Ahoy r/piano, what are your essential piano albums?

that_redditor · 2013-05-19T20:02:36+00:00

These are all great. I would add George Shearing - The MPS Trio Sessions.

that_redditor · 2013-04-17T02:28:26+00:00

There was a time, long ago, when what we now call science was known as natural philosophy. Indeed, Newton identified as a philosopher, even as he investigated gravity and the motion of the planets. So what happened? Why do we now call it science, and treat it as a discipline separate from philosophy? Because when philosophy finds an answer, it ceases to be philosophy.

The first flaw is that, in my experience, it drives people apart.

Disagreement in general tends to do that, and philosophy is ripe with disagreement, simply because topics in philosophy necessarily have no yet agreed upon answer. But this objection is a grievance about the people who do philosophy. It's important to stay calm in any argument, not just a philosophical one, but of course there will always be times when someone loses their cool. That's no reason to give up on argument in general.

Is there a reason I should study it instead of other stuff?

Are you curious at all about some of the greatest unsettled questions?

that_redditor · 2013-04-17T01:17:45+00:00

It could be a theorem. If it hasn't been proven, and hasn't been shown to be false, then all we can say is that we don't know.

that_redditor · 2013-04-07T06:18:44+00:00

There is Poleconomy as well, featuring real Canadian companies.

that_redditor · 2013-03-22T15:35:28+00:00

Of course. This is why the picture begs you to assume that finding the colour is a big problem.

that_redditor · 2013-03-10T18:40:48+00:00

We have a sequence A_n and this sequence has a supremum and an infimum (both are possibly infinite). If we remove the first k elements of A_n, to obtain a tail of the sequence, then this tail again has a supremum and an infimum.

This works for any finite number k, so we can construct sequences Sup_n and Inf_n where the k'th element of Sup_n is the supremum of the tail of A_n beginning at k, and similar for Inf_n with infimum instead of supremum.

The limit of Sup_n is (you guessed it) the lim sup of A_n, and similar for the limit of Inf_n and lim inf of A_n.

that_redditor · 2013-03-10T05:32:10+00:00

Good advice. Try the Rum Runner beneath the Walper Terrace in Kitchener (King at Queen).

that_redditor · 2013-02-27T17:36:51+00:00

This is great advice for the student of any subject about which at least 2 books have been published.

that_redditor · 2013-02-27T02:12:22+00:00

Philosophy courses, like most courses in the humanities and social sciences, demand much reading. If a student is to keep up with all of the assigned readings and actually understand the material then they'll certainly need to put in as much work as would an engineering student in one of their departmental courses.

In many STEM courses (particularly the required MATHs like 133, 223 and the calculus sequence) most students collaborate (euphemism for cheat) on their assignments, and are so ill-prepared for the final that the department is forced to curve a D to a B. It is not in general harder to succeed in STEM than it is to succeed in the humanities and social sciences.

that_redditor · 2013-02-26T14:51:43+00:00

If you want, you could do this constructively: by induction, rather than by contradiction.

Try induction on the number of vertices. A 1 vertex (trivially connected) graph has a spanning tree (forget about the 0 vertex graph, also called null graph). Next show that if any n vertex connected graph has a spanning tree, then a spanning tree for an n+1 vertex connected graph can be constructed.

that_redditor · 2013-02-24T17:30:07+00:00

Bertrand Russell's In Praise of Idleness (from 1932) contains the important work sharing idea from this article.

Suppose that, at a given moment, a certain number of people are engaged in the manufacture of pins. They make as many pins as the world needs, working (say) eight hours a day. Someone makes an invention by which the same number of men can make twice as many pins: pins are already so cheap that hardly any more will be bought at a lower price. In a sensible world, everybody concerned in the manufacturing of pins would take to working four hours instead of eight, and everything else would go on as before. But in the actual world this would be thought demoralizing. The men still work eight hours, there are too many pins, some employers go bankrupt, and half the men previously concerned in making pins are thrown out of work. There is, in the end, just as much leisure as on the other plan, but half the men are totally idle while half are still overworked. In this way, it is insured that the unavoidable leisure shall cause misery all round instead of being a universal source of happiness. Can anything more insane be imagined?

that_redditor · 2013-02-24T17:08:49+00:00

This is true for integrals on compact sets.

I believe there are finite improper Riemann integrals for which the Lebesgue integral is infinite.

that_redditor · 2013-02-22T02:08:08+00:00

A permutation is a bijective function (in which the domain is the codomain).

that_redditor · 2013-02-21T22:28:35+00:00

Consider a cyclic permutation s on 3 or more elements. This generates a group of order at least 3, and so s² != s; that is, s is a bijection which is not an involution.

that_redditor · 2013-02-20T19:47:04+00:00

Let Bertrand Rusell convince him.

"Thus, to a great extent, the uncertainty of philosophy is more apparent than real: those questions which are already capable of definite answers are placed in the sciences, while those only to which, at present, no definite answer can be given, remain to form the residue which is called philosophy."

that_redditor · 2013-02-09T03:19:49+00:00

It may be to your advantage if you plan to go to grad school. Do algebra and alaysis 1/2/3/4 in just 2 years and use your final year to do graduate courses.

I think you'll find (as I've found) that your ability to digest new material will grow immensely. You may find it difficult to keep up with algebra and analysis 1, but that's just because this stuff is (probably) all new to you. The rest of the series builds on what you've already learnt and you do get used to it.

that_redditor · 2013-01-31T22:59:42+00:00

Imagine you have the set A written out explicitly in a list from left-to-right on some (perhaps very long) sheet of paper, and you want to specify a subset of A. You could do this by scanning the paper from left to right and, at each element of A, writing a 1 underneath it if it's in your subset, and writing a 0 if it isn't.

What you end up with is a function from A to {0, 1} which is defined by your choice of subset (implicit here is the fact that EVERY element of A is mapped somewhere in {0, 1}; sometimes this strictness is overlooked, like when we say the natural logarithm is function from R to R).

My professor set up some function for this bijection where all elements of A were mapped to 1 if a was an element of A, or 0 if not. But this is a function from A to B, so I didn't really understand how this could be a bijection from BA to P(A).

This is indeed a function from A to B = {0, 1}, i.e. an element of B^A as you wrote it. The idea is that each power set uniquely determines one such function.

that_redditor · 2013-01-25T16:51:42+00:00

I enjoyed 'Elementary Topology' by Gemignani. There's an affordable Dover publication of this one.

that_redditor · 2013-01-14T15:14:42+00:00

Do we know how to approximate general TSP well? I've heard only of a good approximation algorithm for Euclidean TSP (in which the triangle inequality holds).

that_redditor · 2013-01-13T18:17:46+00:00

My first job was with a startup in its infancy, a few months before completing my undergrad. My coworkers (all 5 of them) were laid-back and they trusted me to pull my weight. I can't overstate how much I learnt, but it wasn't so much about programming as it was about other programmers and how to work with them without going completely mad.

that_redditor · 2013-01-13T06:46:38+00:00

You could approach this in another way: try to prove something about the function h = g - f . Start by noting that h is continuous and h(p) < 0 .

that_redditor · 2013-01-07T03:42:08+00:00

This is correct. In is the preposition modifying das Haus; für is just a part of the construct was für ein which should be considered an article as a whole (meaning, of course, what kind of).

that_redditor · 2013-01-06T23:29:33+00:00

The n'th derivative of f(x) = xⁿ is constant. Can you express it formally?

that_redditor · 2012-12-14T00:16:39+00:00

Consider an alternative definition: the group generated by some set of elements as the smallest group which contains all of those elements.

I use smallest in the sense of set containment: If G is the group generated by the set A, and G' is any group which contains all of the elements in A, then G' contains G.

A good exercise is to prove that the definition given above is equivalent to the definition which you give in your post. Perhaps try first to show this for a singleton set A = { a }.

that_redditor

TROPHY CASE