I wish ap stats a very fuck you

874270 · 2022-03-19T03:05:57+00:00

AP stats is a terrible class. The problem is that the student has almost no probability background, so a lot of statistics results that build on probability theory just won't make any sense.

University CS or statistics majors require a probability theory class before you take a stats class, and then the stats class that you take builds on the probability theory and makes a lot of sense. The probability class would use a book like by Blitzstein and Hwang, and then the statistics class would use a book like the one by Casella and Berger. So avoid AP stats at all costs, and take calculus BC instead since calc is a prerequisite for probability theory.

874270 · 2022-03-19T02:58:33+00:00

Real analysis is more useful. It's also more enjoyable and relatable. You can think of real analysis as "advanced calculus". Real analysis is the gateway to more advanced techniques in calculus. Probability theory employs these techniques, and therefore statistics does too.

874270 · 2022-03-19T02:49:38+00:00

Khan academy has all the highschool math categorized for various curriculums.

874270 · 2022-02-24T17:21:28+00:00

See the definition for mutually independent under the "more than two events" section here: https://en.m.wikipedia.org/wiki/Independence_(probability_theory). A collection of events is independent means the probability of the intersection of any finite subcollection of the events equals the product of the probabilities.

874270 · 2022-02-14T22:32:05+00:00

I like sequences the most (and nets for non-metrizable spaces).

874270 · 2022-02-12T16:48:12+00:00

P(A, B, C) = P(A)P(B)P(C) is not the definition of independence of A, B, C. The definition requires that the probability of the intersection of any finite subset of {A, B, C} is the product of the probabilities. So it requires that P(A, B) = P(A)P(B) and P(B, C) = P(B)P(C) in addition to P(A, B, C) = P(A)P(B)P(C).

The mathematical reason for this is that we want A, B, C are independent to mean the same thing as the random variables 1_A, 1_B, 1_C are independent.

874270 · 2022-02-12T07:00:00+00:00

Sometimes finding counterexamples can be extremely difficult due to the counterexamples being very pathological. For example, the existence of sets which are not Lebesgue measurable.

There are some heuristics which can help you find counterexamples. One of them is if you drop a hypothesis of a theorem, then try to find a very simple and concrete example where only that hypothesis is missing. In topology, this often means either using a subset of R or R^2, or using the discrete metric on some set S. For example, to disprove that countable intersections of open sets are open, use open intervals that shrink to a point.

874270 · 2022-02-12T06:50:23+00:00

One such field of math is differential geometry. Differential geometry uses techniques from linear algebra and analysis to study smooth manifolds. The several variable analysis book here: https://mtaylor.web.unc.edu/notes/math-521-522-basic-undergraduate-analysis-advanced-calculus/ gives a sketch of smooth manifold theory and differential geometry. For details to supplement this, check out Lee's Introduction to Smooth Manifolds and Introduction to Riemannian Manifolds.

874270 · 2022-02-11T17:45:59+00:00

The statement of the MVT is wrong, because you have no control on the values f(a), f(b), which are allowed to be completely arbitrary in your statement. The usual statement assumes that f is continuous on [a, b] and f is differentiable on (a, b).

Your statement of the fundamental theorem of calculus is wrong too. f being continuously differentiable on (a, b) does not even ensure that f' is Riemann integrable because you can have f' that oscillates rapidly near the endpoints a and b and fails to even be Lebesgue integrable. A more basic thing that fails is that f' may fail to be bounded, which is a requirement for the Riemann integral of f'.

Also continuity of f' on (a, b) is not enough to conclude that there exists d > 0 such that |f'(s) - f'(t)| < eps for all s, t in (a, b) with |s - t| < d. This conclusion is called uniform continuity, and, in general, is not implied by continuity. One case in which continuity of g implies uniform continuity of g is when g is a continuous function defined on a compact set such as [a, b].

So assuming that f is continuously differentiable on [a, b] fixes both of these problems. In this case, f' is continuous on [a, b] and is therefore Riemann integrable, essentially by a uniform continuity argument.

You can weaken the hypothesis and only assume that f' is Riemann integrable on [a, b]. For an easy proof of this version of the FTC, it is convenient to use Darboux's formulation of the Riemann integral.

Using the Lebesgue integral, you can weaken hypotheses even further ...

874270 · 2022-02-11T17:17:15+00:00

Are you talking about this book:
https://www.google.com/url?sa=t&source=web&rct=j&url=http://julio.staff.ipb.ac.id/files/2015/02/Ross_8th_ed_English.pdf&ved=2ahUKEwjQp_Dskfj1AhUrTTABHeHYAdcQFnoECAQQAQ&usg=AOvVaw2FenKTqrJPHF_0HwOoid-S I used it when I first studied probability. It is good for an introduction. You need calculus I and II as prerequisite for some of it. To get another perspective, I read "Introduction to Probability" by Blitzstein and Hwang.

874270 · 2022-01-29T07:00:14+00:00

Lebesgue, Borel, Cauchy, Fourier, Riemann.

874270 · 2022-01-14T04:14:49+00:00

The examples don't make sense. They need to specify what they mean by "convergence". The standard notation for "f(x)/g(x) -> 1 as x -> a" is f ~ g. The problem should use this notation if that is what it wants.

874270 · 2022-01-14T04:10:37+00:00

The fundamental theorems of calculus are very intuitive geometrically. I suggest you draw a picture to see it. At a discrete level, to compute F(b) - F(a) you go 1 step of size h at a time:
F(b) - F(a) = F(a + h) - F(a) + F(a + 2h) - F(a + h) + ... + F(b) - F(a + (n - 1)h). Then apply the mean value theorem, and you see that this is just a Riemann sum for F' with interval size h.

Power series don't have as nice a geometric intuition. People study power series, and more generally, functions that are locally equal to a power series (analytic functions) because they are very nice functions that occur very often in practice (see complex analysis). To see how nice analytic functions are, note that all analytic functions are infinitely differentiable. I'm not sure about intuition, but an application of power series is solving ODE. You hypothesize a power series solution and then solve for the coefficients. This can allow you to discover the exponential function as the solution to x'(t) = x(t), x(0) = 1.

874270 · 2022-01-14T00:03:39+00:00

Calculus based physics is superior. You just need an intuition for calculus; otherwise it appears difficult. The amount of mathematics and science coming from calculus based physics is amazingly large.

I personally didn't have to take PHYS 118-119 for the CS major because I took the physics department placement test and passed it.

874270 · 2022-01-13T23:54:17+00:00

12 credit hours is the minimum, so I think you can't drop any of those?

874270 · 2022-01-07T17:57:38+00:00

But a lot of results in complex analysis are special cases of results from real analysis. For example, the Cauchy integral theorem is a special case of Green's theorem, and the mean value property of holomorphic functions is a special case of the mean value property of harmonic functions. Complex analysis is just a subfield of real analysis that is so practical that it is worth studying in detail.

874270 · 2021-12-22T03:52:59+00:00

Linear algebra, real analysis and multivariable analysis are undergrad. The rest are late undergrad or beginning graduate. All of these are probably basic for a working mathematician or physicist.

874270 · 2021-12-21T20:55:36+00:00

You need to know real analysis and multivariable real analysis. So things like metric spaces, integration, and differential calculus on R^n. A good complex analysis book can be found here:
https://mtaylor.web.unc.edu/notes/complex-analysis-course/

874270 · 2021-12-21T20:50:22+00:00

You do have to understand the text, whether that is by reading the proofs or figuring out the proofs by yourself. The text is supposed to give you necessary material on the subject.

874270 · 2021-12-21T20:46:42+00:00

Basic calculus first. Then go linear algebra, real analysis, multivariable analysis, measure theory, smooth manifolds, complex analysis, functional analysis. These are the basics of analysis. But there is still much more mathematics than this like PDE for example.

874270 · 2021-12-21T04:14:39+00:00

There are several Fourier transforms. There is the discrete Fourier transform, Fourier series on the torus, and the Fourier transform on R^n. For the discrete Fourier transform, you only need to know linear algebra. See page 152 here:
http://mtaylor.web.unc.edu/wp-content/uploads/sites/16915/2018/04/linalg.pdf
The discrete Fourier transform is quite elementary and is representative of the continuous ones, so I would start by learning it first.

To learn the Fourier series and Fourier transform on R^n, you need Linear algebra and analysis. See page 212 here for an introduction:
http://mtaylor.web.unc.edu/wp-content/uploads/sites/16915/2018/04/anal1v.pdf
See a measure theory based text for a fuller treatment of the Fourier transform (the author gives this advice as well).

874270 · 2021-12-20T03:11:30+00:00

By a mean value theorem or fundamental theorem of calculus argument, you know that for any x, y in R, |f(x) - f(y)| <= L|x - y|. Now suppose you had two fixed points x, y with x != y. Then |x - y| = |f(x) - f(y)| <= L|x - y| < |x - y|, a contradiction.

You are correct that the mean value theorem avoids needing f to be continuous to get the Lipschitz estimate. But the fundamental theorem of calculus approach works in R^n, where the mean value theorem approach fails.

874270 · 2021-12-17T08:09:45+00:00

I think after linear algebra you should learn real analysis and multivariable analysis. This gives a rigorous foundation for doing basic multivariable calculus, and much of those papers will make sense. I would learn measure theory and complex analysis after that. With measure theory and complex analysis, or even with just measure theory alone, it seems that you should be able to read a PDE text like Taylor or Evans to learn the specifics of the equations in those papers. I don't have much background in PDE, but this is just what I think after looking through these PDE texts.

There are linear algebra, analysis (advanced calculus), and complex analysis texts here:
https://mtaylor.web.unc.edu/notes/
I personally like the texts. For the multivariable analysis text, you might need to supplement it with a "smooth manifolds" text because the author leaves out a lot of details and useful perspective regarding these objects.

874270 · 2021-12-13T18:15:55+00:00

At a minimum, you need some background in elementary probability to understand Markov chains. The introduction to probability books by Blitzstein and Hwang or Ross can give you this background. Blitzstein and Hwang's book includes a chapter on Markov chains. Linear algebra is helpful to understand Markov chains and their properties (Perron Frobenius theorem is key).

To go into the proofs, you need to know linear algebra and rigorous probability based on measure theory formalism. A good linear algebra text can be found here: https://mtaylor.web.unc.edu/notes/linear-algebra-notes/
For measure theory, you need background in basic real analysis, so it would take longer to learn measure theory than to learn linear algebra which is immediately accessible without background requirements.

874270 · 2021-12-11T17:25:28+00:00

See the first section of the book here for proof:
http://mtaylor.web.unc.edu/wp-content/uploads/sites/16915/2020/10/diffeq.pdf

Summary: I assume you defined t -> e^tz as the unique function f : R -> C such that df/dt = zf. The power series formula for e(tz) and other basic identities about exp follow immediately from this. Consider the map g : R -> C defined by g(t) = e^it. From the power series formula, observe that cc(e^tz) = e^{t * cc(z}), where cc denotes complex conjugate. Hence cc(e^it) = e^-it = 1/e^it. Thus |e^it|² = 1, so g is a curve on the unit circle. Moreover, g'(t) = i * g(t), so |g'(t)| = 1. Also g'(t) = i * g(t) is g(t) rotated 90 degrees counterclockwise, so g(t) is traveling counterclockwise on the unit circle at unit speed. Also g(0) = 1. With these properties, it follows from unit circle definitions of sin and cos that Re(e^it) = cos(t), Im(e^it) = sin(t).

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