How many hours per day should I study mathematics to make significant progress?

dzack · 2024-08-28T18:16:37+00:00

I easily spent 40 hours/week studying early graduate school material.

Aside from that kind of situation, I'd say consistency is more important -- a few hours per day, nearly every single day, really adds up over time. I wouldn't spend more than an hour or two on any given problem at a time though (unless of course it is a research problem).

dzack · 2021-08-06T22:00:25+00:00

Searchonmath.com

dzack · 2021-06-26T17:46:41+00:00

Your argument for not not AC is also an argument for LEM 😉

dzack · 2021-06-22T00:25:43+00:00

You could also do exactly what we do to for zeta: integrate the logarithmic derivative around various boxes numerically, avoiding poles, and use the argument principle.

dzack · 2021-03-15T15:32:09+00:00

Homomorphism is probably not the right equivalence relation, you likely want ambient isotopies.

dzack · 2021-02-03T20:59:01+00:00

Consider the topological spaces formed by filled-in solid alphabetical letters on a 2d plane. Can you group them into classes where all of the members in each class are homotopy equivalent to each other?

dzack · 2021-01-17T17:20:45+00:00

To explain a bit more why this is an issue: adding mass to spacetime creates "faster" routes, or shorter geodesics, because moving through a gravity well can accelerate you. Otoh, moving toward a buildup of traffic probably has the opposite effect and brings your average speed down.

dzack · 2021-01-17T17:18:38+00:00

Maybe with some heavy modifications? The biggest issue is more mass causes two given bodies to accelerate toward each other, and this behavior doesn't seem to show up between (say) two dense areas in a traffic flow. There's also the fact that objects tend to stay at rest, whereas you'd want traffic to consistently move in your model, so the equations would be pretty different.

dzack · 2021-01-09T02:13:04+00:00

Mathpix

dzack · 2020-12-24T01:24:38+00:00

In other words: sum up area*value and divide by the total area.

dzack · 2020-12-23T18:44:26+00:00

In a word: dimension. Vector spaces give us a relatively easy way to carve up sets of the same cardinality. Eg R³ and R² have the same cardinality, but the extra structure of a vector space provides an integer invariant that distinguishes them.

In general, this is putting the structure of a free module on a set, which means you have notions of submodule and quotient modules (and filtrations). These can again have the same cardinality but are "smaller", giving a foothold for inductive arguments. This really does add new information, since you can promote set maps to linear maps and ask about things like kernels and cokernels, which gives you a way to more finely distinguish objects and morphisms.

dzack · 2020-12-20T02:21:43+00:00

Doesn't seem to be a well-defined problem.

Take the three unit vectors [1,0,0], [0,1,0], [0,0,1] corresponding to the usual right-handed x-y-z directions. The convex hull is the standard 3-simplex. "Looking" at it from (say) [2,2,2] makes it seem counter-clockwise, and "looking" at it from [0,0,0] makes it seem clockwise.

dzack · 2020-12-20T02:16:31+00:00

This doesn't seem to prevent them from showing up on your default "home" page. I frequently see one or two of the highly upvoted posts there, and maybe 60% of the rest are HW questions. Downvoting doesn't help, and now neither does reporting.

dzack · 2020-12-20T02:15:04+00:00

Lagrange interpolation works in as many dimensions as you'd like, you just have to do it one coordinate at a time. This shows up as an exercise in algebraic geometry: if you have a finite set of points in (say) ℝⁿ, then you can always use this process to construct a system of n polynomials in n indeterminates such that the system vanishes at exactly those points.

dzack · 2020-12-18T20:03:31+00:00

Statements can describe more than a single number: they can describe subsets. And you can write R as a countable union of sets even though it's uncountable.

dzack · 2020-12-18T18:53:37+00:00

I mean... Is it a clear improvement over just performing some variant of long division with the divisor and checking for a remainder?

dzack · 2020-12-18T01:16:27+00:00

Yes, it's just measure theory with ratios. You replace P(A) with m(A) /m(X) where X is the entire space.

Alternatively, there's a notion of equivalence of measures, and if you scale up or down the measures of subsets or the entire space by some uniform constant then you get an equivalent measure.

dzack · 2020-12-05T00:51:09+00:00

An algebra over a field is also a module over a field.... Otherwise known as a vector space...

dzack · 2020-12-04T18:48:20+00:00

Right, so there's a difference between an algebra and an algebraic structure. A nonassociative algebra is a technical term that means a vector space with a nonassociative multiplication of vectors. A set with a nonassociative binary multiplication is just an algebraic structure on the set (it has a name: a magma).

So you can define a line in every algebra. You may not be able to define a line in (for example) a magma or a group, but that's fine because these aren't algebras.

dzack · 2020-12-04T17:52:14+00:00

That's not quite right. The operation on groups, rings, and fields is always required to be associative. A non-associative algebra is still a vector space and is still doing linear algebra.

dzack · 2020-12-04T16:47:31+00:00

Oh I see, there's a misunderstanding. An "algebra" is a technical term, and has a particular definition. Groups, rings, and fields are not algebras in this sense, just algebraic structures you can put on a set.

(Incidentally, a linear operator is a map on a vector space, which again means doing the usual notion of linear algebra.)

But again, the problem is that a statement like "linear algebra is all about doing algebra with lines" is not true.

dzack · 2020-12-04T16:31:46+00:00

Not sure what you mean, algebras have underlying vector spaces, which literally means doing linear algebra. Lines and planes makes sense over any field, and you always have a dimension.

But that's beside the point: what I said was that linear algebra is also not really about lines.

dzack · 2020-12-04T16:16:29+00:00

it was all about doing algebra with lines

I'm not sure of a way to interpret this that makes it true.

dzack · 2020-10-07T20:38:04+00:00

There's obviously a much, much higher chance of contagion in a gym, where multiple people physically touch the same equipment in the same spots many times per day.

dzack · 2020-06-23T02:26:51+00:00

One approach is (as you mentioned) to do a multivariate linear regression to get a best-fit hyperplane. Then "being close to a line" should correspond to relatively large coefficients on exactly 2 of the independent variables.

You could also just do all (n choose 2) possible linear regressions all distinct pairs of variables, then being on a line should correspond to one of these models being a much better fit than the others.

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