Approximate modulus of continuity of L^1 functions

Latex version:

Let f: R -> R be in L1 (loc). Define the approximate left modulus of continuity, L_f (x, e): R x R+ -> R+ union +inf as

L_f (x, e) := sup {d >= 0| 1/r Int [x-r, x] |f(t) – f(x)| <= e for all 0 <= r <= d}

Similarly define the approximate right modulus of continuity by

R_f (x, e) := sup {d >= 0| 1/r Int [x, x+r] |f(t) – f(x)| <= e for all r <= d}

Note that if f = g a.e., then L_f = L_g a.e. and R_f = R_g a.e., by which we mean for almost all x, for all e, L_f (x, e) = R_f (x, e).

Do L_f and R_f determine f almost uniquely almost everywhere? In the following sense:

Question: Suppose f and g in L1 (loc) are such that L_f = L_g a.e. and R_f = R_g a.e. Does it follow that f = g + c a.e. or f = g – c a.e. for some a.e. constant function c?

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Prospective Math, Physics, or CS Major – Opinions for College Class Schedule?

So I’m starting my 4th semester in college, still undecided on what exactly I want to major in. I’ve narrowed it down between math, physics, and computer science. I’d love a combination of the 3 by double majoring in 2 and minoring in the other. I’m still in a position that I have time to take courses from each department to get a feel of each subject. I’m posting on here because I’d love to get some insight as to which courses I should take and which I should drop. I’m primarily interested in applied math grad school, but I might rather do physics or CS which is why I’m trying to see which one I more naturally gravitate towards so I can double major.

I’m currently enrolled in:

-Classical Mechanics

-Linear Algebra

-Theory of Computation


-Complex Variables and Applications (about 50% proofs, 50% computational)

-Basic Real Analysis (big emphasis on proofs – not the advanced Real Analysis, to clarify)

I wish so badly I could take them all, but I need to drop one to avoid overworking myself and failing at everything (even 5 seems scary.. we’ll see how I feel by the drop deadline about actually taking 5).

I have to stay in Classical if I want to minor or major in physics because I’d be behind otherwise. I want to stay in linear algebra because I feel ignorant having not taken it yet.. I just think it’s such a core part of the undergrad math curriculum, I don’t want to wait any longer. Algorithms is probably one of the most useful CS courses – I refuse to graduate without it, and I have friends taking it this semester so it’s probably the best time to take it to get a good grade. Basic Real Analysis seems essential since it will be my first real rigorous proofs course, though I could take it over the summer (though 1 month seems like a rush to develop this way of thinking and see if I like it enough to pursue math). Theory of Computation could wait. Not sure how immediately useful it is. I think it’d be great exposure to the field of theoretical CS, though. It does involve proofs. Complex Analysis seems to be a requirement for all applied math grad schools. I know I’d take it at some point regardless, as it is used in physics. The only reason I wouldn’t take it this semester is because my professor is hard to follow. I think I’m either going to drop complex or theory of comp, though I’m still considering dropping basic real.

I’m just generally curious to hear what people’s thoughts are. If you’re in physics, applied math, or CS, which combination do you think would likely give me adequate exposure to each field so hopefully I’ll know what I’m most interested in by the end of the semester?

I’m also planning to pursue research in one of the fields, though I don’t think I will actually conduct any until the summer. My hope is to at least meet with some professors, find a topic that seems interesting, do a bunch of reading this semester, etc. and go from there.

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What cool math things can I write programs for?

I enjoy playing around with programming in Python and I also have a particular affinity for math, so I like to write little programs that combine the two. If anyone has suggestions of neat things to create, I would very much appreciate it. In the past I’ve done things like Ulam’s Spiral, Langton’s Ant, Sorting Algorithms, etc. I prefer visual stuff, but anything is appreciated!

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struggling with radical expressions. having a hard time knowing where to start in complicated problems and what would be the order of operations. and factors as well.

I am struggling with radical expressions. having a hard time knowing where to start in complicated problems and what would be the order of operations. and factors as well.

I am trying to solve a radical expression and in one of the steps

3+3(sqr root of 5) factors out to 3(1+((squre root of 5))

why though?????

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Very extremely large numbers

Somehow I am extremely interested in extremely huge numbers. I know of Graham’s Number (or however it was called) and the Googolplex, but I need more.

I wonder what number would come out if a Googolplex was taken to the power of a Googolplex, the process was repeated a Googleplex to the power of a Googolplex times, and this would be continued a Googolplex to the power of a Googolplex times. Then then would be continued, each step increasing the number by a Googolplex to the power of a Googolplex times the entire step before that. This process would have a Googolplex to the power of a Googolplex steps. Then all those steps would be continued by the number the final step provided at the first calculation times.

I have no idea if a number that big was bigger than Graham’s number, nor how many zeros it has. I’ll just call that number the Hypergoogolplex for now.

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