http://i.imgur.com/COZJe0p -- looks like the main landing page!
206 karma · joined March 9, 2013
Unfortunately I will probably be not contributing at this current moment, as I am in the camp of "looking for remote work" and not "currently employed doing remote work" :)
Note that for a base 10 normal irrational by the definition of "convergence" in this case means || (number of digits of value a) / (number of digits of value b) ||_\infty < ε (sup norm over all a, b from 0 to 9, a ≠ b) for some ε > 0 such that for some fixed n (the number of digits in the decimal expansion, halted), we have that whenever N > n this inequality holds.
In the case where there are a fixed number m of leading "a"s (whatever digit a is), we have ||(m + dig_a)/(dig_b)||_\infty = ||m/dig_b + dig_a/dig_b||_\infty -- we only need this to be less than SOME fixed ε (not necessarily the same ε as above) for every N > n for some n we choose.
By assumption ||dig_a / dig_b||_\infty < ε whenever n > N. Choose n' > n such that dig_b satisfies m/dig_b < ε
(Note: this is possible as dig_b increases as n' grows: since m is constant and if dig_b were to be constant for n > N' then ||dig_a/dig_b||_\infty would be larger than ε for some n).
Then we have ||m/dig_b||_\infty < ε and ||dig_a/dig_b||_\infty < ε so ||m/dig_b + dig_a/dig_b||_\infty < 2ε.
Thus it converges -- satisfying the same definition as above chooseing n = n' and ε = 2ε.
I don't agree that using this is a "serious handicap" to my capacity for thinking syntactically. It is just another tool to be used when applicable to generate understanding: use an example and see how it follows the rigorous definitions. I lose nothing by doing this when it is useful.
Even in this link, Dr. Tao agrees that it is not a good idea to look at statements on a strictly formal level. In a greater sense this is what I was getting at: play with some examples or with some of the assumptions and see what happens in order to get initial or further understanding. Did I misunderstand the stated point? What was said just prior but in reference to "this[1]" is not supported by Dr. Tao in his post.
Relevant quote: "'fuzzier' or 'intuitive' thinking (such as heuristic reasoning, judicious extrapolation from examples, or analogies with other contexts such as physics) gets deprecated as 'non-rigorous'. All too often, one ends up discarding one’s initial intuition and is only able to process mathematics at a formal level, thus getting stalled at the second stage of one’s mathematical education."
Whenever anyone says something is "trivial" -- I understand that to mean simply "it is obvious to me." I try to NEVER explain things in such ways -- to say things are trivial because it is very clear to me that is NOT trivial to many other people. And there has to be some respect given to this idea. I abhor when professors use this sort of language -- instead of stating it is trivial, explain using an extra sentence or two why it is so. If it takes more than that, it is likely not trivial.
The best way for me to learn new concepts (in my graduate classes, for example) is as follows. And it is the method all of my professors currently use whether they know it or not.
First, they begin with the definition of something. For example (one of my classes last week expressed this idea in particular): in metric spaces they say "A function is upper semi-continuous at x_0 if for all x approaching x_0 the limit supremum of f(x) is smaller or equal to f(x_0)." But what does this mean? You can use the "epigraph" as a way to explain this idea in a different way, explaining what it means to be a u.s.c. and/or a l.s.c. function in a more understandable way. From here, then, after understanding this, I was able to go through the strict mathematical notation of these concepts and follow it more clearly.
Instead of starting from the mathematically "rigorous" notation, I am able to jump to a more familiar idea and proceed to link it back to the notation and rigor described in the definitions and associated theorems. This is a much more "efficient" way, in my opinion, than staring at the notation and trying to decipher it for hours. Having examples makes it easier to understand the context of the symbols and rigor involved: by having a specific example of a general form.
Is there a lack of rigor in American math education? In general I would say yes -- especially in high school/undergraduate mathematics. But how do people most easily learn? Through examples tying back to definitions. This is more of the American way of teaching in my experience.
Furthermore, you state "American textbooks on mathematics" are "softcore." I think this is true for a lot of the lower-level material, but if you pick up Rudin's Real and Complex Analysis I do not think you will feel the same way.
This is interesting: it is one way to allow for a better search experience through personalized search options without using data collected about me.
This does not seem to be a tool to merely judge you as a developer. Rather, it acts as a tool to determine syntax understanding, and to possibly point out new things one may not be aware of. Much more important than assigning a number to a skill.
I am a fresh college graduate. The job posting seemed appropriate for someone like me in my current situation (and was the only one in any whoishiring thread I felt was a great fit and not merely a good fit). I did a six-month internship in the same domain, using many of the same technologies as requested for the job. A perfect fit and the obvious next step, I thought. I would be doing what I had already measurably succeeded at, while also being able to use some of my public speaking experience I gained in school to present the modeled data I would be working on. I submit the application, noting all of this. I hear nothing back. A week later, I follow up with the email listed in the post. Still nothing back. I simply assumed they filled the position, however...
I see them post again the subsequent month in the whoishiring thread, still hiring. I do not get it.
Practically speaking, using ":=" removes any doubt when quickly skimming over old notes. It is nice to be able to easily separate what is defined and what is asserted in each of dozens of statements.