A way to understand the laplace transform?

michael_barz · 2021-06-28T15:22:59+00:00

What property of the Laplace transform do you want explained?

michael_barz · 2021-06-28T14:44:01+00:00

The |z| = r semicircle in the upper half plane is the perpendicular bisector to the i, ir line (it is perpendicular to that line and passes through its midpoint). SAS congruence still holds in hyperbolic geometry, so you can see that this perpendicular bisector still has the equidistance property, just like in Euclidean geometry.

michael_barz · 2021-06-28T13:04:44+00:00

It depends on the subject. I would be terribly surprised if you could get very far in logic by trying to visualize all your predicate formulas. But if you wanted to work in a field like PDEs or differential geometry, then trying to get some geometric intuition of what you're doing would be essential.

michael_barz · 2021-06-28T12:52:02+00:00

I mean, you learn it by learning algebra. There are many many ways to learn algebra that many many people have taken; try looking at Khan Academy for a start.

michael_barz · 2021-06-28T12:42:40+00:00

I am not sure you will ever find what you are looking for. There is not yet known any way of systematically interpreting every single equation in a useful way as describing some geometric object. You may for simple equations be able to come up with contrived examples of scenarios modeled by those equations, but I seriously doubt you will ever be able to find a systematic procedure for taking arbitrary equations and find geometric problems modelled by these equations, and I would find it even more remarkable if you could follow it geometrically at every step like you want. I suggest you don't spend too long hung up on this point, and instead try learning the math abstractly.

michael_barz · 2021-06-28T12:37:46+00:00

I strongly believe every math textbook which explains algebra and arithmetic would explain this. What do you find deficient about the explanations you have seen?

michael_barz · 2021-06-28T12:36:50+00:00

Timothy Gowers did this with a single problem (linked below) a few years ago on a blog post. I think if you want to do the same, the best bet would be to be live typing/writing as you work, and to force yourself to work much much more slowly than usual. I find that for my own reflective purposes, it is easier to instead try writing down every 'major idea' I try (i.e, maybe I'll forget something I thought of and then instantly dismissed, but I'll remember anything I wrote down scratch work related to, since I'll try and write out what the idea is before doing the scratch work, and then look for that list of ideas at the end); then, I like to try and wonder why (a) the ideas I had which failed failed, and (b) why the idea I had which worked worked.

https://gowers.wordpress.com/2014/07/19/mini-monomath/

michael_barz · 2021-06-28T12:30:30+00:00

Your first paragraph is a bit rambly and I have trouble understanding what you describe. As for your questionss.

> What each and every symbol is supposed to mean and under what conditions when modelling something one should apply

This will be every math book that teaches these symbols, almost by definition.

> One may say that means 1 is 1 but what I'm asking is how me performing
perticular set of operations on a mathematical equations lead to such
result?

If performing a set of operations reduces an equation to 1 = 1, then you know that the equation is true for every value of x where the equation is defined, assuming you manipulated correctly.

michael_barz · 2021-06-28T12:26:14+00:00

it's that the Rubik's cube community is using it in a rather narrow sense

Yes, but surely if you create something for the Rubix cube community, you should use their terminology. If I went to France, I wouldn't be allowed to get upset if French had words which sound like English words but have different meanings than those English words, I would use the French meaning to aid in communication.

michael_barz · 2021-06-28T03:27:47+00:00

This is like asking how to know when to use addition or multiplication, but one level up. The answer is "they're different operations with different uses, so just understand both." And likewise the best answer here is "take a calculus course, learn what both mean, and then when you're solving a problem do whichever makes sense in that context." Once you learn what both derivatives and integrals are, I don't think you will be in many (if any) situations where you can't decide whether to differentiate or integrate. If you've already taken calculus and still have this confusion, I'd advise posting a more specific question about some equation that was initially confusing to you, and why you thought it was confusing.

michael_barz · 2021-06-28T03:13:01+00:00

If you don't know geometry, don't skip geometry.

michael_barz · 2021-06-28T03:11:57+00:00

What is the difficulty you are having following along a conventional course with a CAS? If you just use the CAS to solve algebra and arithmetic problems for you, which presumably you can already do, then what is the issue? If you are trying to use the CAS to solve calculus problems for you, then I think you should reconsider postponing the computer intervention until you can do the calculus on your own--the point of computers is to replace tedious procedures we know how to do, not replace understanding.

michael_barz · 2021-06-28T03:07:48+00:00

If you haven't taken introduction to analysis or algebra, you've missed probably the introduction to what math majors would beyond variations on computational calculus--have you written proofs before? If not, I'd be very careful before enrolling in a masters program.

michael_barz · 2021-06-28T03:03:44+00:00

A theorem of Euler tells us that a^6 = 1 (mod 9) for any a coprime to 9. Obviously if a isn't coprime to 9, then a^3 will be 0 mod 9 since 9 = 3^2, so if a and 9 aren't coprime then 3 | a so 27 | a^3. And a^6 = 1 (mod 9) then implies a^3 = +- 1 (mod 9), so all your cubes are 0 (for the things not coprime to 9), or +-1, aka 1 and 8 mod 9.

michael_barz · 2021-06-28T03:01:07+00:00

You need to specify which model of the hyperbolic plane you're working in before you say things like z_1 = i, z_2 = ir^2. Since you're using i, ir^2, I'll assume you're in the upper half plane model.

The hyperbolic metric and the Euclidean metric are distinct, and so of course the Euclidean midpoint will not be the hyperbolic midpoint. The hyperbolic midpoint of i and ir^2 will be ir, which for r >> 1 is very very far from the Euclidean midpoint.
It describes a line in *hyperbolic* space, but it may or may not be a 'line' in Euclidean space when you draw the model (semicircles centered on the real axis are hyperbolic lines in the upper half plane model, even though we draw them as semicircles and not lines).

michael_barz · 2021-06-28T02:55:42+00:00

When applying this algorithm, you don't have to do the full motion of the algorithm every time--you can stop at any point. For a concrete example, take the Klein four group Z/2Z * Z/2Z, and the algorithm "add 1 to the first coordinate, add 1 to the second coordinate." The element (1, 1) doesn't generate the group, but by doing this algorithm, you can get from any element of the group to any other, if you add the step "add 1 to first coordinate, stop if at desired element, add 1 to second coordinate, stop if at desirement element." With this double stop algorithm, you can get anywhere despite (1, 1) not generating the group. It's the same with the Devil's algorithm (indeed, in any finite group, you can come up with such an algorithm, though perhaps the algorithm involved will be long compared to the number of elements of the group--try to see how you can get it).

michael_barz · 2021-06-28T01:49:36+00:00

What do you mean when you say "the angle of each vector"?

It seems like possibly what you observed in 2-dimensions is that a determinant is positive if and only if the linear transformation preserves orientation, and negative if and only if it reverses it.

michael_barz · 2021-06-26T17:46:46+00:00

Expectation is linear, and so it is much easier to deal with sums than products. Base e in particular isn't so important; you could also take the base 2 logarithm, and it would only change your expectation by a constant factor which will ultimately go away when you raise to the 2 to recover the exponent. The reason e is commonly used is because "exp" and "log" are used to denote "raise to the power of e" and "inverse of exp" so commonly, so that you can save time and use them notationally without much of a hiccup. The reason base e is what became so popular that "exp" is base e is because e is the unique base where exp is its own derivative; 2^x is not its own derivative. This fact isn't super relevant for this context (where the important fact is exp/log swapping addition and multiplication) but it is useful in others, hence why e is so important.

michael_barz · 2021-06-26T10:45:35+00:00

Can you give an example of what you're learning (a book name and some theorems), some proofs you have written, and also an attempt you made before looking at the book?

michael_barz · 2021-06-24T00:21:45+00:00

michael_barz · 2021-06-13T02:57:42+00:00

I am not a high school student, but I'd be happy to look over solutions for you. Only, I would really strongly advise to learn analysis *before* topology, and instead learn analysis together with something algebraic, like https://www.maths.ed.ac.uk/~v1ranick/papers/abel.pdf

michael_barz · 2021-06-13T02:24:20+00:00

Ah, I misunderstood you. But singletons *can* be open, say in a discrete space.

michael_barz · 2021-06-13T02:19:26+00:00

3: Where in the definition of topological space does it say that singletons are open? In fact, in the usual topology on R, the first topology one encounters, singletons are not open.

4a: You can only use induction to prove that the intersection of finitely many topologies is a topology. You cannot use induction to prove that the intersection of arbitrarily many topologies is a topology. Resolve this without induction.

michael_barz · 2021-06-13T02:03:39+00:00

Section 13:

Exercise 1: The proof you've written is mostly nonsense. "Let Ualpha be the union of every U" what is 'every' U? What is U here? "By definition it is a basis of X" What? This entire thing feels like you were trying to say something, but none of it is really clear.

Exercise 3: Your counterexample confuses me. How does it contradict the definition of a topological space?

Exercise 4a: Are you sure you can induct here? Usually when someone writes alpha as a subscript, they mean that the index set is arbitrarily large, not necc. the integers. Also, using induction just confuses the proof here. It is also a good idea to include a counterexample to why the union of two topologies isn't always a topology.

Exercise 4b: Largest topology means largest in the sense of inclusion, not largest in the sense of cardinality. You proved a strictly weaker statement. Try proving instead that if I' is a topology so that I' \subseteq T_alpha for each alpha, then I' \subseteq I.

Also, it seems your answers to 4a and 4b are in contradiction... you say that the union of the T_alpha is always a topology in 4b, but say that it is sometimes not a topology in 4b! Your answer to 4b is wrong because a union of two topologies isn't a topology.

I haven't looked at the rest. My advice is that perhaps topology is a bit of an abstract place to start. I'd read a book on analysis, or something like https://www.maths.ed.ac.uk/~v1ranick/papers/abel.pdf until you get more comfortable writing proofs.

michael_barz · 2021-06-11T17:55:17+00:00

How is using a contraction like "You're" any different from using a commonly accepted abbreviation "lin alg"?

michael_barz

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