off senior faculty (with the power to fire or initiate the process of firing) got Coward fired.
Coward's student ratings were consistently high. Based on anecdotes, he sounds like an instructor I would love to have (he spontaneously derived a formula!).
However, this does not matter because Coward refused to participate in the system. What I term 'the system' is the set of departmental norms and standards that exist as they do for very good reasons. These reasons might not be understood by all actors.
My understanding of the state of the system is informed by the following: UCB is an institution that trains thousands of students in mathematics annually. They have a large faculty [2]. The numbers of people involved imply that there is tremendous variation amongst students and instructors - in areas such as raw aptitude, experience, language proficiency and motivation. The benefits of systematizing the process of training in a university of Berkeley's size are manifold. The system has to exist the way it does in order to produce large numbers of adequately (and usually only adequately) educated people. The purpose of disallowing faculty from deviating from such standards is to allow the system to function independently of the people involved in it. There are many disadvantages to this approach, such as stifling lecturer creativity and disabling course-level optimisations (like a particular professor's vivid geometric intuition for something abstract).
I could enumerate the reasons the system exists as it does (at length) but do not have space here, so here is one example: Using a standard textbook means that course content is instructor invariant. This makes the level of training robust to shitty lecturers, who necessarily exist in any sufficiently large system. It gives students recourse to a standard reference they know is correct. It decreases the variation in quality of students. This is particularly important for co-requirements. It is a huge problem when course B depends on course A and students do not have a firm grip on course A. This happened to my class: we arrived in applied mathematics 3 with a totally broken knowledge of multivariable calculus.
Essentially, the system exists as it does to provide a lower bound for the quality of training. Some people will be incompatible with it. UCB should probably have been more gentle in their handling of Coward - he was 'suicidally depressed' - but I say this with the benefit of hindsight.
[1] >> On September 22nd, 2013 he wrote in an email "But I do think it that it [sic] is very important that you not deviate too far from the department norms." On November 12th, 2014 he wrote "I hope that, on the basis of our conversation, you can further adjust to the norms of our department."
[2] https://math.berkeley.edu/people/faculty
[3] One might think that mathematics is mathematics, and any reference material on a topic will do. I can tell you from experience that this is false. A complex analysis novitiate trying to sort out what 'holomorphic', 'analytic' and 'complex differentiable' all mean will inevitably run into equivalent but different definitions.
However, the existence of tricks is enormously useful once understanding the underlying mechanism of a particular tool is not the focus of a problem at hand. Abstraction is a fundamental human cognitive faculty. For instance, if a student understands why the cross-multiplication 'trick' works, then they should be free to use it as they please, provided they can actually explain why it works if prompted to do so. The notion that there was a 'right' way to do something (like use common denominators) was stifling and frustrating during my school years. If I can explain and justify the trick - then let me use it. On the other hand, being boxed into doing things the instructor-sanctioned way can lead to equally vacuous understanding: "Teacher says find a common denominator so that's what I'll do even though I don't know why".
Additionally, I will argue that all methods for doing computations with fractions are 'tricks' at some level. After all, they are just theorems on the field of quotients of the integers embedded in the reals. One should not be precluded from using a 'trick' because one of these theorems ('common denominator method good - your trick bad') is more familiar to an instructor. Replacing one 'trick' with what is actually just another does not facilitate understanding.
Of course, this is predicated on actually understanding the tool in the first place.
I take an opposite viewpoint from the author(s). Students should be encouraged to develop and use tools. The utility of hiding complexity [0] with tooling is part of the very essence of what it means to be a hacker. It is also very useful in other fields. For example, a physicist solving for the flow of some fluid does not need to think about why a particular fraction trick works. This would draw precious cognitive resources better served elsewhere.
Once a concept is understood, tricks become useful tools. In a field such as building construction, short-cuts are often expensive in the long run because the benefit of making some compromise (e.g. use cheap plaster) is outweighed by its consequences (e.g. need to re-plaster after short amount of time). This mode of thinking does not apply to mathematics.
Tricks are not 'bad' and should not be nixed. They should be embraced and presented as tools of great utility.
[0] Such as how fractions work
edit: spelling