Quick Questions: June 08, 2022

We can define a sequence of binary operations B\_n(x, y) for real x, y, where B\_0(x, y) = x + 1 and B\_{n+1}(x,y) = B\_n(x, B\_n(x, B\_n(...) ... )), where B\_n is nested y times. By this definition, B\_0 through B\_4 are the successor function, addition, multiplication, exponentiation and tetration, respectively.

B\_n is well-defined for non-negative integers n. Are there any reasonable/canonical ways to extend the definition of B\_n to non-integer n? For example, is there a sensible definition for B\_1.5, i.e. an operation that's somewhere in between addition and multiplication?
Rather hilariously, this is one of the example questions, but can someone explain the concept of manifolds to me?
If 50% of students who take Phase I of the exam pass on their first try, 40% who take Phase II pass on the first try, and 30% who take Phase III pass on the first try, then would the number of students who pass every phase of the exam on their first try be 50% x 40% x 30% (= 6%)?
Does the math subject GRE matter much post-pandemic for graduate admissions? Is it mostly just necessary for PhD programs or for MS/MA programs as well?
0.11% of the US population has MS.

The average person knows 600 people.

600 * 0.0011 = 0.66

How would you describe the odds of knowing someone with MS? It seems like if the result was 1, you'd say there's a 100% chance you know one person with MS. Would you say there's a 66% chance of knowing someone with MS?
How do schemes allow us to talk about the _values_ a curve takes on? I've been working through Hartshorne but haven't seen this addressed yet. Like, morphisms of schemes are cool and all but how do I know whether my curve intersects with some specific set of points (integers, rationals) or not.

Yes, I do realize arithmetic geometry is a thing but I do not know anything _concretely_, so that's why I'm asking.
Does anyone have the tex files of some math books? Just for use as a reference.
How to check (prove) that a point is generator of modular elliptic curve group?

For example:

P = (4,3)

E: y^(2) = x^(3) \+ 2 over field F\_19.
Are there any master's programs for math/applied math in the US which almost always offer funding? I want to go for a PhD after, so I'd rather not take on a large loan to get the degree :/

(Plus, I can't apply for a PhD rn because I'm graduating with a 3.21 with C+'s/B-'s in some really important core courses, so I likely won't get accepted at any institution even if I did try...)
I have not seen this mentioned in any sources I read. But what does the cohomological dimension of a field tell you about the field, in explicit term?

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